physical chemistry experimentschemistry4.me/curry/pchemlabs.pdf · physical chemistry experiments...
TRANSCRIPT
There is no logical way to the discovery of
elemental laws. There is only the way of
intuition, which is helped by a feeling for the
order lying behind the appearance.
-A. Einstein
Physical Chemistry Experiments Wellesley College, Department of Chemistry
Sandor Kadar
(DRAFT)
Table of Contents TABLE OF CONTENTS ....................................................................................................................................................... 1 TABLE OF FIGURES .......................................................................................................................................................... 3
.................................................................................................................................................. 4
THEORETICAL INVESTIGATION OF THERMODYNAMICAL PROPERTIES OF CO2 ............................................................................... 4 Background ............................................................................................................................................................ 4
Real Gas Behavior ................................................................................................................................................................ 4 The Joule-Thomson Coefficient ........................................................................................................................................... 5 Roadmap for Studying Real Gases ....................................................................................................................................... 5 Computational model of CO2 ............................................................................................................................................... 7 Calculating the heat capacity of CO2 .................................................................................................................................... 8 Calculating the Second Virial Coefficient of CO2 ................................................................................................................ 10 Calculating the Joule-Thomson Coefficient of CO2 ............................................................................................................ 11
Pre-Lab work ........................................................................................................................................................ 12
................................................................................................................................................. 15
IR AND RAMAN STUDIES OF CO2 ..................................................................................................................................... 15 Background .......................................................................................................................................................... 15
The IR Spectrum ................................................................................................................................................................ 16 Raman Spectrum ............................................................................................................................................................... 16 Details of the bending mode ............................................................................................................................................. 19
Experimental Procedure ....................................................................................................................................... 20 IR Spectroscopy ................................................................................................................................................................. 20 Raman Spectroscopy ......................................................................................................................................................... 20
Data Analysis ....................................................................................................................................................... 22
................................................................................................................................................. 23
DETERMINATION OF THE SECOND VIRIAL COEFFICIENT OF CO2 .............................................................................................. 23 Background .......................................................................................................................................................... 23 Experimental Procedure ....................................................................................................................................... 27 Results, Calculations ............................................................................................................................................ 30
................................................................................................................................................. 31
DETERMINATION OF THE JOULE-THOMSON COEFFICIENT OF CO2 ........................................................................................... 31 Background .......................................................................................................................................................... 31 Experimental Procedure ....................................................................................................................................... 33 Results, Calculations ............................................................................................................................................ 36 Appendix: The Derivation of the JT Coefficient .................................................................................................... 37
................................................................................................................................................. 39
DETERMINATION OF THE HEAT CAPACITY RATIO OF CO2 WITH THE RÜCHARDT-METHOD ............................................................ 39 Background .......................................................................................................................................................... 39 Experimental Procedure ....................................................................................................................................... 43 Results, Calculations ............................................................................................................................................ 45 Appendix: Derivation of the equations for adiabatic processes .......................................................................... 46
................................................................................................................................................. 48
DETERMINATION OF THE HEAT CAPACITY RATIO OF CO2 WITH ADIABATIC COMPRESSION ........................................................... 48 Background .......................................................................................................................................................... 48 Experimental Procedure ....................................................................................................................................... 48 Results, Calculations ............................................................................................................................................ 50
................................................................................................................................................. 51
SOLVING AN ORDINARY DIFFERENTIAL EQUATION (ODE) SYSTEM .......................................................................................... 51 Background .......................................................................................................................................................... 51
Ordinary Differential Equation Systems in Chemical Kinetics ............................................................................................ 51 Scientific Modeling ............................................................................................................................................................ 52 Deterministic vs. Stochastic Modeling ............................................................................................................................... 53 A model for Ca2+- oscillations in Paramecium cells ............................................................................................................ 53
The Modeling Environment .................................................................................................................................. 54 The Mathematical/Programing Platform ........................................................................................................................... 54 The Model File ................................................................................................................................................................... 54 Running the modeling ....................................................................................................................................................... 56
Results, Calculations ............................................................................................................................................ 62 Appendix A: model file for SBToolbox for modeling Ca2+-oscillations in Paramecium ......................................... 63 Appendix B: Script file for “Parameter sweep” .................................................................................................... 66
................................................................................................................................................. 68
SOLVING A PARTIAL DIFFERENTIAL EQUATION (PDE) SYSTEM ............................................................................................... 68 Background .......................................................................................................................................................... 68
Partial Differential Equation Systems in Chemical Kinetics ............................................................................................... 68 Numerical Method for Integration .................................................................................................................................... 69 Approximating the Diffusion Terms ................................................................................................................................... 69 Boundary conditions .......................................................................................................................................................... 70 The actual system – Modeling Propagation of Cardiac Potentials with the FitzHugh-Nagumo Model ............................. 70 Objective of the Lab ........................................................................................................................................................... 71
Computational Setup ........................................................................................................................................... 74 Results, Conclusions ............................................................................................................................................. 76
MATLAB CODES AND SAMPLE DATA FOR INSTRUCTORS ...................................................................................................... 78 EXPERIMENT 1: THEORETICAL INVESTIGATION OF THERMODYNAMICAL PROPERTIES OF CO2 .......................... 78
MATLAB Codes .................................................................................................................................................................. 78 Sample data ....................................................................................................................................................................... 81
EXPERIMENTS 2: IR AND RAMAN STUDIES OF CO2 ............................................................................................. 82 MATLAB Codes .................................................................................................................................................................. 82 Sample data ....................................................................................................................................................................... 84
EXPERIMENT 3: DETERMINATION OF THE SECOND VIRIAL COEFFICIENT OF CO2 ............................................... 86 MATLAB Codes .................................................................................................................................................................. 86 Sample data ....................................................................................................................................................................... 92
EXPERIMENTS 4: DETERMINATION OF THE JOULE-THOMSON COEFFICIENT OF CO2 .......................................... 93 MATLAB Codes .................................................................................................................................................................. 93 Sample data ....................................................................................................................................................................... 96
EXPERIMENTS 5: DETERMINATION OF THE HEAT CAPACITY RATIO OF CO2 WITH THE RUCCHARDT-METHOD 100 MATLAB Codes ................................................................................................................................................................ 100 Sample data ..................................................................................................................................................................... 104
EXPERIMENT 6: DETERMINATION OF THE HEAT CAPACITY RATIO OF CO2 WITH ADIABATIC COMPRESSION .. 108 MATLAB Codes ................................................................................................................................................................ 108 Sample data ..................................................................................................................................................................... 113
EXPERIMENTS 7: SOLVING AN ORDINARY DIFFERENTIAL EQUATION (ODE) SYSTEM....................................... 117 EXPERIMENTS 8: SOLVING A PARTIAL DIFFERENTIAL EQUATION (PDE) SYSTEM ............................................. 118 References ......................................................................................................................................................... 119
Table of Figures
FIGURE 1.2 STRATEGY TO STUDY MOLECULAR AND THERMODYNAMICAL PROPERTIES OF CO2 EXPERIMENTALLY AND THEORETICALLY ......... 5 FIGURE 1.3 OVERLAID IR AND RAMAN SPECTRA OF CO2 OBTAINED WITH COMPUTATIONAL MODELING (DFT/B3LYP METHOD/CC-PVTZ
BASIS) ....................................................................................................................................................................... 7 FIGURE 1.4 POTENTIAL SCAN BETWEEN TWO CO2 MOLECULES AS SHOWN IN THE INSERT (DFT/B3LYP METHOD/AUG-CC-PVTZ BASIS) ... 8 FIGURE 1.5 THE LENNARD-JONES POTENTIAL ......................................................................................................................... 11 FIGURE 1.6 B(T) FUNCTION ................................................................................................................................................ 11 FIGURE 1.7 COMBINED PLOT OF THE CALCULATED PHYSICAL QUANTITIES ..................................................................................... 14 FIGURE 2.1 IR SPECTRUM OF SOLID CARBON DIOXIDE .............................................................................................................. 17 FIGURE 2.2 RAMAN SPECTRUM OF CARBON DIOXIDE ...................................................................................................... 17 FIGURE 2.3 DETAILS OF THE BENDING TRANSITION IN THE IR SPECTRUM OF CO2 ........................................................................... 18 FIGURE 2.4 TRANSITIONS IN THE IR/RAMAN SPECTRUM OF CO2 ............................................................................................... 18 FIGURE 2.5 INSIDE VIEW OF RU .......................................................................................................................................... 22 FIGURE 2.6 FRONT VIEW OF RU .......................................................................................................................................... 22 FIGURE 1.1 COMPRESSIBILITY FACTOR OF REAL GASES AS A FUNCTION OF PRESSURE .................................................. 24 FIGURE 3.1 STRATEGY TO MEASURE THE SECOND VIRIAL COEFFICIENT ......................................................................................... 25 FIGURE 3.2 SCHEMATICS OF THE EXPERIMENTAL SETUP TO MEASURE THE SECOND VIRIAL COEFFICIENT .............................................. 29 FIGURE 3.3 THE EXPERIMENTAL SETUP TO MEASURE THE SECOND VIRIAL COEFFICIENT .................................................................... 30 FIGURE 4.1 JOULE-THOMSON EXPERIMENT ........................................................................................................................... 32 FIGURE 4.2 INVERSION TEMPERATURE OF GASES ..................................................................................................................... 32 FIGURE 4.3 PRESSURE VARYING PROTOCOL ............................................................................................................................ 33 FIGURE 4.4 SCHEMATICS OF THE EXPERIMENTAL SETUP TO MEASURE THE JOULE-THOMSON COEFFICIENT .......................................... 35 FIGURE 4.5 THE EXPERIMENTAL SETUP TO MEASURE THE JOULE-THOMSON COEFFICIENT ................................................................ 35 FIGURE 5.1 SCHEMATICS OF THE RÜCHARDT EXPERIMENT ........................................................................................................ 40 FIGURE 5.2 THE EQUILIBRIUM (LEFT) AND STARTING (RIGHT) POSITION OF THE PISTON (P0=EXTERNAL PRESSURE, P=PRESSURE IN THE
RESERVOIR, A=SURFACE AREA OF PISTON, M=MASS OF PISTON) ......................................................................................... 40 FIGURE 5.3 SAMPLE MEASUREMENT DATA, FITTED VALUES, AND THE EXPONENTIAL TERM OF THE DAMPED OSCILLATION ...................... 43 FIGURE 6.1 ADIABATIC GAS LAW APPARATUS ........................................................................................................................ 49 FIGURE 6.2 THE SCHEMATICS OF AGLA ................................................................................................................................ 49 FIGURE 6.3 CONNECTIONS TO THE CONTROLLER BOX ............................................................................................................... 49 FIGURE 7.1 BIFURCATION DIAGRAM. KG: BIFURCATION PARAMETER, [CACY]CY: CYTOSOLIC CA2+-CONCENTRATION ............................... 52 FIGURE 7.2 EXPERIMENTAL CACY CONCENTRATION IN A PC12 CELL (A ) AND THE RESPECTIVE FFT SPECTRUM (B) .............................. 57 FIGURE 7.3 THE G-PROTEIN CASCADING MECHANISM AND CICR-FACILITATED CA2+-DYNAMICS ....................................................... 59 FIGURE 7.4 SCHEMATICS OF THE MODEL OF CA2+-DYNAMICS IN PARAMECIUM CELLS ..................................................................... 60 FIGURE 7.5 MATHEMATICAL MODEL FOR CA2+-OSCILLATION ..................................................................................................... 60 FIGURE 8.1 REACTION-DIFFUSION SYSTEM DOMAIN ................................................................................................................ 70 FIGURE 8.2 ACTION POTENTIAL PROPAGATION IN TIME IN CARDIAC TISSUE ................................................................................... 72 FIGURE 8.3 TIME EVOLUTION OF ACTION POTENTIAL WITH POINT-LIKE EXCITATION (TOP ROW) AND WITH PERTURBATION THAT RESETS THE
ACTION POTENTIAL IN THE BOTTOM HALF OF THE DOMAIN (BOTTOM ROW) .......................................................................... 72
Theoretical Investigation of Thermodynamical
Properties of CO2
Background
In the complex laws-driven Nature it was a refreshing discovery that the behavior
of gases, regardless of their chemical identity, can be described with a very simple
mathematical relationship, the Ideal Gas Law:
m
pV nRT
pV RT
(1.1 )
The complications arose when gases were cooled down or compressed to high
pressure, when they started to show their individual nature. Efforts to describe gases
under these conditions resulted in semi-empirical models, such as the van der Waals
equation; however, these yielded more questions than answers. A slew of physical
constants were measured, such as the constants a and b in the van der Waals
equation, without offering explanation to their values beyond some qualitative
considerations. Part of this effort was Joule and Thomson’s work, which ultimately
laid the path for the industrial processes based on liquefied gases (such as using
supercritical carbon dioxide to extract caffeine from coffee beans or to replace
highly toxic chemicals in dry-cleaning processes). It wasn’t until Statistical
Mechanics evolved that quantitative, molecular-level explanation could be offered
for the behavior of gases.
Real Gas Behavior
When gas molecules are far apart from each other, the interaction among them is
negligible. However as they get close to one another by the force of pressure or
lower temperature (which in turns lowers their average kinetic energy), two types of
interactions prevail. While the average distance among them is a few times their
diameter, attraction forces will become significant. However if the molecules are
forced further closer, strong repelling forces prevail. Therefore, as the distance
between molecules decreases, first long range attracting forces, then short range
repelling forces will determine their behavior. Since the ideal gas law does not
account for any interaction among molecules or their finite sizes, it is expected that
under these conditions (high pressure and/or low temperature) it will fail.
The Joule-Thomson Coefficient
Roadmap for Studying Real Gases
The importance of experimentation and theoretical studies cannot be
emphasized enough. We learn about Nature through experimentation and the
experimental data is interpreted by theoretical models founded on results from
statistical mechanics. In turn, those models can be then used to predict processes,
phenomena, etc. During these labs, you will collect experimental data showing the
deviation from the ideal behavior and use existing theoretical models to support
them. The roadmap of this approach is shown in Figure….. The model molecule for
this adventure will be CO2, although for some of the experiments can be performed
with other molecules as well for comparison.
During the first lab, you will calculate the respective thermodynamical
quantities for CO2. Using statistical mechanics tools (included in Gaussian), you
will estimate the vibrational transitions in a CO2 molecule, from which the heat
capacity, CV, can be estimated. You will also estimate the potential between two
CO2 molecules as a function of the separation (also with Gaussian). From these
quantities the second virial coefficient, B2, and in turn the Joule-Thomson
coefficient, JT, can be estimated.
During the subsequent labs you will measure those physical quantities and
compare your findings to your theoretical predictions and to accepted literature
values.
Figure 1.1 Strategy to study molecular and thermodynamical properties of CO2 experimentally
and theoretically
IR/Raman spectroscopyComputational modelingVibrational transitions
(ni)
Adiabatic process(adiabatic compression/expansion
and Ruchhardt method)
Heat capacity(Cp, CV)
2 /4
2/
1
5,
2 1
i
i
T
iV
Ti
ii
B
eC R
T e
hc
k
n
High/low pressure measurements ( )/ 2
0
12 6
( ) 2 1
( ) 4
BU r k T
AB T N e r dr
U rr r
Second virial coefficient(B)
Isenthalpic expansion measurements
Joule-Thomson coefficient (JT)
,
1JT
p mH
T BT B
p c T
Experimental(Classical Thermodynamics)
Theoretical(Statistical Mechanics)
Computational model of CO2
The molecular properties of CO2, including vibrational spectra and intermolecular
interactions, can be predicted with computational modeling quite well. Here,
Gaussian was used to create computational models of these two properties.
Figure 1.2 shows an overlay of the IR and Raman spectra obtained with a DFT/
B3LYP method and cc-pVTZ basis. As expected, the vibrational transitions that
change the dipole moment of the molecule will show up in the IR spectrum, and the
transition that doesn’t impact the dipole moment will show up in the Raman
spectrum.
The extent of the interaction between two molecules as a function of the distance
between them can also be estimated with Gaussian. The Lennard-Jones potential
function will be used to describe the potential function, for which the values of
and can be estimated: the depth of the potential well is and the distance where
the potential is zero is (see Figure 1.3). You are to perform the potential scan with
two method/basis combinations of your choice to estimate the Lennard-Jones
constants for CO2 and you will be provided with the results from Figure 1.3 (this
model was running for over a week on a server, so be mindful of the time it takes to
run these models).
Figure 1.2 Overlaid IR and Raman spectra of CO2 obtained with computational modeling
(DFT/B3LYP method/cc-pVTZ basis)
050010001500200025003000
Ab
sorb
ance
/Co
un
ts (
AU
)
Wave number (cm-1)
IR spectrum
Raman spectrum
n2
(bending)
n3
(asymmetric strech) n1
(symmetric strech)
Figure 1.3 Potential scan between two CO2 molecules as shown in the insert (DFT/B3LYP
method/aug-cc-pVTZ basis)
Calculating the heat capacity of CO2
The constant volume heat capacity (CV) of a molecule has contributions from its
translational, rotational, and vibrational modes:
, , ,V V trans V rot V vibC C C C (1.2)
Specifically, the CO2 molecule has 3 translational and 2 rotational freedoms, each
contributing 1/2RT to CV. Since linear molecules have (3N-5) vibrational degrees of
freedom, the CO2 molecule has 4 vibrational modes contributing to CV. These are a
symmetric, an asymmetric, and two bending modes (the bending modes are doubly
degenerate and they show up at the same frequency):
Fundamental
frequency
(cm-1)
Mode
667.3
Bending (doubly degenerate)
1340 Symmetric stretching
2349.3
Asymmetric stretching
The vibrational frequencies were obtained from the Gaussian simulation.
The total vibrational contribution can be calculated from:
2 /4
, 2/
1
,1
i
i
T
i iV vib i
Ti B
hceC R
T ke
n
(1.3)
where kB = Boltzmann constant (J/K)
ni = ith fundamental frequency (m-1)
ivibrational temperature (of the ith vibrational mode)
c = speed of light (m/s)
T = temperature (K)
(Remember, the bending mode is to be counted twice).
Therefore, the overall CV value can be estimated from:
,
,
2 /4
, 2/
1
, , ,
2 /4
2/
1
2 /4
2/
1
3
2
2
2
,1
3 2
2 2 1
5
2 1
i
i
i
i
i
i
V trans
V rot
T
i iV vib i
Ti B
V V trans V rot V vib
T
i
Ti
T
i
Ti
C R
C R
hceC R
T ke
C C C C
eR R R
T e
eR
T e
n
(1.4)
Calculating the Second Virial Coefficient of CO2
To model the behavior of nonideal gases, we must modify the familiar ideal gas
law to account for real behavior. One equation used to model real gases is the virial
equation of state, which is a power series expansion of the compressibility
coefficient Z in terms of (1/Vm) or p.
2 3
2 3 4 2
1 1 1 11 ... 1m
m m m m
pVZ B B B B
RT V V V V
2 3
2 3 4 21 ... 1mpVZ A p A p A p A p
RT
(1.5)
where Z is the compressibility coefficient, An is the nth virial coefficient in terms
of p, Bn is the nth virial coefficient in terms of (1/Vm), . We will only consider the
second virial coefficient, A2 and B2.
From Statistical Mechanics we know that an intermolecular potential, U(r), and
the corresponding virial coefficients are related by eqn. (1.6).[2, 3]
( )/ 2
0( ) 2 1BU r k T
AB T N e r dr
(1.6)
where r = intermolecular separation (m)
U(r) = intermolecular potential (J)
T = temperature (K)
NA = Avogadro number (mol-1)
A commonly used potential function is the Lennard-Jones potential function:
12 6
( ) 4U rr r
(1.7)
where = separation where the repulsion turns to attraction between molecules
= depth of the potential well (J)
Figure 1.4 shows a typical Lennard-Jones potential and Figure 1.5 shows the calculated
temperature dependence of the second virial coefficient of CO2.
Calculating the Joule-Thomson Coefficient of CO2
The Joule-Thomson coefficient of a real gas describes the change in temperature
over change in pressure in an isenthalpic process. For an ideal gas, this coefficient
is zero, because the effect is caused by intermolecular forces.
The second virial coefficient is related to the Joule-Thomson coefficient (see
eqn. (1.8 )), which is not surprising since they both describe the behavior of real
gases, and they both vanish for ideal gases (for the expression used in (1.8) for the
Joule-Thomson coefficient please refer to the Appendix).
Figure 1.4 The Lennard-Jones potential
Figure 1.5 B(T) function
-5.00E-21
-1.00E-35
5.00E-21
1.00E-20
1.50E-20
2.00E-20
2.50E-20
4.00E-10 5.00E-10 6.00E-10 7.00E-10 8.00E-10 9.00E-10 1.00E-09
U(r
) /J
r /m
Lennard-Jones Potential
-400
-350
-300
-250
-200
-150
-100
-50
0
50
0 100 200 300 400 500 600 700 800 900
B(T
) /
mLm
ol-1
T /K
B(T)
( )
#
1
1
1
p
p
p
p
nRTpV nRT nB T p V nB
p
V nRT nR BnB n
T T p p T
nR B nRTT n nB
c p T p
nRT B nRTnT nB
c p T p
BnT nB
c T
JT
pp
1 Vμ - -T +V
c T
,
#
1
see the Appendix for the derivation of this representation of the JT Coefficient
JT
p m
BT B
c T
(1.8)
Pre-Lab work
1. Calculate Cv for CO2 using eqn. (1.4).
a. Devise your function or use the CVCO2.m function to evaluate the value of CV
between 100-1000K with 0.1 K step.
The usage of the CVCO2.m: function [Cv] = CVCO2(T,v)
%CVCO2a calculates the heat capacity of CO2 at different temperatures
%T = vector with temperature values
%v = vector with vibrational frequencies (in cm^-1)
%Cv = returned matrix with translational, rotational, and vibrational
%contribution in each column evaluated at the temperatures in T in each row
b. If you used your own function, return the CV values in a matrix with the
different contributions in each column evaluated at each temperature per row.
2. Calculate the Lennard-Jones potential between two CO2 molecules based on the and
values obtained from the Gaussian simulation and eqn. (1.7) using your own
function, or the Uljv.m function.
a. The usage of the Uljv.m function:
function U = Uljv(epsilon,sigma,r)
%Calculates the potential using the Lennard-Jones function as a function of
distance
%U = Ulj( epsilon, sigma, r)
%epsilon = depth of potential well (in J)
%sigma = r(U=0) (in m)
%r = distance vector (in m)
b. If you used your own function, return the U(r) values in a row vector.
3. Calculate the value of B(T) at the same temperature values as the CV values using the
Lennard-Jones potential and eqn. (1.6) using your own function or the BTp.m
function.
a. The usage of the BTp.m function: function BTi1 = BTp( epsilon,sigma,r,T)
%Calculates the second virial coefficient as a function of temperature
%BTi = BTp( epsilon,sigma,r,T)
%epsilon = depth of potential well (in J)
%sigma = r(U=0) (in m)
%r = distance vector
%T = temperature vector
b. If you used your own function, return the B(T) values in a row vector.
4. Calculate the Joule-Thomson coefficient at the same temperature values as the CV
values (1.8) using your own function or the JT.m function.
a. The usage of the JT.m function: function mu = JT( Cp,B,T )
%Calculates the Joule-Thomson coefficient as a function of temperature from
%Cp(T) and B(T)
b. If you used your own function, return the JT(T) values in a row vector.
5. Create a function that calculates the heat capacity, the second virial coefficient, and
the Joule-Thomson coefficient at room temperature, or use the roomT.m function:
function [Cv_room BT_room muT_room] = roomT(T,BT,Cvt,muT, T_room)
%Finds the Cv, B(T), and mu(T) values at temperature T_room
6. Create a function that generates a plot with four graphs as shown in Figure 1.6 or use
the marked_plot.m function: function
marked_plot(x,y,x0,y0,xmin,ymin,x_label,y_label,graph_title,point_text,pos_flag)
%x = independent variable (vector)
%y = dependent variable (vecttor)
%x0 = x coordinate for the point (number)
%y0 = y coordinate for the point (number)
%xmin = x axis minimum flag:
% 0 = x axis starts at 0
% 1 = x axis starts at min(x)
%ymin = y axis minimum flag:
% 0 = y axis starts at 0
% 1 = y axis starts at min(y)
%x_label = x axis label (string)
%y_label = y axis label (string)
%graph_title = title of the graph (string)
%point_text = test for the showed point, eg. the y value (string)
%pos_flag = position of text around the point:
% 0=no text
% 1=above point
% 2=below point
7. Create a control script that calculates all of the physical quantities listed above and
generates the plot as shown in Figure 1.6, and obtain the room temperature values
(Note: you may need to rerun the calculation if the experiments were performed at a
different temperature).
.
Figure 1.6 Combined plot of the calculated physical quantities
IR and Raman Studies of CO2
Background
While carbon dioxide is a relatively simple molecule, it has a relatively
complex spectroscopic behavior. The main features and some of the details will be
examined in this lab to the extent that will help to explore the thermodynamical
properties of CO2.
Considering that the CO2 molecule is linear, 3N-5=4 transitions are expected:
Bending (doubly degenerate)
(n2)
Symmetric stretching
(n1)
Asymmetric stretching
(n3)
Table 2.1 Vibrational modes of CO2
The numbering of the transitions is as follows (following numbering in the
literature):
1: Symmetric stretching
2: Bending
3: Asymmetric stretching
Furthermore, indicating the rotational transitions, the niJ(n) nomenclature will
be used, where n is the vibrational level (n = 0, 1, ….), J is the rotational level (J =
0, 1, 2,….), and i is the identification of the transition as listed above (i = 1, 2, 3).
Energetically these transitions all fall into the IR range, however there are
some differences. While for a transition to be IR active it has to change the dipole
moment of the molecule, a transition that changes the polarizability of a molecule
will be Raman active. This fact reveals two important points about IR and Raman
methods: one, they complement each other, and two, they leverage different
properties of molecules. That said, it is important to keep in mind, that regardless of
the detecting technique (as to what properties the method is based on), transitions
are present regardless. The result of this is quite surprising: transitions that are not
IR active can and will interact with IR active transition yielding new IR or Raman
active transitions.
In comparison to the HCl/DCl system that you already studied before, there
are some further differences. In the HCl/DCl there was only a “parallel mode” (that
is a transition parallel with the axis of the molecule), the J = 0 transition was
prohibited, therefore the fundamental frequency of the transition was missing from
the spectrum. However, in the CO2 molecule there are “perpendicular modes” (that
is modes perpendicular to the axis of the molecule, namely the two bending modes),
the J = 0 selection rule is allowed for other than the ni0(0) ni
0(1) transition. The
transitions with the J = 0 selection rule are referred to as the Q branch (and as
before the J =+1 transitions constitute the R branch, and the J = -1 the P
branch). Because the transitions associated with the Q branch have only a
vibrational component, they have the same energy and they all show up at the same
frequency, that is, at the respective fundamental frequencies.
Another noticeable difference is the missing odd numbered rotational levels
for n = 0. The explanation for this “odd” phenomenon is beyond the scope of this
lab (see Ref. [1]).
The IR Spectrum
A typical IR spectrum of CO2 is shown in Figure 2.1. The bending (n2) and
asymmetric stretch (n3) transitions are prevailing and some of the combination
bands also are identifiable Most interesting is that they are a result of an IR inactive
band (symmetric stretch, n1) and an IR active band (n2 or n3). For example …at
3500 nu1 (1500) and 2*nu2…
Raman Spectrum
A typical Raman spectrum is shown in Figure 2.2. Instead of the expected one
transition associated with the symmetric stretch at about n1=1336 cm-1, there are
two distinct peaks. The reason is that the first harmonic of the bending transition (n2
= 667 cm-1) is at almost at the same wave number as the symmetric stretch, and
therefore they interact to yield two split peaks due to the phenomenon called Fermi
resonance. The coupling constant between the two peaks is 110 cm-1. The
interesting fact about this phenomenon is that a Raman inactive transition interacts
with a Raman active one. The conclusion here to be drawn is that transitions do
exist regardless of our detecting methods; for example, one of the peaks around
3500 cm-1 is a result of an IR active and a Raman active transition.
Figure 2.1 IR spectrum of solid carbon dioxide
Figure 2.2 Raman spectrum of carbon dioxide
1,281.28
1,391.03
1,336.15
0
50
100
150
200
250
300
350
400
450
500
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500
Inte
nsi
ty
n (cm-1)
CO2 Raman Spectrum
n = 55cm-1 n = 55cm-1
Figure 2.3 Details of the bending transition in the IR spectrum of CO2
Figure 2.4 Transitions in the IR/Raman spectrum of CO2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
610630650670690710730
Ab
sorb
ance
Wave number (cm-1)
667.4 cm-1
1 1
1 2(1) (1) 2076.9cmn n
0
2 (0) 0n
1 1
2 (1) 667cmn
0 1
2 (2) 1285.4cmn
2 1
2 (2) 1335.2cmn
1
1 (1) 1388.2cmn
1 1
2 (3) 1932.5cmn
3 1
2 (3) 2003.3cmn
1
3 (1) 2349.2cmn
618.0 cm-1
667.8 cm-1
720.8 cm-1
647.1 cm-1
791.4 cm-1
597.3 cm-1
668.1 cm-1
741.4 cm-1
961.0 cm-1
1063.7cm-1
688.7 cm-1
Details of the bending mode
The details of the IR spectrum of the bending transition is shown in Figure 2.3
and the most prevailing transitions are shown in Figure 2.4 and some of the
transitions involving this mode are in Figure 2.4 with the respective energy levels,
except the J = 0 transitions. Since the J = 0 transitions do not have a rotational
component, they all show up at the frequency of the vibrational transition, that is at
the same frequency.
Recall that the rotational energy levels are given by the following equation:
2 2( 1) ( 1),
8 8
h hE J J BJ J B
Ic Ic
(2.1 )
where h is the Planck constant, I is the rotational inertia of the molecule, c is the speed
of light, B is the collection of all the constants. Based on this equation, the energy
level can be characterized in terms of B and the rotational quantum number, J. The
table below lists first few transitions in the P and R branches (keep in mind that the
odd number of rotational levels are missing for n=0):
P branch
Initial level Final level E
J E J E
n22(0) n2
1(1) 2 6B 1 2B -4B
n24(0) n2
3(1) 4 20B 3 12B -8B
n26(0) n2
5(1) 6 42B 5 30B -12B
n28(0) n2
7(1) 8 72B 7 56B -16B
R branch
Initial level Final level E
J E J E
n20(0) n2
1(1) 0 0 1 2B 2B
n22(0) n2
3(1) 2 6B 3 12B 6B
n24(0) n2
4(1) 4 20B 5 30B 10B
n26(0) n2
7(1) 6 42B 7 56B 14B
Table 2.2 Rotational transition energy levels in terms of B
From Table 2.2 it is clear, that the rotational levels are staggered 4B apart, which
means that the rotational peaks are 4B apart in both branches. Therefore using the
experimental spectrum and eqn. (2.1 ) the equilibrium bond length of C=O can be
estimated:
2
2 2 2 2 22 2
2
34 23 1
10
2 8 1
( )8 8 2 8 2 ( )8 2
16 ( )
6.62 10 6.022 101.16 10
16 3.14 0.016 3 10 39.0
a
a
a
kgm
s
kg mmol s
hNh h hB
Aw OIc mr c Aw O r cr c
N
hNr
Aw O cB
ms molm
m
(2.2 )
where 10.39 0.04B cm obtained from average distance between two neighbor
peaks in the experimental the spectrum, r is the bond length, I=2mr2 is the
rotational inertia of the molecule, m is the mass of the oxygen atom, and Aw(O) is
the atomic weight of oxygen.
Experimental Procedure
IR Spectroscopy
1. Start the software that operates the Bruker IR Spectrophotometer (the one on the
left) and login as CHEM330L (password: pchem).
2. Remove the gas cell from the desiccator are remove both rubber septa.
Note: the cell has to be stored in the desiccator between experiments as the
windows of the cell are made from KBr, which is hydroscopic, and the windows
get foggy (impairing the optical properties of the windows), and potentially
destroyed if exposed to moisture for a long period of time.
3. Flush the cell by allowing He gas to flow in one port and out on the other port
for about 30 s.
4. Immediately after stopping the flushing process replace both septa.
5. Take the background scan with the gas cell as prepared (with He in it).
6. Remove the stopper from the needle of the 10-mL gas syringe and fill it with
CO2 by placing the needle into CO2 gas flow (e.g. from a tubing attached to the
gas cylinder) and withdrawing the plunger slowly.
Note: after filling the syringe, put the rubber stopper back on the needle to
prevent air and moisture getting in. Before every time you use the syringe to
deliver a sample push the plunger forward about 0.2 mL to force the air out that
potentially entered the syringe.
7. Find the pin hole on one of the septa and slowly insert and push in the needle of
the gas syringe into the pin hole.
Note: the needle of the gas syringe is dull, the end is closed and there is a pin
hole at the end of the needle on the side to allow the gas to flow in and out.
8. Inject about 2 mL gas into the gas cell.
9. Obtain the IR spectrum.
10. Save the spectrum in ASCII, MATLAB, and image format.
11. Replace the cell to the desiccator.
Raman Spectroscopy
The inside view of the Raman Spectrometer (Raman Unit, RU) is shown Figure 2.5.
The sample placed in the sample holder is exposed to a 532nm laser beam. The
scattered light at 900 is directed to the spectrophotometer detector through a 540nm
longpass filter to minimize the stray-beam effect from the exciting beam.
1. Start MATLAB on the attached computer.
2. The function called OO_Grab captures a single spectrum from the RU. :
function [w s] = OO_Grab(xflag, iTime, iPass, iBox) %xflag: flag for type of return scale: % 'wl' = wavelength % 'wlCO2' = wavelength range relevant to CO2 % 'wn' = wavenumber % 'wnCO2' = wavenumber range relevant to CO2 %iTime = integration time (in s) %iPass = number of spectra to be averaged %iBox = number of points to be averaged in one spectrum
3. The controls are on the front (see Error! Reference source not found.).Turn
n the RU by
i. Turning on the Power Switch.
ii. Turn the Key Switch to the ON position.
iii. Enter the 4 digit code on the keypad. The green Laser On light should
come on.
iv. Pressing “1” on the keypad disables the laser.
4. With the laser enabled, collect the background spectrum.
5. Disable the laser by pressing “1” on the keypad.
Note: The opening of the sample holder will disengage the laser, however
this is a safety feature and not intended for operating the unit. Always
disengage the laser by pressing “1” on the keypad.
6. Take several rods of DyIce® from the storage box in the thermostat box.
7. Pick one that fits the sample holder, scrape off the outer layer to remove
moisture.
Note: use thermos-gloves handling DryIce® to avoid freeze burns!
8. Place it in the sample holder so that the laser can effectively scatter off to the
opening of the detector.
9. Close the lid.
10. Enter the four-digit code to enable the laser.
11. Collect the spectrum of the sample.
12. When done, turn off the RU:
i. Press “1” on the keypad.
ii. Turn the key to OFF. Remove and return the key.
iii. Turn off the power switch.
Figure 2.5 Inside view of RU Figure 2.6 Front view of RU
Data Analysis
1. Obtain the wavenumbers for the IR -active transitions from the IR spectrum.
2. For the Raman spectrum process the background and the spectrum of the
sample in the most effective way:
i. Try to simply subtract the background from the spectrum of the sample.
ii. If that is not sufficient, apply the smooth function to both spectra before
the subtraction.
iii. You can also try to fit a 7-11 order polynomial on the IR spectrum, and
use it as a background that you subtract from the actual spectrum.
3. Compare the experimental spectra to the ones obtained with Gaussian.
4. Identify the most prevailing combination bands in the IR spectrum.
Determination of the Second Virial Coefficient of CO2
Background
The virial1 equation was developed in an effort to match experimental data with
existing mathematical models, just like the van der Waals equation; however, as
you will see, Statistical Mechanics can shed some light on the meaning of the
empirical results.
The validity of the Ideal Gas Law usually is characterized by the compressibility
coefficient, Z:
m
pVZ
nRT
pVZ
RT
(3.1)
The compressibility coefficient is Z = 1 for ideal gases. The Ideal Gas Law is a
limiting law; the limiting value of the compressibility coefficient is 1 as the
pressure approaches zero and the temperature approaches infinity. At moderately
low pressures the compressibility coefficient of gases usually is less than 1 (except
the noble gases and the hydrogen gas) and at higher pressure it is higher than 1 (see
Figure 3.1).
The deviation of the compressibility coefficient can be approximated with the
typical power series that is often used in science when experimental data is fitted to
an arbitrary mathematical model2.
2 3
2 3 4 21 ... 1mpVZ A p A p A p A p
RT
2
2 3 2
1 1 11 ... 1m
m m m
pVZ B B B
RT V V V
(3.2)
where Ai and Bi are second, third, fourth virial coefficients in terms of p and
1/Vm. As pressure increases and temperature decreases, more of the higher-order
terms have to be considered. In this experiment the pressure of the gas will be
between 1-35 bar, and in this range only the second virial coefficient has to be
considered:
1 The word “virial” comes from the Latin force referring to the interation among molecules 2 Remember, the potential of a thermocouple as a function of temperature is given by
2
1 2 3( )V T A A T A T where Ai are constants
Figure 3.1 Compressibility factor of real gases as a function of pressure
2
2
11
1
m
Z BV
Z A p
(3.3)
We will be using the second form in this experiment as we will monitor the
pressure of the gas. The relationship between A2 and B2 is:
2 2B A RT (3.4)
The strategy that we will follow is that we will determine the compressibility factor
as a function of pressure (see Figure 3.2). The schematics of the experimental setup
are shown in Figure 3.3. We load a “Sample tank” with CO2 at about 35 bar which is
high enough for the gas to show deviation from the ideal behavior. Then we release
a small sample of the gas into the “Expansion tank” to a low enough pressure so
that using the Ideal Gas Law the amount of released sample can be calculated. After
evacuating the “Expansion tank,” we release another small sample from the
“Sample tank.” We continue the process until the pressure in the two tanks is the
same. The sum of the released samples makes up for the total initial number of
moles.
Figure 3.2 Strategy to measure the second virial coefficient
The amount of the first sample released from the Sample tank:
11
e ep Vn
RT (3.5)
where p1e is the pressure in the “Expansion tank” after releasing the first sample, Ve
is the volume of the “Expansion tank,” and the pressure in the “Sample tank” drops
to p1s. The Expansion tank then evacuated, and a second sample is released from the
“Sample tank”:
22
e ep Vn
RT (3.6)
The process is continued until the pressure in the two tanks is the same after the mth
sample: pme = pm
s. The total amount of gas is the sum of the amounts of the
samples released from the “Sample tank”:
0 1 2
1
0 1 2
0
1
...
1...
1
ms e e e s
m m m i
i
e s e e e e e e
m m
me s e e
m i
i
n n n n n n n
n p V p V p V p VRT
n p V V pRT
(3.7)
where ni is the number of moles in the i th release. We need to devise a method to
calculate the amount of gas remaining in the sample tank after each release.
Consider the r th release, after which the remaining gas in the Sample tank is:
n1
n2
nr
nr+1
nm
0
1
r
r i
i
n n n
1
r
i
i
n
n0
p1s
p2s
prs
pr+1s
pms
s s
rr
r
p VZ
n RT
p0s
Measured
Calculated from pie values
Between pre and pm
e
Pressure in Sample tankAmounts of gas releasedInto the Expansion tank
0
1
r
r i
i
n n n
(3.8)
The combination of Equations (3.7) and (3.8) yields:
1 1
1
1
1
m re s e e e e
r m i i
i i
me s e e
r m i
i r
n p V V p V pRT
n p V V pRT
(3.9)
Substituting Equation (3.9) into Error! Reference source not found. results in:
1
1
s s
rr
r
s s
rr m
e s e e
m i
i r
s
rr e m
e e
m isi r
p VZ
n RT
p VZ
p V V p
pZ
Vp p
V
(3.10)
where Zr is the value of the compressibility factor after the r th release at the
pressure of prs. Equation (3.10) is the basis of the experiment. The pressure has to
be measured at every release in both the “Expansion tank.” and the “Sample tank.”
The summation has to be performed after the experiment was completed: for the r
the release is the sum of all “Expansion tank” pressure readings from reading r+1 to
m.
Notice, that the actual volume of either tank is not relevant, only the ratio of the
two volumes, which will also be determined in the experiment. When the
experiment is done after the mth reading, once again close the “Sample tank” and
evacuate the “Expansion tank,” then allow the gas to expand into the “Expansion
tank,” when the pressure will be pf.
s s e s e s
m f f f
s s s e
m f f
s s e
m f f
sem f
s
f
p V p V V p V p V
p V p V p V
p p V p V
p pV
V p
(3.11)
Therefore we can use the Ideal Gas Law to calculate Ve/Vs.
Experimental Procedure
1. The schematics of the experimental setup is shown in Figure 3.3. For the used
functions see below.
2. Close manual valves (MV1 - MV3) and start vacuum pump manually (red button, SW).
3. Connect the CO2 tank and set the pressure to 400PSI on the regulator.
4. Write a control script to execute the following steps:
a. Execute the VC_setup.m script.
b. Load Sample tank with gas to 450 PSI pressure with the VC_LoadGas
function.
c. Run the experiment using the VC_RunExperiment function, with 90000 Pa
pressure in each expansion and a tolerance of 1200 Pa. The temperature, the pe
and ps readings will be returned in an array (m+1 rows, 3 columns); save the
data in the variable psave.
5. Close the main valve on the gas cylinder.
6. Release the pressure with the butterfly valve on the regulator.
7. Disconnect the gas hose.
Functions used by the control script: VC_setup Configures the MODuck data acquisition interface for
control and data acquisition, creates the sNI object through
which the control/data acquisition is performed
Input channels (data acquisition):
ai0: temperature sensor
ai1: 500 PSI pressure sensor (sample tank)
ai2: 30 PSI pressure sensor (expansion tank)
Output channels (control):
line0: Loading solenoid valve
line1: Expansion solenoid valve
line2: Vacuum solenoid valve
VC_LoadGas(p,punit,sNI)
Loads the sample holder with gas to pressure p (given in unit
'punit')
p: target pressure for the sample tank
punit: unit in which the pressure is given ('Pa','PSI','bar')
sNI: data acquisition object
psave = VC_RunExperiment(psample, tol,
sNI)
Returns psave in the form of = [temp pe ps] (m+1 rows)
psample: Sample pressure withdrawn at a time (Pa) tol: Tolerance for accepting equilibration (Pa)
sNI: data acquisition object
a) Read initial pressure and temperature values
b) Save initial readings
c) Load Expansion tank with psample (Pa) gas
d) Check if the pressure is enough (loaded = 1)
e) If loaded = 0, take a final reading (this is the
mth data point).
After the mth reading, evacuate expansion
tank one last time, open the valve between the
tanks, and take one more reading to determine
the volume ratio.
f) If there is enough gas, allow the system to
equilibrate.
g) Read pressure and temperature values in the
Sample and in the Expansion tanks.
h) Pump down Expansion tank.
i) Repeat steps c-h until there is not enough gas.
j) Evacuate the system.
Do NOT disconnect the tubing under pressure! The free end of the tubing may break lose due to
the violent gas flow resulting in personal injury! De-pressurize the system by opening all valves
BEFORE disconnecting the tubing!
Figure 3.3 Schematics of the experimental setup to measure the second virial coefficient
SV1= solenoid to Sample Tank, SV2= solenoid to Expansion Tank, SV3= solenoid to vacuum, PT1=
pressure transducer for Sample Tank, PT2= pressure transducer for Expansion Tank, PG1= pressure
gauge for Sample Tank, PG2= pressure gauge for Expansion Tank , MV1-MV3 = manual valves, SW =
switch to turn on vacuum pump
Load
ing
valv
e
Ven
t 2
Sam
ple
tan
k
Sam
ple
val
ve
Exp
ansi
on
tan
kVen
t 1
Vac
uu
m v
alve
Pressure transducer(ps)
Pressure sensor(pe)
Sample pressuregauge
Loading pressuregauge
Figure 3.4 The experimental setup to measure the second virial coefficient
Results, Calculations
1. Write a script to process the psave array.
a. Calculate the sum in (3.10)
b. From the final pressure readings (taken when the last sample was expanded) calculate
Ve/Vs (it should be around 2.71)
c. Calculate Zr as a function of pressure in the Sample tank.
2. Plot Zr as a function of the pressure in the Sample Tank and obtain A (second virial
coefficient in terms of pressure). Discard data points at low pressure that do not fit the trend
(and explain your reasoning in your report).
3. Calculate B (the second virial coefficient in terms of volume) in mL/mol unit.
a. Compare your results to the literature value.
b. Compare your results to the theoretical value that you obtained before.
4. Compare your Zr vs. ps graph with Error! Reference source not found.. Which part of
REF _Ref283032344 \h Error! Reference source not found. is covered by your
experimental results? Explain your reasoning.
Determination of the Joule-Thomson Coefficient of
CO2
Background
When a real gas is driven through a porous disk (see Figure 4.1 Error! Reference
source not found.) and enters a lower pressure, molecules break apart from each
other and the potential energy between them will change. The nature of the change
will depend on the potential energy before the expansion, and if the process is
adiabatic, the potential energy change will affect the thermal energy of the
molecules, and therefore the temperature of the gas. If molecules were under
conditions where attractive forces dominated, the potential energy of the molecules
will increase when the gas enters the lower pressure. This energy is converted from
the energy of the molecules, and therefore the gas will cool down. If the molecules
begin under conditions where the repelling forces prevail, the potential energy will
decrease, passing the extra energy to the molecules; therefore the gas warms up.
The process through the porous medium is isoenthalpic3, and change of
temperature as a function of the pressure under these circumstances is called the
Joule-Thomson coefficient:
JT
H
T
p
(4.1)
Since the expansion can occur from states when the attractive and when the
repulsing forces dominate, the Joule-Thomson coefficient can be positive or
negative, which depends on the pressure and the temperature of the gas. At high
enough pressure, the repulsion forces will always dominate, hence m< 0 and the gas
warms up. Lowering the temperature will reach a point where the attractive and
repulsion forces are equal, the gas behaves like an ideal gas, µ=0. Further lowering
the pressure, at low temperature values the repelling forces (µ<0), higher
temperature values the attractive forces (µ>0) , and at very high temperature again
the repelling forces will dominate (see Figure 4.2).
While the process of gases going through a porous disk is adiabatic, distinction
has to be made from the classical adiabatic process. In both cases there is no heat
exchange between the environment and the system. However, although µ=0 for
ideal gases, ideal gases always cool down in a regular adiabatic process.
.
3 See Appendix for details
Figure 4.1 Joule-Thomson experiment
Figure 4.2 Inversion temperature of gases
The experiment will be performed with a setup much like Joule and Thomson’s original
experiment. The schematic of the experimental setup is shown in Figure 4.4 and the actual setup is
shown in Figure 4.5. The gas is allowed to pass through a porous disk from the high pressure
side and leave on the low pressure side (which is open to the atmosphere). The temperature of
the gas is measured with thermistor temperature probes on both sides. The pressure is measured
with a pressure transducer on the high-pressure side which measures the difference between the
pressure in the system and the atmospheric pressure; therefore, it directly measures the pressure
difference between the two sides of the frit (since the low-pressure side is open to the
atmosphere). The pressure on the high pressure side is controlled by an electronic pressure
regulator which can set the pressure between 0-120 PSI based on a signal between 0-10V. The
MODuck interface can provide 0-5V control signal only, therefore there is a circuit that can
provide an additional 0-5V bias signal which can be adjusted with the knob in the lower right
corner of the setup. This bias signal provides approximately 0 – 60 PSI initial pressure. You will
write a function JT_Run which performs the experiment by adjusting the initial pressure from
min_pres to max_pres over the time of ramp_time while allowing the system to
equilibrate for dwell_time between adjustments (see Figure 4.3).
Figure 4.3 Pressure varying protocol
Experimental Procedure
1. The schematic of the experimental setup is shown in Figure 4.4 and the actual setup is
shown in Figure 4.5.
2. Connect the gas supply to the setup and turn on the heating on the gas regulator.
Carbon dioxide cools down considerably upon expansion and the regulator has
to be heated to prevent freezing!
3. Turn on the fan of the heat exchanger to High.
4. Open the gas supply to the setup and set the supply pressure to about 10 bar.
5. Verify that the MODuck box is connected to the pressure transducer and two
temperature sensors, as well as to the pressure regulator.
6. Open the gas flow.
7. Execute the JT_setup script.
8. Write the JT_Run function as specified below to perform the experiment. [dT dp] = JT_Run(ramp_time, dwell_time,
min_pres, max_pres, sNI) ramp_time = time to ramp up pressure from min_pres
to max_pres (s)
dwell_time = time between changing pressure for
equilibrating (s)
min_pres = starting pressure (PSI)
max_pres = ending pressure (PSI)
sNI = data acquisition object
Returns: Array with the T = Tf - Ti and p = pf - pi values
a) Determines number and size of steps in pressure,
based upon the input parameters.
b) Measures the pressure offset provided by the 0-5V
bias signal (which is controlled by the external
knob).
c) Sets the pressure using the provided function.
d) Waits for dwell_time.
e) Reads temperature and pressure data using the
provided function.
f) Stores a data point in an array of the form [dT
dp].
g) Repeats steps c-f until max_pres has been
reached.
h) Reduces the pressure in the system at the end of the
experiment.
Functions used in the control function:
JT_setup Configures the MODuck data acquisition interface for
control and data acquisition, creates the sNI object through
which the control/data acquisition is performed
Input channels (data acquisition):
ai0: initial temperature sensor
ai1: final temperature sensor
ai3: Initial pressure sensor
Output channels (control):
ao0: 0-5V signal for pressure controller (regulator is biased
with 5V initial signal)
JT_set_p_controller(pressure, offset,
sNI) Sets the voltage of the pressure controller to reach the
desired pressure. pressure: desired pressure in PSI
offset: pressure provided by manual 0-5V offset sNI: data acquisition object
[ti, tf, p] = JT_read_sensors(sNI)
Reads initial and final temperature, and initial pressure,
averaged over 1000 readings. Temperatures given in Kelvin;
pressure given in PSI. sNI: data acquisition object
.
Figure 4.4 Schematics of the experimental setup to measure the Joule-Thomson coefficient
Figure 4.5 The experimental setup to measure the Joule-Thomson coefficient
Results, Calculations
1. Make a plot of T vs. p and calculate the Joule-Thomson coefficient from the graph, in
units of K/bar.
2. Using the experimental data, calculate the Joule-Thomson coefficient at each point where
readings were taken, and calculate an average of the obtained values. Compare the average
value with the value obtained from the graph.
3. Compare the experimental value of JT with the value that was obtained from theoretical
calculations and to the accepted literature value.
Appendix: The Derivation of the JT Coefficient
( , )
:
0
1
:
0
pT
p
p pT T
p p
T
T
S f T p
S SdS dp dT
p T
S
T
dp
S S S SdH TdS Vdp T dp dT Vdp T dp T dT Vdp
p T p T
S H
T T T
S
p
dT
SdH T dp Vd
p
2 2
2
2
1
1 1 1
T T
T T
pT T T
p
H ST V
p p
S HV
p T p
S S H H H VV V
T p T p T T p T p T p T T
S
p T p
2
2 2
2 2
2
1 1
1 1 1
p p
pT
S H H
T p T T T T p
S S
p T T p
H H H VV
T T p T p T p T T
2 21
10
Euler relation : 1
1
/
pT
pT
pT
T pH
H p
H H H VV
T p T p T p T
H VV
T p T
H VV T
p T
T p H
p H T
T H
p H T p
1
1
pT T
pT
JT
ppH
H
c p
H VT V
p T
T VT V
p c T
Determination of the Heat Capacity Ratio of CO2 with
the Rüchardt-method
Background
The heat capacity ratio (γ=Cp/CV) is typically measured with an adiabatic process.
The simplest method is performing adabatic compression or expansion on a gas and
measuring p, T, and V. In this experiment the Rüchardt-method will be used, which
is based on the compression and expansion of a trapped gas sample by an
oscillating mass (piston). The experimental setup for this approach consists of a gas
reservoir with a Plexi® glass tube. In the tube a piston can move quasi-
frictionlessly (see Figure 5.1).
At equilibrium, the piston is at rest and the pressure inside the reservoir is:
0
mgp p
A (5.1)
where p0 is the atmospheric pressure, A is the area of the cross section, and m is the
mass of the ball (see Figure 5.2, left). When the piston is removed from its
equilibrium, the force returning the piston to its equilibrium is given by:
2
2
d xA dp m
dt (5.2)
where dp is the pressure change due to the displacement of the piston. Since the
oscillation of the ball has relatively high frequency, the compression-expansion of
the gas in the reservoir can be considered for practical purposes adiabatic:
.pV const (5.3)
Figure 5.1 Schematics of the Rüchardt experiment
Figure 5.2 The equilibrium (left) and starting (right) position of the piston (p0=external
pressure, p=pressure in the reservoir, A=surface area of piston, m=mass of piston)
Taking the derivative of both sides yields:
( 1)
( 1)
( 1)
0V dp p V dV
V dp p V dV
p V pdp dV dV
V V
(5.4)
The displacement of the ball (x) will change the volume of the gas with dV:
dV Ax (5.5)
Combining eqns. (5.5), and (5.6) yields:
22
2
2
2
p d xA x m
V dt
d xkx m
dt
(5.6)
This equation is Hook’s Law which characterizes a simple harmonic motion, where
k is the Hook constant. The Hook constant is related to the angular frequency () of
the oscillation. The angular frequency is also related to the period (T) of the
oscillation:
2
2
22
k p A
m mV
mVT
p A
(5.7)
From (5.7) one can obtain:
2
2 24
mV
pT A (5.8)
According to (5.8), by measuring the period of the oscillation, one can calculate the
ratio of the heat capacities.
However, the oscillation is damped (specifically under damped) which can be
characterized by the following equation:
0
2 2
2
2
0
22
0
( ) cos( )
2, ,
2
2( ) cos
( ) cos
t
t
t
x t x e t
b
T m
x t x e tT
p Ax t x e t
mV
(5.9)
where x(t) is the displacement of the oscillator as a function of time, x0 is the
amplitude of the oscillation, is the initial phase (which will be ), b is the
damping coefficient, m is the mass of the oscillator. The last of Eqn. (5.9) will be
used to fit experimental data and to obtain .
The experimental setup consists of a water jug (V = 19.9 L) with a Plexi® tube
attached vertically, which has the length of l = 1.3 m and the diameter of d = 0.037
m. There is a Plexi® piston with the mass of m = 0.318 kg which can move nearly
frictionlessly in the Plexi® tube. The cylinder rests on a bumper at the bottom of
the Plexi® tube (to prevent it from falling into the jug) under rest conditions. The
cylinder will travel up in the Plexi® tube when the pressure inside the system is
increased and it is caught by a powered electromagnet at the top of the Plexi® tube.
When the electromagnet is powered down, the cylinder is dropped into the tube and
performs the damped oscillation for the measurement until it comes to a rest on the
bumper. The last few oscillations during which the cylinder is bouncing off the
bumper will have to be disregarded. The jug and the Plexi® tube is filled with the
gas to be examined through a solenoid valve with a tubing going down to the
bottom inside the jug (for effective flushing). The pressure inside the system is
monitored by a pressure transducer. Since the displacement of the oscillator is
proportional with the pressure inside the system, and the pressure is proportional to
the signal from the transducer (V(t)), the signal from the transducer will be fitted
directly to obtain . Since there is an offset signal from the pressure transducer at
atmospheric pressure at the beginning of the experiment (about 0.5 V), the collected
V(t) data will be offset with this value allowing the points centered around zero.
2
2
0
22
0
( ) ( )
2( ) cos
( ) cos
off
t
off
t
off
V t V t V
V t V e tT
p AV t V e t
mV
(5.10)
where V(t) is the signal collected from the transducer over time, V is the average of
the V(t) values, which in fact is the offset of the signal, Voff(t) is the signal from the
transducer over time centered around zero.
For the experiment the reservoir has to be filled with the gas and the piston has to
be held by the electromagnet. Under these conditions the pressure inside the
reservoir and outside are the same (see Figure 5.2 right).
Figure 5.3 Sample measurement data, fitted values, and the exponential term of the damped
oscillation
Experimental Procedure
1. Verify the connections to the MODuck interface (see Appendix) and connect the
interface to the computer.
2. Connect the gas supply, and set the pressure to about 7 PSI with the regulator on
the solenoid valve.
3. Connect to the MODuck interface to the Controller box as follows
Sensor or Solenoid valve MODuck port
Pressure sensor AI0
Solenoid valve P0.0
Electromagnet P0.1
+5V Any 5V port
4. Execute the R_Setup.m script to configure the interface for data collection.
5. Execute the R_Flush(1,120,sNI) function to flush the system with the
studied gas (where 1 is a flag to perform the flush, 120 is the time in s for which
the flushing lasts, and sNI is the data acquisition object).
Note: if the same gas was used before, the flush time can be reduced to about 10s.
During the flushing, the cylinder will travel up, and the electromagnet will hold it
allowing the gas to escape between the wall of the tube and the cylinder. At the
end of the flushing, the cylinder will recess to the bumper.
6. The control code should perform the following operations:
Open the solenoid to make the cylinder travel up to the electromagnet.
Power the electromagnet.
Close the solenoid valve.
Wait until the pressure inside the system will resume atmospheric
pressure.
Turn off the electromagnet to initiate the experiment (drop the cylinder).
Immediately start to collect data (1000 points).
Functions: R_Control(flag, sNI) Operates the solenoid and the electromagnet based
on the value of flag:
Value of flag Solenoid valve Electromagnet
0 Off Off
1 On Off
2 Off On
3 On On
r = R_Read(n, sNI) Reads n points from the pressure transducer and
returns r, which is a matrix with n rows and the
time values in the first column, the signal values in
the second column
7. Shut off and disconnect the gas supply when done.
Results, Calculations
4. Make a plot of the raw V(t) data. Determine the portion that was collected before the cylinder
hit the bumper (the oscillations that were collected after the cylinder hit the bumper have
noisy maxima and minima). Load this portion of data points into new arrays, e.g. x for the
signal, t for time.
5. Calculate the mean of the useful portion of the collected signal and offset the V(t) values.
6. Create a function to be fitted (Eqn. (5.10)) with initial values of the amplitude (a),
(lambda), the period, (T), phase (phase) placed into a parameter array. Also set the values
for the atmospheric pressure (patm), surface area of the piston (A), and the volume of the
reservoir (V):
p0=[a lambda gamma phase]; f = @(p,t) p(1).*exp(-p(2) .* t).* cos( sqrt(patm*p(3)*A^2/(m*V)-
p(2)^2) .* t + p(4));
7. Perform a non-linear parameter fitting using nlinfit:
p = nlinfit(t,x,f,p0);
The array p will have the new fitted value of .
8. Make a plot of the raw and fitted data (signal vs. time).
9. Compare the experimental value of with the value that was obtained from theoretical
calculations and to the accepted literature value.
Appendix: Derivation of the equations for adiabatic processes
For ideal gases the heat capacity is defined as:
V
V
UC
T
(5.11)
The First Law of Thermodynamics is:
dU dq dw (5.12)
For an adiabatic process there is no heat exchange between the system and its
environment.
0 V
V
dU pdV C dT
pdV C dT
(5.13)
Assuming that the gas obeys the Ideal Gas Law, the following derivation holds:
1
pV nRT
pdV Vdp nRdT
pdV Vdp dTnR
(5.14)
Substituting of (5.13) into (5.14) yields (using the Cp-CV = nR relationship):
( 1)
0
v
v
p v
p v
v
CpdV Vdp pdV
nR
CpdV Vdp pdV
C C
C CpdV Vdp pdV
C
pdV Vdp pdV
Vdp pdV
dV dp
V p
(5.15)
where = Cp/Cv, the ratio of the specific heats at constant pressure and constant
volume, respectively. Integrating (5.15) yields:
ln lnV p k
pV k
(5.16)
Determination of the Heat Capacity Ratio of CO2 with
Adiabatic Compression
Background
Recall, that for adiabatic processes the following relationships hold:
1
1
.
.
.
pV const
p T const
TV const
(6.1)
In this experiment CO2 will be driven through an adiabatic process during which the
V, p, and T data will be collected.
The Adiabatic Gas Law Apparatus (AGLA) setup consists of a Plexi® glass
cylinder equipped with a piston which can be moved with a lever very quickly
allowing quick change of the volume of the trapped gas inside the cylinder (see
Figure 6.1). The piston is operated by a pneumatic actuator utilizing the same gas as
the gas being studied. The cylinder is also equipped with a fast response
temperature probe, pressure sensor and a sliding resistor which monitors the
position of the piston very accurately (which position in turn is converted to volume
with a calibrating equation). The photo and the schematics of the setup are shown in
Figure 6.1and Figure 6.2. The solenoid valves and the sensors are accessed through the
Controller box (Figure 6.3). The Controller box is connected to the MODuck data
acquisition and controller interface operated by MATLAB.
Experimental Procedure
1. Connect the gas source to the setup. Set the regulator on the gas source to 50PSI
and open the butterfly valve on the regulator.
Note: Using higher pressure may damage the equipment!
2. Connect to the MODuck interface to the Controller box as follows
Sensor or Solenoid valve MODuck port
Volume sensor AI0
Temperature sensor AI1
Pressure sensor AI6
Actuator control valve P0.0
Loading valve P0.1
Release valve P0.2
3. Execute the AC_Setup script to configure the MODuck interface for control and
data collection.
4. Execute the AC_Flush script to flush the cylinder with the gas.
5. The script AC_Compression performs the adiabatic compression and captures
the data.
Figure 6.1 Adiabatic Gas Law Apparatus Figure 6.2 The schematics of AGLA
Figure 6.3 Connections to the controller box
.
Functions:
AC_setup Configures the MODuck data acquisition interface for
control and data acquisition, creates the sNI object through
which the control/data acquisition is performed
Input channels (data acquisition):
ai0: volume sensor
ai1: temperature sensor
ai2: pressure sensor
Output channels (control):
line0: Actuator solenoid valve
line1: Loading solenoid valve
line2: Release solenoid valve
AC_Flush(sNI) Flushes the cylinder with the gas [raw_data]= AC_Compression(n,sNI) Performs the adiabatic compression and captures the raw
data into the raw_data array with n rows with [V T p]
columns [T, p, V] = AC_SensorCals(raw_data)
Converts the raw data to physical quantities, volume (m3),
temperature (K), and pressure (Pa)
Results, Calculations
1. Convert the raw_data array to physical quantities with the AC_SensorCals function.
2. Derive the logarithmic version of each of the three equations in (6.1), and perform a
linear fit on each representation to get .
3. Calculate an average value of
4. Perform a nonlinear fit on one of the three equations and obtain another value for
5. Graph the four representations (three linear, one nonlinear) side-by-side.
6. Calculate the values of CV and Cp.
7. Compare your experimental with the literature value and with the value that you
obtained from the theoretical calculations. Also, compare your experimental value
with the one derived from the Rüchardt method.
Note: In the absence of temperature readings, use only the first equation in (6.1) and perform
both linear and nonlinear fits on this equation.
Solving an Ordinary Differential Equation (ODE)
System
Background
Ordinary Differential Equation Systems in Chemical Kinetics
The rate equations of chemical kinetics are first-order ordinary differential
equations. Some examples that you have seen previously are shown below:
A Product(s) [A][A]
dr k
dt 0[A] [A] kt
t e
2A Product(s) 21 [A][A]
2
dr k
dt
0
1 1
[A] [A]kt
In chemical reactions that involve more species and/or complex reactions (as
opposed to elementary reactions as shown above), rate equations become very
complex and in those cases typically there is no analytical solution available.
A typical ordinary differential equation of the ith species involving n species
can be written in the form
1 ,0( ,..... ) ( 0)ii n i
dcf c c c t c
dt
(7.1 )
where ci is the concentration of the ith species at time t, and ci,0 is the concentration
of the ith species at time t=0 (initial conditions). The function fi(ci,…cn) is typically
a combination of terms that are made up from concentration(s) of species and some
constants (e.g. rate constant). When there is the product of concentrations of more
than one species (or the higher power of the concentration of a single species), the
equation is non-linear. Equations or equation systems involving non-linear kinetic
terms have very rich and unique dynamical behavior, discussion of which goes
beyond the scope of this
lab. The few of these
properties (related to
this lab) are discussed
below.
If there is no apparent
change over time to the
concentrations, and
after small perturbation
to any of the
concentration the
system returns instantly
to the same state, the
system is in stable
steady state, and if it
returns to the same state after some “roundtrip” (e.g. some concentrations initially
move away from the steady state value, but ultimately the system returns to the
steady state then the system is excitable. If variation of some parameter results in
the loss of the stability of the steady state, the system becomes oscillatory, where
some or all of the concentrations go through periodic changes over time between
two values. A parameter that can invoke this behavior is called a bifurcation
parameter, and the value of the parameter where that occurs is a bifurcation point
(see Figure 7.1). Further change of the bifurcation parameter can yield the return of
the steady state or further bifurcations, and potentially chaotic behavior. Systems
with non-linear dynamics are particularly sensitive to the variation of the
bifurcation parameter(s), however they show much less dramatic changes to the
variation of other parameters.
When there is no analytical solution available for the differential equation system,
the problem has to be solved numerically. Some of such methods will be reviewed
in class. While these methods take different approaches, they are similar in the
sense that they attempt to evaluate the fi(ci,…cn) functions in discrete time interval
in order to obtain the ci(t) functions, (that is the concentration of the involved
species as a function of time) via numerical integration. Some of these methods
maintain constant time steps (e.g. Euler method), others adjust the time step based
on the nature of the fi(ci,…cn) functions (e.g. “predictor/corrector” methods). The
latter type of methods are particularly useful integrating systems with
concentrations changing several orders of magnitude in a very short time, such as in
case of oscillating chemical reactions.
Scientific Modeling
Scientific models are developed to help us to explain experimental results, to gain
a better understanding of phenomena in Nature or a subset of Nature which is an
experimental configuration or setup. These models are typically some sort of
mathematical constructs. There are two, often conflicting expectations towards
Figure 7.1 Bifurcation diagram. kg: bifurcation
parameter, [CaCy]cy: cytosolic Ca2+-concentration
models: on one hand, they should be able to provide explanations for the
observation in an expected depth, on the other hand they should be simple enough
that they are tractable, and mathematically feasible. This latter expectation has been
hindered by limitations by available computational power. As more computational
power became available, more complex, realistic models prevailed.
The objective of this lab will be to implement and verify a mathematical model,
which is suitable to provide a plausible explanation for Ca2+-oscillations in
Paramecium cells and support some relevant experimental data, yet simple enough
that it can be managed mathematically.
Deterministic vs. Stochastic Modeling
Deterministic models always generate the same outcome. However, a real cell has
many species and many processes, and as you learnt in class, each type of molecule
can have many different microstates yielding a variety of macro states. In addition,
agonists (such as dopamine, DA) bind to the cell wall and the signal is carried
through the cell to its destination, plowing through a very much heterogeneous
environment. Therefore experimental data is subject to these variations. To address
these random variations, there is a number of approaches, one of which is
intuitively leveraging the random variations, called stochastic resonance. While the
experimental data is “noisy”, the underlying periodic signal (oscillation) prevails
after the data is processed with a fast Fourier transform (FFT) algorithm (see Figure
7.2).
A model for Ca2+- oscillations in Paramecium cells
Like a good homeowner fixing up his house constantly, implementing new and
replacing outdated things, evolution kept adding, removing, changing functionality
to cells which resulted in the many different types of cells, each being a fine tuned
machine with many and complex chemical processes. In these complex
environments, Ca2+ in the cytosol of the cell has been identified as one of the
responsible entities to carry signals, through the oscillation of its concentration, to
many cellular processes and between cells.[5] Specifically, the frequency of the
oscillation, as rudimentary as it may sound, has been shown to serve as such a
signal which is decoded by the target processes.
Each cell implements the Ca-signaling somewhat differently, and many more
processes are involved even in one cell than we include in our model, but
remember, the objective is to provide a model that is sophisticated enough to
explain some experimental results, yet simple enough to manage numerically.
In this lab, we’ll test a combined model assembled from two simpler models, each
of which individually captures a subset of the processes. One of the models focuses
on the agonist binding process,[6, 7] “CC-model,” and the other one captures the
calcium-induced Ca2+-release from the ER,[8-10] “BDG-model” (both abbreviated
after the authors). The experimental results that will be used to match against is
shown in Figure 7.2
The schematics of the model is shown in Figure 7.3 and Figure 7.4, and the
respective differential equations are shown in Figure 7.5. Table 7.4 shows the
considered species (dynamical variables) and Table 7.5 shows the explanations for
all the other variables, constants, etc. in the model. Please note, constants and
variables referring to the schematic model are in the italics type (e.g. kg), references
to the Matlab code are in block type (e.g. kg).
The differential equations are expressed in terms of variables for readability, and
also some of the terms individually can be of an interest. For example, V3, is Hill-
functions describing the binding of multiple Ca2+ and IP3 to the IP3-receptor (IP3R),
with each step having a different rate (cooperativity effect).
The objective is to test the response of the model to the variation of the
bifurcation parameter (by constructing the bifurcation diagram), kg, which
represents the rate determining step of the binding process of the agonist to the
receptor, specifically, if the model can reproduce the experimental 0.036Hz
frequency identified in the experimental data, and also to predict at what values
of the bifurcation parameter the oscillation would cease to exist.
The Modeling Environment
The Mathematical/Programing Platform
The modeling will be performed in MATLAB with the Systems Biology Toolbox
(SBTB) installed. The SBTB allows one to create a text file with a specific format
which resembles the mathematical model in a much more human-friendly way than
the programing environment of MATLAB. While the text file is very convenient, it
has to have a specific format/syntax that the SBTB engine can read and process.
Once the model is setup in a text file, the SBTB engine can be invoked manually
from the MATLAB prompt, or it can be invoked from a MATLAB script. The latter
option also allows varying a parameter of the model (which is also present in the
text file) and rerun the model with the new value of the parameter automatically.
This way the behavior of the model can be tested automatically in response to the
variation of a parameter.
The Model File
The structure of the model file is shown in Table 7.1.
********** MODEL NAME
ZQX Model and CC-BDG Model
The name of your model
(not used in the
simulation) ********** MODEL NOTES
Combined model
Notes for yourself
********** MODEL STATES The differential
equations for the rates
d/dt(CaCy) = R1+R2-R3+R4+R5-R6 {isSpecie::concentration} %state
d/dt(CaER) = R3-R4-R5 {isSpecie::concentration} %state
d/dt(IP3) = R7-R8-R9 {isSpecie::concentration} %state
d/dt(GaGTP) = R11-4*R12-R13 {isSpecie::concentration} %state
d/dt(DAG) = R7-R14+R15 {isSpecie::concentration} %state
d/dt(aPLC) = R12-R16 {isSpecie::concentration} %state
for each species in a new
line. Best is using rate
variables (R1, R2,…)
defined later for
visibility . The names in
parenthesis represent the
concentration of the
respective species. CaCy(0) = 250
CaER(0) = 100
IP3(0) = 0.8
GaGTP(0) = 60
DAG(0) = 40
aPLC(0) = 8
Initial conditions for
each species (in mM)
********** MODEL PARAMETERS
kg0 = 0.02 {isParameter} %parameter
V0 = 400 {isParameter} %parameter
k = 70 {isParameter} %parameter
……
Parameters (e.g. rate
constants) of the model.
These are not updated
automatically during the
modeling ********** MODEL VARIABLES
khp = khp0*DAG^2/(KD1^2+DAG^2) {isParameter} %variable
kDAG = kDAG0*DAG^2/(KD1^2+DAG^2) {isParameter} %variable
kPLC = kPLC0*DAG^2/(KD1^2+DAG^2) {isParameter} %variable
…..
Variables are used to
keep the model clean
and easy to read. Here,
Hill-functions are
defined which typically
describe reactions
involving proteins ********** MODEL REACTIONS
R1 = V0 %reaction
R2 = V1 %reaction
R3 = V2 %reaction
……
The definition of the
reaction variables in
terms of parameters
********** MODEL FUNCTIONS
********** MODEL EVENTS
********** MODEL MATLAB FUNCTIONS
function m = modulation(time,kgbase,kgpeak,tbase,tpeak)
per = tbase + tpeak
residue = mod(time,per)
if residue <= tbase
m = kgbase
else
m = kgpeak
end
return
Custom functions. We
will use this option to
generate the time-
dependent stimulation y
the agonist
Table 7.1 Structure of the model file
Running the modeling
There are several ways of running the modeling:
1. A short script can be executed that generates a plot of the [Ca2+]Cy vs. time
and FFT vs. frequency plots.
model = SBmodel('CombCa.txt')
iTime = 1024; %327.68;
t = (0:1.0:iTime);
np=iTime/1.0;
'CombCa.txt': text file with the model
model: binary representation of the model
iTime: the duration of the reaction (in s)
t: vector containing the time points where the integration
has to be performed (points between 0 and iTime with 1.0
s step)
np: number of time points output =
SBsimulate(model,'ode23s',t); output: data structure with results
fftc=abs(fft(output.statevalues(
:,1))).^2;
nn = length(fftc);
dt=1/iTime;
fs=np/iTime;
f = (0:dt:fs);
output.statevalues(:,1): [Ca2+]Cy array with
concentration values as a function of time
fftc: array with the FFT values of the [CaCy]
nn: length of the fftc vector
fs: number of frequency points for FFT
f: array with the frequency points for FFT figure(1);
plot(t,output.statevalues(:,1));
axis([0 iTime 0 1000 ]);
figure(1): figure for [Ca2+]Cy vs. time
plot, axis: making the plot and set the axes
figure(2);
plot(f,fftc);
axis([0 0.2 0 1e12]);
figure(2): figure for FFT vs. frequency
plot, axis: making the plot and set the axes
Table 7.2 Structure of a basic script to perform the modeling
2. Using the graphics interface to the SBTB. Construct the model and invoke
the editor:
model = SBmodel('CombCa.txt');
SBedit(model);
3. Constructing a more involved scripts that varies the desired parameter or
parameters over a range with a given step size (“parameter sweeping”). For a
sample and commented script please refer to the Appendix.
Figure 7.2 Experimental CaCy concentration in a PC12 cell (a ) and the respective FFT spectrum (b)
Step Representation in model
1.
Agonist (such as deltamethrin, DA) binds to
its receptor on the membrane of the cell
(fast), typically in a periodic fashion.
G-protein [Gp()] gets activated (using
the energy from a GTP molecule), which is
also fast.
G-protein splits into it’s Gp() and Gp()
components. This is a slow process, hence it
is the rate determining step of the activation
process, with a rate constant kg (kg), which
varies in time (see on the right) and it is
scaled with kg0.
The binding increases when treatment is
added to kgadj at time t1.
kg is set with a function called modulation.
Gp() is regenerated (khd)
kg =
kg0*modulation(time,t1,kgadj,
kgbase,kgpeak,tbase,tpeak
kg
t
kgbase
kgpeak
tbase tpeak
Agonist
R
Gp(GTP -
GTP
GDPkg Gp(B
Gp(GDP -
Gp(GTP -
Gp(GDP -khp
kg1
2.
The Gp( portion of this protein activates
phospholipase C (PLC) which in turn,
cleaves (kDAG) phosphatidylinositol 4, 5
bisphospate (PIP2) to inositol (1,4,5)
trisphosphate (IP3) and diacylglycerol
(DAG).
DAG is responsible indirectly for the Ca2+-
channels to open on the cell membrane
allowing the Ca2+ to flood into the cell,
which is present at 20,000-fold higher
concentration outside the cell (kdin).
Activated PLC can get deactivated (kPLCr)
3.
1. IP3 binds to the IP3-receptor (IP3R) of the
endoplasmic reticulum (ER) and along with
the cytosolic Ca2+ (CaCy) activates the
receptor.
4.
2. Upon proper activation of IP3R, Ca is
released from the ER (Ca2+-induced Ca2+
release, CICR), V3
5.
3. Smooth endoplasmic reticular Ca2+ ATPase
(SERCA) pumps replete the ER with Ca2+
(V2). Ca2+ also leaks out from the ER (kf).
6.
4. IP3 is removed via Ca2+-dependent (V5) and
Ca2+-independent () pathways.
Table 7.3 Details of the model
kPLCr
kPLC
PLC
PIP 2
IP 3
DAG
kDAG
Ca
~
V1kdin
V4
Gp( Gp( PLC *
IP 3
Ca Cy
Ca ER
kz
kx
ER
Ca ER
V 3
ER
Ca Cy
Ca ER
V2
~
ERkf
IP 3
Ca Cy
V5
Figure 7.3 The G-protein cascading mechanism and CICR-facilitated Ca2+-dynamics
Figure 7.4 Schematics of the model of Ca2+-dynamics in Paramecium cells
Figure 7.5 Mathematical model for Ca2+-oscillation
Agonis t
R
G p(G T P -
G T P
G DPkg
kP L C r
kP L C
P L C
P IP 2
IP 3
DAG
kDAGkhd
C aC y
C aE R
V 3
kz
kx
V 2
~
V 0
~
V 1
kdin
V 5
E R
k
kf
V ld
V 4
IP3R
G p( G p( P L C *
G p(G DP -
G p(G T P -
G p(G DP -khp
kg1
=CC Model
=BDG Model
0 1 2 3
2 3
34 5 3
4
4
[ ][ ]
4 [ ] [ ] [ ]
[ ][ *] [ ]
[ *][
[ ][ ] [ ]
[ ][ ]
]
]
[
][
]
[
g
PLC hg
D
Cy
f ER Cy
ERf
AG hd l
PLC
R
d
E
d G G
d CaV V V V k Ca k Ca
dt
d C
TPk G GDP
dt
k G GTP PLC k G GTP
d
aV V k Ca
dt
d IPV V IP
dt
DAGk PLC k DAG V
dt
d PLCk G GTP PLC k
dt
[ *]hp PLC
1
2 2
2 2 2 2 2
2
2 2 42 2
33 3 2 2 2 2 2 2
4
2
35
4 4
3
5
5
0
2
3
0
[ ]
[ ]
[ ]
[ ] [ ][ ]
[ ] [ ] [
[
[ *]
[ ] [ ]
[ ] [ ]
] [ ] 4
[ ]
]
[ *]
DAG
n pCy
M n n
din
Cy
M
Cy
Cy
p p
ERM
x Cy y ER z
d Cy
V k PLC
Ca IPV V
k Ca k I
V k DAG
CaV V
k Ca
Ca IPCaV V
k Ca k
G GDP G G GTP PLC
PLC
Ca k IP
P
P
2'
2 2
1
[ *]
[ ], , ,
[ ]n n hp DAG PLC
D
PLC
DAGk k n k k k
K DAG
Parameter Model
representation
Maximum value Description
[Ca]Cy CaCy Concentration of Ca2+ in the cytosol
[Ca]ER CaER Concentration of Ca2+ in the ER
[IP3] IP3 Concentration of IP3 in the cytosol
[Ga-GTP] GaGTP Concentration of activated G-protein on the inner side of the
agonist receptor
[DAG] DAG Concentration of DAG in the cytosol
[PLC*] aPLC Concentration of aPLC (activated PLC) in the cytosol
Table 7.4 Species (dynamical variables) for the model of Ca2+-oscillations
Parameter Model
representation
Maximum value Description
kg kg0 Stimulation by agonist
kPLC kPLC kPLC0 Rate of formation of PLC*
khg khg Rate of hydrolysis of GTP to GDP
kDAG kDAG kDAG0 Production of DAG
khd khd Hydrolysis of DAG
Vld Vld Passive leak of DAG
khp Khp Hydrolysis of PLC*
G0 G0 Total concentration of G-proteins
P0 P0 Total concentration of PLC
kdin kdIN Rate of Calcium entry from extracellular stores
k k Passive efflux of Ca from cytosol
V0 V0 Passive leak of Ca
V1 V1 Active influx of ca to cytosol from extracellular stores
V2 V2 VM2 Pumping rate into ER
V3 V3 VM3 Pumping rate out of ER
ky ky Threshold for Ca release from ER
kz kz Threshold for activation by IP3
V4 V4 Agonist dependent production of IP3
V5 V5 VM5 Ca-dependent IP3 degradation
kf kf Passive efflux of Ca from ER to cytosol
ε eps Ca-independent degradation of IP3
kx kx Threshold for activation by Ca
(β) (Production of IP3 –not explicitly included in this model,
replaced by kg)
n,m,p,q n,m,p,q Cooperativity factors (integers)
Table 7.5 Parameters used in the mathematical model for Ca2+-oscillation
Results, Calculations
For your lab report:
1. Perform the modeling with varying kg over a range that captures the two
bifurcation points (or the amplitude of the oscillation falls below 100 nM)
2. Construct the bifurcation diagram.
3. Interpret the meaning of the bifurcation diagram.
4. Identify the value of kg that matches the experimental data (that is generates
oscillating signal with approximately the same frequency.
5. Construct the FFT diagram and identify the value of the oscillation matching the
experimental data.
6. Based on the bifurcation diagram, argue if the experimental data was collected
near either steady state (either bifurcation point, upper or lower) or in the
.middle of the oscillatory range, and predict the effect of small variation to kg
(e.g. some environmental effect).
Appendix A: model file for SBToolbox for modeling Ca2+-oscillations in
Paramecium
********** MODEL NAME
ZQX Model and CC1 Model
********** MODEL NOTES
Combined model
********** MODEL STATES
d/dt(CaCy) = R1+R2-R3+R4+R5-R6 {isSpecie::concentration}[6] %state
d/dt(CaER) = R3-R4-R5 {isSpecie::concentration} %state
d/dt(IP3) = R7-R8-R9 {isSpecie::concentration} %state
d/dt(GaGTP) = R11-4*R12-R13 {isSpecie::concentration} %state
d/dt(DAG) = R7-R14+R15 {isSpecie::concentration} %state
d/dt(aPLC) = R12-R16 {isSpecie::concentration} %state
CaCy(0) = 30
CaER(0) = 100
IP3(0) = 0.80000000000000004
GaGTP(0) = 60
DAG(0) = 40
aPLC(0) = 8
********** MODEL PARAMETERS
kg0 = 0.017000000000000001 {isParameter} %parameter
V0 = 800 {isParameter} %parameter
k = 15000 {isParameter} %parameter
kf = 1 {isParameter} %parameter
eps = 1 {isParameter} %parameter
VM2 = 700 {isParameter} %parameter
VM3 = 150 {isParameter} %parameter
VM5 = 5700 {isParameter} %parameter
k2 = 0.10000000000000001 {isParameter} %parameter
k5 = 1 {isParameter} %parameter
kx = 0.5 {isParameter} %parameter
ky = 0.20000000000000001 {isParameter} %parameter
kz = 0.20000000000000001 {isParameter} %parameter
kd = 0.40000000000000002 {isParameter} %parameter
kl = 1 {isParameter} %parameter
m = 2 {isParameter} %parameter
p = 2 {isParameter} %parameter
n = 4 {isParameter} %parameter
q = 1 {isParameter} %parameter
G0 = 200 {isParameter} %parameter
P0 = 10 {isParameter} %parameter
KD1 = 25 {isParameter} %parameter
kgbase = 0.5 {isparameter} %parameter
kgpeak = 1.1000000000000001 {isparameter} %parameter
kgadj = 12.199999999999999 {isparameter} %parameter
tbase = 24.699999999999999 {isparameter} %parameter
tpeak = 2 {isparameter} %parameter
khd = 100 {isParameter} %parameter
khg = 0 {isParameter} %parameter
Vld = 1000 {isParameter} %parameter
kdIN0 = 10 {isParameter} %parameter
kdIN0adj = 900 {isParameter} %parameter
khp0 = 0.5 {isParameter} %parameter
kDAG0 = 1000 {isParameter} %parameter
kPLC0 = 1.9999999999999999e-007 {isParameter} %parameter
t1 = 100 {isParameter} %parameter
t2 = 300 {isParameter} %parameter
********** MODEL VARIABLES
khp = khp0*DAG^2/(KD1^2+DAG^2) {isParameter} %variable
kDAG = kDAG0*DAG^2/(KD1^2+DAG^2) {isParameter} %variable
kPLC = kPLC0*DAG^2/(KD1^2+DAG^2) {isParameter} %variable
kdIN = mod_dIN2(time,t2,kdIN0,kdIN0adj) {isParameter} %variable
V1 = kdIN*DAG {isParameter} %variable
V2 = VM2*CaCy^2/(k2^2+CaCy^2) {isParameter} %variable
V3 =
VM3*CaCy^m*CaER^2*IP3^4/((kx^m+CaCy^m)*(ky^2+CaER^2)*(kz^4+IP3^4))
{isParameter} %variable
V4 = kDAG*aPLC {isParameter} %variable
V5 = VM5*IP3^p*CaCy^n/((k5^p+IP3^p)*(kd^n+CaCy^n)) {isParameter}
%variable
PLC = P0-aPLC {isParameter} %variable
GaGDP = G0-GaGTP-4*PLC {isParameter} %variable
kg = kg0*modulation(time,t1,kgadj,kgbase,kgpeak,tbase,tpeak)
{isParameter} %variable
********** MODEL REACTIONS
R1 = V0 %reaction
R2 = V1 %reaction
R3 = V2 %reaction
R4 = V3 %reaction
R5 = kf*CaER %reaction
R6 = k*CaCy %reaction
R7 = V4 %reaction
R8 = V5 %reaction
R9 = eps*IP3 %reaction
R11 = kg*GaGDP %reaction
R12 = kPLC*GaGTP^4*PLC %reaction
R13 = khg*GaGTP %reaction
R14 = khd*DAG %reaction
R15 = Vld %reaction
R16 = khp*aPLC %reaction
********** MODEL FUNCTIONS
********** MODEL EVENTS
********** MODEL MATLAB FUNCTIONS
function m = modulation(time, t1, kgadj, kgbase,kgpeak,tbase,tpeak)
per = tbase + tpeak;
residue = mod(time,per);
if residue <= tbase
m = kgbase;
elseif (time > t1)
m = kgadj;
else
m = kgpeak ;
end
return
function v = mod_dIN2(time,t2, kdIN0,kdIN0adj)
if (time >= t2)
v = kdIN0adj * (time - t2) ;
else
v = kdIN0;
end
return
Appendix B: Script file for “Parameter sweep”
model_file = 'CombCa.txt'; %Defining the model file
numpar_1 = 100; %Number of steps of first parameter for sweeping
(Parameter 1)
numpar_2 = 1; %Number of steps of second parameter for sweeping
(Parameter 2)
output_file_root = 'raw/2014_01_'; %Raw Excel data output folder and root name for the
data files
output_image_root = 'images\'; %Image output folder and root name for the data files
par_1 = 0.02; %Starting value of Parameter 1
par_2 = 700.0; %Starting value of Parameter 2
parrun = 1;
par_1_step = 0.001; %Stepsize of Parameter 1
par_2_step = 25.00; %Stepsize of Parameter 1
par_2_initial = par_2; %Saving the initial value of Parameter 2
iTime = 1000; %Length of simulation
dt = 0.1; %Time interval between integrations
t = (0:dt:iTime); %Time vector with time steps dt between 0 and iTime
np=iTime/dt; %Number of time points (real)
for kdx = 1:numpar_1 %Loop to sweep Parameter 1
par_2 = par_2_initial; %Reseting the initial value of Parameter 2
for idx = 1:numpar_2 %Loop to sweep Parameter 2
Z=[]; %Array to save [CaCy] vs. t values
ZEr = []; %Array to save [CaER] vs. t values
ZFFT=[]; %Array to save FFT of [CaCy] vs. frequency values
model = SBmodel(model_file); %Creating the binary version of the model
modelstructure = SBstruct(model); %Extracting a data structure from the model
modelstructure.parameters(1).value = par_1; %Setting the new value of Parameter 1
modelstructure.parameters(2).value = par_2; %Setting the new value of Parameter 2
model1 = SBmodel(modelstructure); %Creating the binary version of the modified model
output = SBsimulate(model1,'ode23s',t); %Performing the integration
fftc=abs(fft(output.statevalues(:,1))).^2; %Calculating the FFT values of [CaCy]
nn = length(fftc); %Number of points in FFT (power of 2 typically)
dt_fft=1/iTime;
fs=np/iTime; %Number of frquency points in the FFT spectrum
f = (0:dt_fft:fs); %Vector holding the frequency points
Z = [Z output.statevalues(:,1)]; %Filling Z helper array with the [CaCy] vs. t values
ZEr = [ZEr output.statevalues(:,2)]; %Filling ZEr helper array with the [CaER] vs. t values
ZFFT = [ZFFT fftc]; %Filling ZFFT helper array with the FFT of [CaCy] vs.
frequency values
par_1str = num2str(par_1); %Creating a string with the value of Parameter 1
par_2str = num2str(par_2); %Creating a string with the value of Parameter 2
output_file = [output_file_root '(' par_1str ')(' par_2str ').csv']; %Creating a name of the output Excel file for [CaCy]
from the root name, and the strings of Parameter 1 and 2
output_file2 = [output_file_root '(' par_1str ')(' par_2str ')FFT.csv']; %Creating a name of the output Excel file for FFT data
from the root name, and the strings of Parameter 1 and 2
output_file3 = [output_file_root '(' par_1str ')(' par_2str ')ER.csv']; %Creating a name of the output Excel file for [CaER]
from the root name, and the strings of Parameter 1 and 2
csvwrite(output_file, Z); %Writing [CaCy] data into an Excel file
csvwrite(output_file2, ZFFT); %Writing FFT data into an Excel file
csvwrite(output_file3, ZEr); %Writing [CaER] data into an Excel file
fig1 = figure(1);
set(fig1,'position', [50 50 950 950]); %Creating the figure with the 3 plots
subplot(2,2,1);
plot(t,output.statevalues(:,1));
axis([0 iTime 0 1000 ]);
title(['CaCy (par_1=' par_1str ',par_2=' par_2str ')']);
subplot(2,2,2);
plot(t,output.statevalues(:,2));
axis([0 iTime 0 1000 ]);
title('CaER');
subplot(2,2,3);
plot(f,fftc);
fft_max = 1.05*max(fftc(100:length(fftc)));
axis([0 0.1 0 fft_max]);
title('CaCy FFT');
disp(['par_1=' num2str(par_1) ] )
saveas(fig1,[output_image_root '(' par_1str ')(' par_2str ')CaCy.jpg']); %Saving the figures in MATLAB and in .jpg format
saveas(fig1,[output_image_root '(' par_1str ')(' par_2str ')CaCy.fig']); %Saving the figures in MATLAB and in .fig format
par_2 = par_2 + par_2_step;
end
par_1 = par_1 + par_1_step;
end
Solving a Partial Differential Equation (PDE) System
Background
Partial Differential Equation Systems in Chemical Kinetics
We saw that a homogenous chemical system with multiple chemical reactions
can be described with an ODE system. However, if the system is not homogenous,
the special variation of the concentrations have to be taken into account. Spatially
varying concentrations represent concentration gradients which can be taken into
account with considering diffusion in the reaction domain.
The general equation for a non-homogenous reaction-diffusion system is
2
1 ,0
2 22
2 2
( ,..... ) ( 0)kk k k n k
k kk
cD c f c c c t c
t
c cc
x y
(8.1 )
where ck is the concentration and Dk is the diffusion coefficient of the kth species,
and ck,0 is the concentration of the kth species at time t=0 (initial conditions), the 2
term has the spatial variables (in the x and y dimension). The function fk(ci,…cn)
contains the reaction rate terms. Mathematically these equations constitute a partial
differential equation system (PDE).
The solution of such systems can be performed numerically, much like we saw with
ODE, however the diffusion term has to be approximated as well. A further
complication is that while homogeneous systems could be characterized with one
set of concentration values, in systems with diffusion each point of the system has a
different set of concentrations therefore the PDE has to be solved for each point of
the system at every time step. One can imagine how much more tedious such a job
is in comparison with the solution of an ODE! Therefore we will consider only a
small part of the system with the assumption that it is representative for the entire
system.
Numerical Method for Integration
While in the previous lab the ODE system was solved under the hood, now
performing the numerical integration is part of the adventure. The Euler-method
will be employed, which is perhaps one of the least computationally tasking
approach. The method can be summarized with eqns. (8.2).
1
, , 1
, , 1
( ,..... )
( ,..... )
( ,..... )
k kk n
k k t t k t k n
k t t k t k n
c cf c c
t t
c c c f c c t
c c f c c t
(8.2 )
where t is the time step, ck,t is the concentration of the kth species at time t, ck,t+t
is the concentration of the kth species at time t+t. The time step is chosen
arbitrarily; however, care has to be taken. The smaller it is chosen, the more
accurate evaluation of the f(c1,….cn) function is possible, however from the
perspective of the calculation, the larger it is the faster the calculation goes.
Approximating the Diffusion Terms
For the purpose of this lab, we will consider two spatial dimensions, x and y. We
will divide the reaction domain into small “cells”, and assume that each cell
communicates with its neighbor cell via diffusion (Figure 8.1). If the cells are small
enough, the diffusion term can be approximated with the concentration differences
2
2 2
1, , 1, , , 1 , 1, ,
2
1, 1, , 1 1, ,
2
4
k kk k k
i j i j i j i j i j i j i j i j
k
i j i j i j i j i j
k
c cD c D
x y
c c c c c c c cD
h
c c c c cD
h
(8.3 )
where h = x = y is the size of a cell (assuming square cell it is the same in
both directions), and Dk is the diffusion coefficient of the kth species.
Figure 8.1 Reaction-diffusion system domain
Boundary conditions
Since we included only small part of the actual system, we have to assume that
the cells at the boundary will communicate with cells not included in the treatment.
The appropriate boundary condition to reflect this assumption can be approximated
with setting the concentrations in the first and last rows and columns to the values
of the concentrations in the second to last rows and column. For example the
concentrations in the first column are set to the concentrations in the third column,
concentrations in the last row (nth row) are set to the concentrations in the (n-2)th
row, etc. (see Figure 8.1 grey area). Therefore the cells in the first and last columns
and rows communicate with cells outside of the domain the same way as their do
with their immediate neighbor cells.
The actual system – Modeling Propagation of Cardiac Potentials with the
FitzHugh-Nagumo Model
We saw, that the signal carried by Ca2+-ions inside a cell can be modeled with an
ODE system. The same approach can be used to model how the signal is delivered
to the cellular mechanism which is responsible for releasing for example dopamine
which ultimately carries the signal to the next cell. To model a network of these
cells would require the solution of the ODE system for each cell at every time step,
and then solving the respective PDE system accounting for the signal propagation
(i.e. diffusion of the agonist) among cells. One can imagine the computational task
x
y
i=1 i=n
j=n
j=1
i, j i+1, ji-1, j
i, j+1
i, j-1
h
that such an enterprise would entail. Alternatively, we could conceivably bypass
what is happening in the cells with the notion that it is triggered by a potential
change on the membrane and it ultimately results in a potential change on the
membrane, and it is sufficient to follow the potential change across the domain of
cells.
To employ this approach we will use a relatively simple model called the
Fitzhugh-Nagumo model, which models the potential propagation across a cell
domain. This simplified model has two variables: the action potential in the domain
(v) and the so-called recovery variable (r).
21v
v v a v r d vt
re bv r
t
(8.4 )
where a,b, and e are constants, and d is the “diffusion coefficient”, which in this
case the conductance (i.e. coupling strength between cells). The model in this form
is dimensionless. Notice that the recovery variable doesn’t propagate between cells,
as it is an inherent property of the domain.
Objective of the Lab
The objective of the lab is to show how the action potential propagates through a
medium using Fitzhugh-Nagumo model, when the medium is not homogenous.
It has been argued that minor inhomogeneity in the medium, such as a damaged
segment in the heart tissue can result in breaking the propagating action potential
and ultimately yield a self-sustaining potential wave which in turn can lead to
cardiac arrest. If one considers the propagating action potential as a wave of
potential change that sweeps through the medium as it carries the signal, a damaged
portion of the tissue can break the wave, leading to a wave with a free end which
starts to rotate locally and will never leave the tissue, constantly triggering an action
potential which is a known precursor to cardiac arrest (see Figure 8.3 bottom row).
Another source of free end in the action potential is early action potential
propagating in the wake of the previous one. The cell needs some time after passing
an action potential to recover (this is incorporated into the model as r), however an
early excitation can initiate an action potential. The generation of a normal action
potential (AP) is shown in Figure 8.2. If a stimulation greater than the threshold
arrives between A and D, a new action potential is generated before the resting
potential is restored. Spatially, this kind of excitation would occur right behind the
previous AP (between B and D on the time scale), where the medium is not
homogeneous, therefore the new action potential will not be uniform, therefore free
ends may form (see Figure 8.3 top row).
Figure 8.2 Action potential propagation in time in
cardiac tissue
AP = action potential; RP = refractory period; T = AP threshold
(~-60 mV); A = resting potential, stimulus arrives, Na+ channels
open; B = Na+ channels close, K+ channels open; C = Ca2+
channels open; D = resting potential restored, Ca2+ and K+
channels close
The model therefore should:
Show that undisturbed action potential propagates through the medium
without interruption.
Demonstrate that action potential which is interrupted in a manner that a
free end of the action potential wave is formed will sustain itself
indefinitely.
Create a scenario where a new action potential is formed in the wake of
the previous action potential resulting free-ended in rotating potential
wave.
Figure 8.3 Time evolution of action potential with point-like excitation (top row) and with perturbation that
resets the action potential in the bottom half of the domain (bottom row)
Computational Setup
1. Basic parameters
The snippet below shows basic parameters. The size of the domain is 128x128
(nrows, ncols). The time step is dt=0.05, and dur = 25000 step will provide
sufficient information. The size of a cell h = 2.0 (h). The value of the constants in
the model can be changed (a, b, e, d); use these as starting values. Arrays for
the potential (v), recovery (r), diffusion terms (d), perturbation/excitation (iex)
have to be initialized.
ncols=128; % Number of columns in domain
nrows=128; % Number of rows in domain
dur=25000; % Number of time steps
h=2.0; % Grid size
h2=h^2;
dt=0.05; % Time step
Iex=35; % Amplitude of external current
a=0.1; b=0.8; e=0.006; d=1.0; % FHN model parameters
n=0; % Counter for time loop
k=0; % Counter for movie frames
done=0; % Flag for while loop
v=zeros(nrows,ncols); % Initialize voltage array
r=v; % Initialize refractoriness array
d=v; % Initialize diffusion array
iex=zeros(nrows,ncols); % Set initial stim current and pattern
2. Setting the perturbation
There are two perturbations: one initially generates the action potential, which is
applied on for the first n1e = 50 time steps, and the second sometime during the
course of reaction (between n2b and n2e steps) to generate a second perturbation
which will yield the wave with a free end. If the excitation is point-like
(StimProtocol==1), the second perturbation should be off-center so that the
closer part of the excitation to the previous excitation will die where leaving a
crescent-shaped excitation with two free ends. If the first excitation is wave-like
(StimProtocol==2), that is it is applied to the first 5 columns, the second step
should be resetting v in some of the cells (e.g. the lower 30 rows) to zero until the
excitation dies, leaving the excitation wave in the top 30 rows unchanged, but with
a free end: if StimProtocol==1
iex(62:67,62:67)=Iex; % If point-perturbation is chosen
else
iex(:,1:5)=Iex; % If strip-perturbation is chosen
end
n1e=50; % Step at which to end 1st stimulus
switch StimProtocol
case 1 % Two-point stimulation
n2b=3800; % Step at which to begin 2nd stimulus
n2e=3900; % Step at which to end 2nd stimulus
case 2 % Cross-field stimulation
n2b=5400; % Step at which to begin 2nd stimulus
n2e=5580; % Step at which to end 2nd stimulus
end
3. Setting up the display
The snippet below will setup a 128x128 pixel size display with the potential
showing as a function of time in the spatial domain (x and y directions):
ih=imagesc(v); set(ih,'cdatamapping','direct')
colormap(hot); axis image off; th=title('');
set(gcf,'position',[500 100 512 512],'color',[1 1 1],'menubar','none')
% Create 'Quit' pushbutton in figure window
uicontrol('units','normal','position',[.45 .02 .13 .07], ...
'callback','set(gcf,''userdata'',1)',...
'fontsize',10,'string','Quit');
4. Setting up the time loop for integration
The time loop will include the actual calculation at each time step.
a. Checking if excitation has to be turned on or off:
if n == n1e % End 1st stimulus
iex=zeros(nrows,ncols); % Resetting excitation to zero
end
if n == n2b % Step to begin 2nd stimulus
switch StimProtocol
case 1
iex(62:67,49:54)=Iex; % Excitation in off-center
case 2
v(60:end,:)=0; % Resetting bottom half of the domain to v=0
end
end
if n == n2e % End 2nd stimulus
iex=zeros(nrows,ncols);
end
b. Setting up the boundary conditions Setting the potential along the perimeter. For example in first column (v(1,:)) to the values in
column 3 (v(1,:)). v(1,:)=v(3,:);
v(end,:)=v(end-2,:);
v(:,1)=v(:,3);
v(:,end)=v(:,end-2);
c. Calculating the diffusion terms
Iterating through non-boundary cells to calculate the potential gradient. For each
cell the 4 adjacent cells are considered.
for i=2:nrows-1
for j=2:ncols-1
d(i,j)=(v(i+1,j)+v(i-1,j)+v(i,j+1)+v(i,j-1)-4*v(i,j))/h2;
end
end
d. Calculating the changes to the potential and recovery values
This can be done with matrix operations (instead of iterating through each cell)
which makes the calculations much faster. The excitation is added (iex) which is a
set value if excitation is on and zero otherwise. Also, the diffusion terms are added
(stored in d): dvdt=v.*(v-a).*(1-v)-r+iex+d;
v_new=v + dvdt*dt; % v can NOT be updated before r is calculated, so the
% calculated v is stored in v_new temporarily
drdt=e*(b*v-r);
e. Updating the potential and recovery values
r=r + drdt*dt;
v=v_new; clear v_new
f. Updating the display and saving movie frames
Use the snippet as is below to update the display and save every 500th display for
the movie file
m=1+round(63*v); m=max(m,1); m=min(m,64);
% Update image and text
set(ih,'cdata',m);
set(th,'string',sprintf('%d %0.2f %0.2f',n,v(1,1),r(1,1)))
drawnow
% Write every 500th frame to movie
if rem(n,500)==0
k=k+1;
mov(k)=getframe;
end
n=n+1;
done=(n > dur);
if max(v(:)) < 1.0e-4, done=1; end %If activation extinguishes, quit early.
if ~isempty(get(gcf,'userdata')), done=1; end % Quit if user clicks on
'Quit' button.
end
5. Saving the movie
Prompts you for a folder and a name for the movie. Use it as is. sep='\';
[fn,pn]=uiputfile([pwd sep 'SpiralWaves.avi'],'Save movie as:');
if ischar(fn)
% movie2avi(mov,[pn fn],'quality',75)
movie2avi(mov, [pn fn], 'compression', 'none');
else
disp('User pressed cancel')
end
close(gcf)
Results, Conclusions
1. Assemble the code to perform the modeling
2. Model both types of perturbation to demonstrate the nature of the free end
of the potential wave.
3. Generate images from the movie to show your points.
4. Write up a short, “letter-type” lab report with the objective, method used,
your findings, and conclusions.
5. Include the code that you used in your submission (in a separate .m file)
Appendix
MATLAB Codes and Sample Data for Instructors
EXPERIMENT 1: THEORETICAL INVESTIGATION OF
THERMODYNAMICAL PROPERTIES OF CO2
MATLAB Codes
RunCO2.m:
clear;
clc;
roundsf = @(n,x) round(x./(10.^(floor(log10(abs(x)) - n + 1)))).*10.^(floor(log10(abs(x)) - n +
1)); %function to round x to n sig figs
T_exp = 294; %Temperature of the experiment
epsilon = 190; %epsilon/kB (J) - From Gaussian
sigma = 4.5e-10; %mv = [667.3 667.3 1340 2349.3]; - From Gaussian
v = [667.3 667.3 1340 2349.3]; %cm^1 - From Gaussian
constants; %Set common constatnts
T = (100:1:1000); %Temperature vector
r = 1e-11:1e-11:1e-8; %Distance vector for the LJ potential
Cv = CVCO2(T,v); %Calculating the Cv values at each temperature and for each contribution
Cvt = sum(Cv,2); %Calculating the total Cv values at each temperature
BT = BTp( epsilon,sigma,r,T); %Calculating the second virial coefficient values at each
temperature
muT = JT( Cvt+8.314,BT,T ); %Calculating the JT-coefficient values at each temperature
U = Uljv(epsilon,sigma,r); %Calculating the LJ-potential values as a function of intermolecular
separation
index_T = find(T == T_exp); %Finding the index of temperature matching the experimental
temperature
Cv_exp = Cvt(index_T); %Find;ng the expected Cv value
BT_exp = BT(index_T);%Finding the expected B(T) value
muT_exp = muT(index_T); %Finding the expected mu(T) value
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
%plot Cv:
subplot(2,2,1)
marked_plot(T,Cvt,T_exp,Cv_exp,0,0,'Temp (K)', 'C_v (J/molK)','Heat capacity @constant volume vs.
T of CO_2',['C_v=' num2str(roundsf(4,Cv_exp)) 'J/molK'])
%plot B
subplot(2,2,2)
marked_plot(T,BT,T_exp,BT_exp,0,1,'Temp (K)', 'B (cm^3/mol)','Second virial coefficient vs. T of
CO_2', ['B =' num2str(roundsf(4,BT_exp)) 'cm^3/mol'])
%plot mu
subplot(2,2,3)
marked_plot(T(2:end),muT,T_exp,muT_exp,0,0,'Temp (K)', '\mu_{JT} (K/bar)','Joule-Thomson coeff.
vs. T of CO_2',['\mu_{JT}=' num2str(roundsf(4,muT_exp)) 'K/bar'])
%plot U
subplot(2,2,4)
marked_plot(r,U,r(find(U==min(U))),min(U),1,1,'r (A)', 'U(r) (J)','Lennard-Jones
potential',['\epsilon = ' num2str(epsilon) 'J'])
axis([4e-10 1e-9 -5e-21 2e-20]); %focus on well of potential energy curve
gtext(['\epsilon = ' num2str(epsilon) 'J']); %label epsilon
CVCO2.m
function [Cv] = CVCO2(T,v)
%CVCO2a calculates the heat capacity of CO2 at different temperatures
%T = vector with temperature values
%v = vector with vibrational frequencies (in cm^-1)
%Cv = returned matrix with translational, rotational, and vibrational
%contribution in each column evaluated at the temperatures in T in each row
constants
theta = h .* c .* v ./ kB .* 100; %vibrational temperatures, including conversion from cm-1 to m-
1
Cvt = 3/2 * R; %translational component
Cvr = R; %rotational component
Cv = [T' T' T']; %initialize matrix with as many rows as temperatures in T
for i=1: length(T)
thetaT = theta ./T(i);
ex = exp(thetaT);
Cvv = R * thetaT .^2 .* ex ./(ex - 1) .^2; %vector with Cv contribution from each mode
Cv(i,:) = [Cvt Cvr sum(Cvv)]; %write row of matrix with trans, rot, and vib components at
this temperature
end
end
Uljv.m
function U = Uljv(epsilon,sigma,r)
%Calculates the potential using the Lennard-Jones function as a function of distance
%U = Ulj( epsilon, sigma, r)
%epsilon = depth of potential well (in J)
%sigma = r(U=0) (in m)
%r = distance vector (in m)
constants;
sigma_r = sigma ./ r;
U = 4* epsilon*kB*(sigma_r .^12 - sigma_r .^6);
end
BTp.m
function BTi = BTp( epsilon,sigma,r,T)
%Calculates the second virial coefficient as a function of temperature
%BTi = BTp( epsilon,sigma,r,T,f )
%epsilon = depth of potential well (in J)
%sigma = r(U=0) (in m)
%r = distance vector
%T = temperature vector
constants;
dr = 1e-11; %step size for calculating potential energy curve
BTi = T; %initialize vector for values of the virial coefficient at each T
U = Uljv(epsilon,sigma,r); %calculate Lennard-Jones potential
%Two methods of calculating B: nested for loops or matrix form
%method 1: nested for loops
% N = length(r); %index for iterating over r
% B = r; %initialize vector for each value of r
% for j=1:length(T) %iterate over temperature
% for i=1:N %at each temperature, iterate over r
% B(i) = (exp(-U(i)/kB/T(j))-1) *r(i) *r(i) * dr;%calculate integrand
% end
% BTi(j) = -2*pi*NA * sum(B)*1e6; %integrate, multiply by constants, convert to cm^3
% end
%method 2: matrix form
for j=1:length(T)%iterate over temperature
BTi(j)=-2*pi*NA *1e6*sum((exp(-U/kB/T(j))-1) .*r.^2) * dr; %integrate and convert to cm^3
end
end
JT.m
function mu = JT( Cp,B,T )
%Calculates the Joule-Thomson coefficient as a function of temperature from
%Cp(T) and B(T)
dB = diff(B,[],2);
dT = diff(T,[],2);
dBdT = dB ./ dT;
mu = (dBdT .* T(2:end) - B(2:end)) ./ Cp(2:end)' *.1; %calculation of mu and conversion to hPa
end
marked_plot.m
function marked_plot(x,y,x0,y0,xmin,ymin,x_label,y_label,graph_title,point_text)
%x = independent variable (vector)
%y = dependent variable (vecttor)
%x0 = x coordinate for the point (number)
%y0 = y coordinate for the point (number)
%xmin = x axis minimum flag:
% 0 = x axis starts at 0
% 1 = x axis starts at min(x)
%ymin = y axis minimum flag:
% 0 = y axis starts at 0
% 1 = y axis starts at min(y)
%x_label = x axis label (string)
%y_label = y axis label (string)
%graph_title = title of the graph (string)
%point_text = text for the showed point, eg. the y value (string)
%Set limits of dashed line, either at 0 (xmin, ymin = 0) or minimum x/y:
xmin = xmin*min(x);
xmax = max(x);
ymin = ymin*min(y);
ymax = max(y);
%Establish horizontal and vertical dotted lines from xmin, and ymin to the
%point (x0,y0):
x_hor = linspace(xmin,x0,10);
y_hor = linspace(y0,y0,10);
x_vert = linspace(x0,x0,10);
y_vert = linspace(ymin,y0,10);
plot(x,y); %plot the data
hold on
plot(x0,y0,'ro'); %highlight the point
plot(x_hor,y_hor,'r--'); %plot horizontal line
plot(x_vert,y_vert,'r--'); %plot vertical line
%label axes, title
xlabel(x_label);
ylabel(y_label);
title(graph_title);
gtext(point_text); %prompt user to place text
hold off
end
constants.m
kB = 1.3806e-23; %J/K
h = 6.626e-34; %JK
NA = 6.022e23; % mol^-1
R = NA*kB;
c = 3e8; %m/s
Sample data
Data file: N/A
Image/Figure file(s): RealGases.fig (T = 294 K)
Results
Experimental
Literature
Error N/A
EXPERIMENTS 2: IR AND RAMAN STUDIES OF CO2
MATLAB Codes
Collect_CO2.m:
disp('Press any key to capture background!(Make sure that the laser is on!'); pause; [w bg] = OO_Grab('wnCO2',320,40,1); disp('Press any key to capture a spectrum!(Make sure that the laser is on!'); pause; [w sp] = OO_Grab('wnCO2',320,40,1); s = sp - bg; p = polyfit(w,s,11); s_c = polyval(p,w); s_final = smooth(s - s_c,3); plot(w,s_final); title('Raman Spectrum of CO_2'); ylabel('Intensity (counts)'); xlabel('\nu (cm^{-1})'); grid on [peaks] = peakfinder(s_final(50:100)) + 49; peak1 = w(peaks(1)); peak2 = w(peaks(2)); coupling = peak2 - peak1; disp('==================================='); disp(['Peak 1: ' num2str(peak1) ' cm^{-1}']) disp(['Peak 2: ' num2str(peak2) ' cm^{-1}']) disp(['Coupling constant: ' num2str(coupling) ' cm^{-1}']) disp('===================================');
OO_Grab.m:
function [w s] = OO_Grab(xflag, iTime, iPass, iBox) %xflag: flag for type of return scale: % 'wl' = wavelength % 'wlCO2' = wavelength range relevant to CO2 % 'wn' = wavenumber % 'wnCO2' = wavenumber range relevant to CO2 %iTime = integration time (in s) %iPass = number of spectra to be averaged %iBox = number of points to be averaged in one spectrum
spectrometerObj = icdevice('OceanOptics_OmniDriver.mdd'); connect(spectrometerObj); disp(spectrometerObj); %integrationTime = 20000000; iTime = iTime*1e6; spectrometerIndex = 0; channelIndex = 0; enable = 1; numOfSpectrometers = invoke(spectrometerObj, 'getNumberOfSpectrometersFound'); display(['Found ' num2str(numOfSpectrometers) ' Ocean Optics spectrometer(s).']); spectrometerName = invoke(spectrometerObj, 'getName', spectrometerIndex); spectrometerSerialNumber = invoke(spectrometerObj, 'getSerialNumber', spectrometerIndex); display(['Model Name : ' spectrometerName]) display(['Model S/N : ' spectrometerSerialNumber]); invoke(spectrometerObj, 'setIntegrationTime', spectrometerIndex, channelIndex, iTime); invoke(spectrometerObj, 'setScansToAverage', spectrometerIndex, channelIndex, iPass ); invoke(spectrometerObj, 'setCorrectForDetectorNonlinearity', spectrometerIndex, channelIndex,
enable); invoke(spectrometerObj, 'setBoxcarWidth', spectrometerIndex, channelIndex, iBox); % invoke(spectrometerObj, 'setCorrectForElectricalDark', spectrometerIndex, channelIndex,
enable); wavelengths = invoke(spectrometerObj, 'getWavelengths', spectrometerIndex, channelIndex); disp('=============================================');
if strcmp(xflag, 'wl') w = wavelengths; disp(['Wavelength range: ' num2str(min(w)) ' - ' num2str(max(w)) 'nm']); elseif strcmp(xflag, 'wlCO2') w = wavelengths(934:1136); disp(['Wavelength range: ' num2str(min(w)) ' - ' num2str(max(w)) 'nm']); elseif strcmp(xflag, 'wn') w = (1 ./(532*1e-9)-1 ./(wavelengths * 1e-9))/100; disp(['Wavenumber range: ' num2str(min(w)) ' - ' num2str(max(w)) 'cm^-1']); elseif strcmp(xflag, 'wnCO2') w = wavelengths(934:1136); w = (1 ./(532*1e-9)-1 ./(w * 1e-9))/100; disp(['Wavenumber range: ' num2str(min(w)) ' - ' num2str(max(w)) 'cm^-1']); end disp('Capturing requested type of spectrum.....'); spectralData = invoke(spectrometerObj, 'getSpectrum', spectrometerIndex); if strcmp(xflag,'wlCO2') || strcmp(xflag,'wnCO2') s = spectralData(934:1136); else s = spectralData; end disp('Requested type of spectrum captured'); disp('============================================='); %%
end
Sample data
Raman spectrum
Data file: CO2-3-11-2015.mat
Image/Figure file(s): CO2-3-11-2015.fig
IR spectrum
Data file: Gas Cell CHEM330.129.mat
Image/Figure file(s): CO2_IR.jpg
EXPERIMENT 3: DETERMINATION OF THE SECOND VIRIAL
COEFFICIENT OF CO2
MATLAB Codes
VC_control_script.m:
%Control script to setup and run VC experiment clear;
clc;
VC_setup;
psample = 90000; %Sample pressure withdrawn at a time (Pa)
tol = 1200; %Tolerance for accepting equilibration (Pa)
VC_LoadGas(450,'PSI',sNI);
psave = VC_RunExperiment(psample, tol, sNI)
VC_RunExperiment.m:
function psave = VC_RunExperiment(psample, tol, sNI)
%Returns psave in the form of = [temp pe ps] (m+1 rows)
%psample = Sample pressure withdrawn at a time (Pa)
%tol = Tolerance for accepting equilibration (Pa)
%sNI = data acquisition object
%Runs the apparatus as outlined in the lab manual:
% a) Read initial pressure and temperature values
% b) Save initial readings
% c) Set a flag (expflag) to control the experiment (1 if there is enough
% gas to load the Expansion tank to psample, 0 otherwise to stop)
% d) Load Expansion tank with psample (Pa) gas
% e) Check if the pressure is enough (loaded = 1)
% f) If loaded = 0, take a final reading (this is the mth data point).
% a. After the mth reading, evacuate expansion tank one last time,
% open the valve between the tanks, and take one more reading to
% determine the volume ratio.
% g) If there is enough gas, allow the system to equilibrate.
% h) Read pressure and temperature values in the Sample and in the
% Expansion tanks.
% i) Pump down Expansion tank.
% j) Repeat steps d-i until there is not enough gas (set expflag = 0).
% k) Evacuate the system.
equil=10; %time in seconds to equilibrate pressure
%a),b) read initial pressure and temperature values
psave=VC_ReadCal(10,100,sNI);
%c)Set a flag (expflag) to control the experiment (1 if there is enough gas
%to load the Expansion tank to psample, 0 otherwise to stop)
expflag=1; %1 = continue
lastflag=0; %1=pressures are the same; take one more point
while (expflag==1)
%d) load expansion tank
disp('Loading expansion tank')
loaded=VC_LoadSample(psample, 'Pa',sNI);
%e)check if tank is loaded
if (loaded==0)
%f) if loaded=0 take a final reading and stop
if (lastflag==0) %this is the first time the pressure is insufficient
lastflag=1; %take just one more data point (to determine volume ratio)
disp('Pressures are equal.')
else %we have taken all the experimental points; end the experiment
expflag=0;
disp('Final measurement for volume ratio.')
end
end
%g) check for equilibration
equilflag=0;
while (equilflag==0) %Until the system equilibrates...
pe0=VC_ReadCal(10,100,sNI); %initial pressure
pe0=pe0(1); %initial pressure in expansion tank
pause(equil) %wait for the equilibration time
pe1=VC_ReadCal(10,100,sNI); %check pressure again
pe1=pe1(1); %pressure in expansion tank
if (abs(pe0-pe1)<= tol) %if the change is less than the tolerance:
equilflag=1;
disp('equilibrated')
end
end
%h)record a data point
datapoint=VC_ReadCal(10,100,sNI)
psave = [psave; datapoint];
%i)pump out the expansion tank
disp('Pumping expansion tank')
VC_PumpDown('Sample', tol, equil,sNI)
end %j)this while loop repeats d)-i) until the experiment is over
%k) experiment is over, last data point has been taken
VC_PumpDown('All', tol, equil,sNI)
VC_setup.m:
devs = daq.getDevices;
dev = devs.ID;
sNI = daq.createSession('ni');
sNI.addAnalogInputChannel(dev,'ai0','Voltage');
aiNI1=sNI.Channels(1);
aiNI1.TerminalConfig='SingleEnded';
aiNI1.Range=[-5 5];
sNI.addAnalogInputChannel(dev,'ai1','Voltage');
aiNI2=sNI.Channels(2);
aiNI2.TerminalConfig='SingleEnded';
aiNI2.Range=[-5 5];
sNI.addAnalogInputChannel(dev,'ai2','Voltage');
aiNI3=sNI.Channels(3);
aiNI3.TerminalConfig='SingleEnded';
aiNI3.Range=[-5 5];
sNI.addDigitalChannel(dev,'port0/line0','OutputOnly');
sNI.addDigitalChannel(dev,'port0/line1','OutputOnly');
sNI.addDigitalChannel(dev,'port0/line2','OutputOnly');
sNI.addDigitalChannel(dev,'port0/line3','OutputOnly');
sNI.outputSingleScan([1 1 1 1])
sNI.inputSingleScan;
VC_LoadGas.m
function [loaded] = VC_LoadGas(p,punit,sNI)
%LoadGas(p,punit,sNI)
%Loads the sample holder with gas to pressure p (given in unit 'punit')
%p = target pressure for the sample tank
%punit = unit in which the pressure is given ('Pa','PSI','bar')
%sNI = data acquisition object
sNI.outputSingleScan([1 1 1 1])
mean_data = VC_ReadData(10,100,sNI);
pload = VC_PresCal500(mean_data(2),punit);
loaded = 0;
p_before = 0;
while loaded == 0;
sNI.outputSingleScan([0 1 1 1]);
pause(.2);
sNI.outputSingleScan([1 1 1 1]);
pause(5);
mean_data = VC_ReadData(10,100,sNI);
pload = VC_PresCal500(mean_data(2),punit);
disp(['The pressure currently is: ' num2str(pload) ]);
if pload >= p
pause(10);
mean_data = VC_ReadData(10,100,sNI);
pload = VC_PresCal500(mean_data(2),punit);
if pload >= p
loaded = 1;
end
if pload - p_before < 5
disp('There is not enough pressure in source cylinder');
return
end
end
p_before = pload;
end
disp(['Loaded to ' num2str(pload) punit]);
sNI.outputSingleScan([1 1 1 1]);
disp('DONE! :)');
end
VC_LoadSample.m
function [loadflag] = VC_LoadSample(p,punit,sNI)
%LoadSample(p,punit,sNI)
%Loads the expansion tank with gas to pressure p (given in unit 'punit')
%p = target pressure for the expansion tank
%punit = unit in which the pressure is given
%sNI = dataacquisition object sNI.outputSingleScan([1 1 1 1])
%loadflag = indicates if there was enough gas to load to pressure p
% 1 = there was enough gas
% 0 = there was not enough gas
mean_data = VC_ReadData(10,100,sNI);
pload = VC_PresCal30(mean_data(3),'Pa');
if pload >= 200
disp('Pumping down needed....')
VC_PumpDown('Sample',1300,10,sNI);
end
mean_data = VC_ReadData(10,100,sNI);
loaded = 0;
pprev = 1e5;
while loaded == 0;
pst = 40/VC_PresCal500(VC_ReadData(1,100,sNI),'PSI');
sNI.outputSingleScan([1 0 1 1]);
pause(pst);
sNI.outputSingleScan([1 1 1 1]);
pause(5);
ai_data = VC_ReadData(10,100,sNI);
pload = VC_PresCal30(ai_data(3),punit);
disp(['Pressure in Expansion tank is ' num2str(pload) ' Pa']);
if pload > p
pause(10);
loaded = 1;
ai_data = VC_ReadData(10,100,sNI);
pload = VC_PresCal30(ai_data(3),punit);
loadflag = 1;
elseif abs(pprev-pload)< 100
loadflag = 0;
loaded =1;
return;
end
pprev = pload;
end
disp(['Loaded to ' num2str(pload) punit]);
sNI.outputSingleScan([1 1 1 1]);
disp('DONE! :)');
end
VC_PumpDown.m
function VC_PumpDown(flag,tol,equil,sNI)
%PumpDown(flag,tol,equil,sNI)
%flag = 'Sample' only Expansion tank is pumped down
% 'All' entire system is pumped down (including the Sample holder)
%tol = tolerance in Pa (fluctuation)
%equil = time to wait after pumping down (tol met) in s
sNI.outputSingleScan([0 0 0 0])
switch flag
case 'Sample'
sNI.outputSingleScan([1 1 0 1]);
case 'All'
sNI.outputSingleScan([1 0 0 1]);
end
mean_data = VC_ReadData(10,100,sNI);
p1 = VC_PresCal30(mean_data(3),'Pa');
disp(['Pumping down ' flag '....']);
while p1 > tol
mean_data = VC_ReadData(10,100,sNI);
p1 = VC_PresCal30(mean_data(3),'Pa');
end
disp(['Waititng ' num2str(equil) 's....']);
pause(equil);
sNI.outputSingleScan([1 1 1 1]);
disp('DONE! :)');
end
VC_ReadCal.m
function caldata = VC_ReadCal(channel, n, sNI)
%reads data and uses the three calibration functions to return calibrated
%pressures and temperature in the form [pe (Pa), ps (Pa), T (K)]
%ch = channel number to read data from
% ch > 10 reads allchannel
%n_sample = number of samples to be read and averaged
%sNI = name of datacollection object
ai_data = VC_ReadData(channel,n,sNI);
pe = VC_PresCal30(ai_data(3),'Pa');
ps = VC_PresCal500(ai_data(2),'Pa');
temp = VC_TempTherm(ai_data(1),'K');
caldata = [pe ps temp];
VC_ReadData.m
function [raw_data] = VC_ReadData(ch, n_sample, sNI)
%ch = channel number to read data from
% ch > 10 reads allchannel
%n_sample = number of samples to be read and averaged
%sNI = name of datacollection object
for j=1:n_sample
rawdataNI(j,:) = sNI.inputSingleScan;
end;
mean_data = mean(rawdataNI,1);
if ch >= 10
raw_data = mean_data;
else
raw_data = mean_data(ch+1);
end
end
VC_PresCal30.m
function p = VC_PresCal30(pvolt, punit)
%pvolt = voltage signal from 30PSI pressure transducer
%punit = unit of returned pressure value (string)
% 'Pa' = Pascals
% 'PSI' = PSI
% 'bar' = Bar
a = 1.3146e-5;
b = 0.472; %0.50476;%0.472
pressure = @(V) (V-b)/a;
switch punit
case 'Pa'
p = pressure(pvolt);
case 'PSI'
p = pressure(pvolt)/6894.76;
case 'bar'
p = pressure(pvolt)/1e5;
end
end
VC_PresCal500.m function p = VC_PresCal500(pvolt, punit)
%pvolt = voltage signal from 30PSI pressure transducer
%punit = unit of returned pressure value (string)
% 'Pa' = Pascals
% 'PSI' = PSI
% 'bar' = Bar
a = 8.6572e+5;
b = -7.6961e5;
pressure = @(V) a*V + b;
switch punit
case 'Pa'
p = pressure(pvolt);
case 'PSI'
p = pressure(pvolt)/6894.76;
case 'bar'
p = pressure(pvolt)/1e5;
end
end
VC_TempTherm.m
function t = VC_TempTherm(varargin)
%TempTherm(signal,temp_unit)
% signal = voltage signal from the thermistor probe
% temp_unit = temperature unit in which the result is returned (optional, defailt is 'C')
% 'C' = result in Celsius
% 'K' = result in Kelvin
% 'F' = result in Fahrenheit
if rem(nargin,2) == 0
TempUnit = varargin{nargin};
VSignal = varargin{nargin-1};
else
TempUnit = 'C';
VSignal = varargin{nargin};
end
Temp = @(K,R) 1./(K(1)+K(2).*log(R)+K(3).*log(R).^3)-273.15;
K = [1.02119E-03 2.22468E-04 1.33342E-07];
R0 = 15500;
Vex = 5.08;
Rt = VSignal*R0/(Vex - VSignal);
if strcmp(TempUnit,'K')
t = Temp(K,Rt)+273.15;
elseif strcmp(TempUnit,'F')
t = 9/5*Temp(K,Rt)+32;
else
t = Temp(K,Rt);
end
end
VC_Data_Process.m
% Processes VC data in form [pe ps temp; pe ps temp;]
R = 8.314;
data_m = rawdata((end-1),:); %data point where pressures are equal
extra_point = rawdata(end,:);%data point to determine volume ratio
data = rawdata(1:(end-1),:);%remove very last "extra" point so data(end)=mth point
pe = data(:,1); %array of pressures in expansion tank
ps = data(:,2); %pressures in sample tank
T = data(:,3); %temperature at each step
m = length(pe); %number of "steps" in the experiment
%volume ratio can be calculated or given.
%given:
Ve_Vs = 2.7;
%calculated:
%pf = extra_point(1); %final pressure after extra step (in expansion tank, since low pressure
reading)
%Ve_Vs =(ps(m)-pf)/pf; %ratio of volume of expansion and sample tanks
Z=zeros(m-1,1); %array to be filled with (m-1) Z values
for r = 1:m-1 %calculate Z for m-1 samples (last sample doesn't count)
pressure_sum = sum(pe(r+1:m)); %sum of "remaining" expansion tank pressures
Z(r)=ps(r)/(pe(m)+(Ve_Vs*pressure_sum)); %compressibility coefficient
end
ps = ps(1:(end-1)); %remove last point from sample pressure array, to match dimension with Z
T = T(1:(end-1)); %remove last point
cutpoints = 1; %number of low pressure points to remove (to be varied depending on data quality)
%perform linear fit on compressibility coefficient vs sample tank pressure:
params = polyfit(ps(1:(end-cutpoints)),Z(1:(end-cutpoints)),1);
Zcalc = polyval(params,ps);
%plot Z vs sample pressure with linear fit
plot(ps, Z, '.');
hold on
plot(ps,Zcalc);
xlabel('Pressure (Pa)');
ylabel('Z');
A2=params(1); %Slope of line is A2
B2=A2*R*T(1); %convert A2 to B2
B2_mL=B2*1e6; %convert to mL/mol
gtext(['B_2 = ' num2str(B2_mL) ' mL/mol'])
hold off
Sample data
Data file: CO2 (5-28-2015).mat
Image/Figure file(s): CO2 (5-58-2015).fig
Results
Experimental B = -125.6 mL/mol
Literature4 BmL/mol (296 K)
Error 0.16%
4 http://www.ddbst.com/en/EED/PCP/BII_C1050.php
EXPERIMENTS 4: DETERMINATION OF THE JOULE-THOMSON
COEFFICIENT OF CO2
MATLAB Codes
JT_setup.m
devs = daq.getDevices;
dev = devs.ID;
sNI = daq.createSession('ni')
sNI.addAnalogInputChannel(dev,'ai0','Voltage')
aiNI1=sNI.Channels(1)
aiNI1.TerminalConfig='SingleEnded'
aiNI1.Range=[-5 5]
sNI.addAnalogInputChannel(dev,'ai1','Voltage')
aiNI2=sNI.Channels(2)
aiNI2.TerminalConfig='SingleEnded'
aiNI2.Range=[-5 5]
sNI.addAnalogInputChannel(dev,'ai3','Voltage')
aiNI3=sNI.Channels(3)
aiNI3.TerminalConfig='SingleEnded'
aiNI3.Range=[-5 5]
sNI.addAnalogOutputChannel(dev, 'ao0', 'Voltage')
JT_Run.m
function[dT, dp] = JT_Run(ramp_time, dwell_time, min_pres, max_pres, sNI)
%increases pressure from min_pres to max_pres over time ramp_time, spending
%dwell_time at each pressure, and returning delta-T and delta-p values
%ramp_time = time to ramp up pressure from min_pres to max_pres (s)
%dwell_time = time between changing pressure (s) for equilibrating
%min_pres = starting pressure (PSI)
%max_pres = ending pressure (PSI)
%sNI = data acquisition object
num_steps = floor(ramp_time/dwell_time); %number of pressure steps (integer)
pres_range = max_pres-min_pres;
pres_step = pres_range/num_steps;
%determine pressure offset:
[t1, t2, p1] = JT_read_sensors(sNI); %measure pressure in tube
offset = p1 %pressure provided by manual voltage offset
%initialize arrays for dT and dp
dT = zeros(num_steps, 1);
dp = zeros(num_steps, 1);
pressure = min_pres;
for n=1:num_steps
%set pressure
disp(['Setting pressure to ' num2str(pressure) ' PSI'])
JT_set_p_controller(pressure,offset,sNI);
%wait dwell_time
disp('Equilibrating...');
pause(dwell_time);
%measure Ti, Tf, Pf
[Ti, Tf, delta_p]=JT_read_sensors(sNI);
delta_T = Tf-Ti;
%record data point
disp(['Pressure: ' num2str(-delta_p) ' Temp: ' num2str(delta_T) ' \mu =' num2str(-
delta_T*14.5/delta_p)])
dp(n)=-delta_p;
dT(n)=delta_T;
%increment pressure
pressure = pressure + pres_step;
end
%lower pressure at end of experiment
JT_set_p_controller(0, offset, sNI); %only really sets pressure to "offset" since only 5V are
variable
JT_set_p_controller.m
function JT_set_p_controller(pressure, offset, sNI)
%sets the voltage of the pressure controller
%pressure = desired pressure in PSI
%offset = pressure provided by manual 0-5V offset
%sNI = data acquisition object
%convert pressure to 0-5V signal. V=0 : P=offset, V=5: P=offset+60
pres_to_volts = @(p,offset) (p-offset)/12 + 0.25; %voltage from 0 to 5V to provide 0 to 60 psi
volts = pres_to_volts(pressure,offset);
%cases where desired pressure is outside of available range:
if volts > 5
volts=5;
else if volts<0
volts=0;
end
end
sNI.outputSingleScan(volts);%set controller to desired voltage
JT_read_sensors.m
function [ti, tf, p] = JT_read_sensors(sNI)
%reads initial and final temperature, and initial pressure, averaged over
%1000 readings. Temperatures given in Kelvin; pressure given in PSI
n_sample = 1000; %number of samples to average
%pressure sensor calibration to PSI:
PresFunc150 = @(V) (254998*V-125735.8)*1e-5*14.5; %in PSI
%temperature sensor calibration to Kelvin:
K = [1.02119E-03 2.22468E-04 1.33342E-07];
R0 = 15532;
Vex = 5.08;
Rt = @(VSignal, R0, Vex) VSignal*R0/(Vex - VSignal);
Temp = @(K,data) 1./(K(1)+K(2).*log(data)+K(3).*log(data).^3);
%t = Temp(K,Rt(4.5,R0,Vex))+273.15;
rawdataNI = zeros(n_sample,3);
%take n_sample raw voltage readings and store in rawdataNI
for j=1:n_sample
rawdataNI(j,:) = sNI.inputSingleScan;
end
%average to a single set of raw voltage data
mean_data = mean(rawdataNI,1);
raw_ti = mean_data(1);
raw_tf = mean_data(2);
raw_p = mean_data(3);
%convert voltage to units
ti = Temp(K,Rt(raw_ti,R0,Vex));
tf = Temp(K,Rt(raw_tf,R0,Vex));
p = PresFunc150(raw_p);
JT_analysis_KL.m
%run experiment:
JT_setup;
ramp_time = 1200;
dwell_time = 30;
min_pres = 65;
max_pres = 120;
[dT, dp] = JT_Run(ramp_time, dwell_time, min_pres, max_pres, sNI);
%Data analysis
dp = dp/14.5; %convert to bar from PSI
%plot dp vs dT
plot(dp,dT,'o')
ylabel('\Delta T (K)')
xlabel('\Delta p (bar)')
hold on
%determine muJT from slope of graph
params = polyfit(dp,dT,1); %polynomial fit, order 1 (linear)
muJT=params(1); %muJT = slope of graph
calcT =polyval(params,dp); %calculate line of best fit
plot(dp,calcT) %Plot line of best fit
R=corrcoef(dp, dT);%determine correlation coefficients
R=R(1,2);%select R
gtext(['\mu_{JT} = ' num2str(muJT) ' K/bar ' char(10) 'R = ' num2str(R) ]) %char(10) is a line
break
hold off
title('Joule-Tomson Coefficient of Ar')
%determine dT/dp at each data point and average
pt_JT = dT./dp; %point-by point array of dT/dP
avg_JT = mean(pt_JT);
Sample data
Data file: JT_CO2_KL_6-15-2015.mat
Image/Figure file(s): JT_CO2_KL_6-15-2015.fig
Results
Experimental JT = 1.18 K/bar
Literature5 JT = 1.16 K/bar
Error 1.70%
5
http://webbook.nist.gov/cgi/fluid.cgi?T=293&PLow=0&PHigh=1&PInc=.01&Applet=on&Digits=5&ID=C124389
&Action=Load&Type=IsoTherm&TUnit=K&PUnit=MPa&DUnit=mol%2Fl&HUnit=kJ%2Fmol&WUnit=m%2Fs
&VisUnit=uPa*s&STUnit=N%2Fm&RefState=DEF
Data file: JT_Ar_SK-5-3-2015a.mat
Image/Figure file(s): JT_Ar_SK-5-3-2015-.fig
Results
Experimental JT = 0.395 K/bar
Literature6 JT = 0.375 K/bar
Error 4.40%
6
http://webbook.nist.gov/cgi/fluid.cgi?T=293&PLow=0&PHigh=1&PInc=0.01&Applet=on&Digits=5&ID=C744037
1&Action=Load&Type=IsoTherm&TUnit=K&PUnit=MPa&DUnit=mol%2Fl&HUnit=kJ%2Fmol&WUnit=m%2Fs
&VisUnit=uPa*s&STUnit=N%2Fm&RefState=DEF
Data file: JT_He_SK-4-30-2015-.mat
Image/Figure file(s): JT_He_SK-4-30-2015-.fig
Results
Experimental JT = -0.0637 K/bar
Literature7 JT = -0.0623 K/bar
Error 2.25%
7
http://webbook.nist.gov/cgi/fluid.cgi?T=293&PLow=0&PHigh=1&PInc=0.01&Applet=on&Digits=5&ID=C744059
7&Action=Load&Type=IsoTherm&TUnit=K&PUnit=MPa&DUnit=mol%2Fl&HUnit=kJ%2Fmol&WUnit=m%2Fs
&VisUnit=uPa*s&STUnit=N%2Fm&RefState=DEF
Data file: JT_N2_SK-5-3-2015a.mat
Image/Figure file(s): JT_N2_SK-5-3-2015a.fig
Results
Experimental JT = 0.256 K/bar
Literature8 JT = 0.222 K/bar
Error 15.3%
8
http://webbook.nist.gov/cgi/fluid.cgi?T=293&PLow=0&PHigh=1&PInc=0.01&Applet=on&Digits=5&ID=C772737
9&Action=Load&Type=IsoTherm&TUnit=K&PUnit=MPa&DUnit=mol%2Fl&HUnit=kJ%2Fmol&WUnit=m%2Fs
&VisUnit=uPa*s&STUnit=N%2Fm&RefState=DEF
EXPERIMENTS 5: DETERMINATION OF THE HEAT CAPACITY
RATIO OF CO2 WITH THE RUCCHARDT-METHOD
MATLAB Codes
R_Setup.m:
f = mfilename(); disp([f ': Setting up data collection interface']);
devs = daq.getDevices; dev = devs.ID; sNI = daq.createSession('ni'); sNI.addAnalogInputChannel(dev,'ai0','Voltage'); % Data channel for the
pressure sensor aiNI1=sNI.Channels(1); aiNI1.TerminalConfig='SingleEnded'; aiNI1.Range=[-5 5]; sNI.addDigitalChannel(dev,'port0/line0','OutputOnly'); %Control signal for
the solenoid valve sNI.addDigitalChannel(dev,'port0/line1','OutputOnly'); %Control signal for
the electromagnet sNI.addDigitalChannel(dev,'port0/line2','OutputOnly'); %Not used sNI.addDigitalChannel(dev,'port0/line3','OutputOnly'); %Not used sNI.outputSingleScan([1 1 1 1]); sNI.inputSingleScan; disp([f ': Data collection interface setup complete']);
R_Run_Experiment.m:
clear
R_setup; %Setting up MODuck
R_Flush(1,30,sNI); %Flushing with new gas
R_Control(0,sNI); %0 = Solenoid off/Electromagnet off
n = 1000; %number of poinbts to be collected p = zeros(n,1); t = zeros(n,1);
r = R_Read(100,sNI); %Read 100 points for initial pressure pi = mean(r,2); %pi = initial pressure R_Control(3,sNI); %3 = Solenoid on/Electromagnet on - Drive cylinder up pause(5); R_Control(2,sNI); %2 = Solenoid off/Electromagnet on - Hold cylinder until
pressure equilizes r = R_Read(100,sNI); %Read 100 points for pressure p = mean(r,2); %p = pressure system
while p > 1.05*pi %wait until pressure drops to atmospheric pressure
within 5% r = R_Read(100,sNI); %Read 100 points for pressure p = mean(r,2); %p = pressure system end R_Control(0,sNI); %0 = Solenoid off/Electromagnet off - Drop cylinder to
perform experiment r = R_Read(n,sNI); %Read n points for pressure
R_Data_Process_damped.m
m = 0.318 ;%mass of cylinder (kg) patm = 101325; %atmospheric pressure V=0.0199+(.037/2)^2*pi*1.3;%volume of continer (L) A = (.037/2)^2*pi; %surface of cylinder
x = r(:,2); t = r(:,1); plot(r(:,1),r(:,2)); disp('==================================================================='); disp('Click on last point to be considered for fitting, then right click!'); disp('==================================================================='); [q1 q2] = getpts; q = find(t < q1(1)); n1 = q(end); %Index of last point considered for fitting
x = r(1:n1,2); t=r(1:n1,1);
a = 1; %Initial value of amplitude lambda = 0.1; %Initial value of lambda gamma = 1.3; phase = pi; %Initial value of phase offset = mean(x); %Centering the displacement around 0 xoff = x - offset; p0=[a lambda gamma phase]; f = @(p,t) p(1).*exp(-p(2) .* t).* cos( sqrt(patm*p(3)*A^2/(m*V)-p(2)^2) .* t
+ p(4)) ; % function for damped oscillation f2 = @(p,t) p(1)*exp(-p(2) .* t); %upper exponential function f3 = @(p,t) -p(1)*exp(-p(2) .* t); %lower exponential function p = nlinfit(t,xoff,f,p0); v2p = @(v) (v-0.5)./1.1603e-4; p2v = @(p) 0.5+1.1603e-4 .*p; plot(t,xoff,'.'); hold on; xd = f(p,t); xe1 = f2(p,t); xe2 = f3(p,t); plot(t,xd,'r'); plot(t,xe1,'--'); plot(t,xe2,'--');
gamma = p(3) text(0.75*(max(t)-min(t)),0.75*max(xoff),['\lambda = ' num2str(p(2))
char(10) '\gamma = ' num2str(gamma)])
R_Control.m
function [raw_data]= R_Control(flag,sNI)
%function R_Control(flag,sNI) %flag = flag for action to be performed: % 0 = Solenoid off/Electromagnet off % 1 = Solenoid on/Electromagnet off % 2 = Solenoid off/Electromagnet on % 3 = Solenoid on/Electromagnet on %sNI = data acquisition object
f = mfilename();
switch flag case 0 sNI.outputSingleScan([1 1 1 1]); disp([f ': 0 = Solenoid off/Electromagnet off']); case 1 sNI.outputSingleScan([0 1 1 1]); disp([f ': 1 = Solenoid on/Electromagnet off']); case 2 sNI.outputSingleScan([1 0 1 1]); disp([f ': 2 = Solenoid off/Electromagnet on']); case 3 sNI.outputSingleScan([0 0 1 1]); disp([f ': 3 = Solenoid on/Electromagnet on']); otherwise disp([f ': Use of R_Control:']); disp([f ': flag = flag for action to be performed:']); disp([f ': 0 = Solenoid off/Electromagnet off']); disp([f ': 1 = Solenoid on/Electromagnet off']); disp([f ': 2 = Solenoid off/Electromagnet on']); disp([f ': 3 = Solenoid on/Electromagnet on']); return end end
R_Flush.m
function R_Flush(flag,tflash,sNI) %function R_Flash(flag,tflashI) %flag % 1= flash container % 0= do not flash cntainer %tflash = time to flash if flag == 0 return; end f = mfilename(); pi = []; for i=1:100 pi = [pi; sNI.inputSingleScan]; end pi = mean(pi)
disp([f ': Flashing container for ' num2str(tflash) ' s....']); start_flash = clock; sNI.outputSingleScan([0 0 1 1]); while etime(clock,start_flash) < tflash end sNI.outputSingleScan([1 1 1 1]); disp([f ': Waiting....']); while sNI.inputSingleScan > 1.05*pi end disp([f ': Done flashing container']);
R_Read.m
function [raw_data]= R_Read(n,sNI) %function [raw_data]= R_Read(n,sNI) % raw_data = returned matrix with raw sensor reagings (n rows, 2 % columns, t,V(t)) %n = numberof ponts to be collected during compression %sNI = data acquisition object
f = mfilename(); disp([f ': Reading pressure transducer....']); raw_data = []; start_time = clock; for i=1:n raw_data = [raw_data; etime(clock,start_time) sNI.inputSingleScan]; end end
Sample data
Data file: CO2 (5-18-2015).mat
Image/Figure file(s): CO2 (5-18-2015).fig
Results
Experimental = 1.30
Literature9 = 1.29
Error 0.80%
9 http://webbook.nist.gov/cgi/cbook.cgi?ID=C124389&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
Data file: Ar (5-18-2015).mat
Image/Figure file(s): Ar (5-18-2015).fig
Results
Experimental = 1.59
Literature10 = 1.67
Error 4.80%
10
http://webbook.nist.gov/cgi/cbook.cgi?ID=C7440371&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
Data file: He(5-19-2015).mat
Image/Figure file(s): He(5-19-2015).fig
Results
Experimental = 1.52
Literature11 = 1.67
Error 8.9%
11
http://webbook.nist.gov/cgi/cbook.cgi?ID=C7440597&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
Data file: N2 (5-18-2015).mat
Image/Figure file(s): N2 (5-18-2015).fig
Results
Experimental = 1.38
Literature12 = 1.40
Error 1.40%
12
http://webbook.nist.gov/cgi/cbook.cgi?ID=C7727379&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
EXPERIMENT 6: DETERMINATION OF THE HEAT CAPACITY RATIO
OF CO2 WITH ADIABATIC COMPRESSION
MATLAB Codes
AC_Setup.m:
f = mfilename(); disp([f ': Setting up data collection interface']);
devs = daq.getDevices; dev = devs.ID; sNI = daq.createSession('ni'); sNI.addAnalogInputChannel(dev,'ai0','Voltage'); % Data channel for the volume
sensor aiNI1=sNI.Channels(1); aiNI1.TerminalConfig='SingleEnded'; aiNI1.Range=[-5 5]; sNI.addAnalogInputChannel(dev,'ai1','Voltage'); % Data channel for the
temperature sensor aiNI2=sNI.Channels(2); aiNI2.TerminalConfig='SingleEnded'; aiNI2.Range=[-1 1]; sNI.addAnalogInputChannel(dev,'ai6','Voltage'); % Data channel for the
pressure sensor aiNI2=sNI.Channels(3); aiNI2.TerminalConfig='SingleEnded'; aiNI2.Range=[-1 1]; sNI.addDigitalChannel(dev,'port0/line0','OutputOnly'); %Control signal for
the actuator valve sNI.addDigitalChannel(dev,'port0/line1','OutputOnly'); %Control signal for
the loading valve sNI.addDigitalChannel(dev,'port0/line2','OutputOnly'); %Control signal for
the release valve sNI.addDigitalChannel(dev,'port0/line3','OutputOnly'); %Not used sNI.outputSingleScan([1 1 1 1]); sNI.inputSingleScan; disp([f ': Data collection interface seyup complete']);
AC_Flush(sNI):
function AC_Flush(sNI) %1. Loads cylinder with gas through the loading valve (piston up) %2. Flushes gas out through the release valve (piston down) %3. Equilibrates
f = mfilename(); disp([f ': Piston down, Open release valve']); sNI.outputSingleScan([0 1 0 1]); %Piston down, Open release valve pause(5); disp([f ': Piston up, Open loading valve, close release valve']);
sNI.outputSingleScan([1 0 1 1]); %Piston up, Open loading valve, close
release valve pause(5); disp([f ': Close loading valve, open release valve']); sNI.outputSingleScan([1 1 0 1]); %Close loading valve, open release valve pause(5); disp([f ': Close all valves and equilibrating for 20s']); sNI.outputSingleScan([1 1 1 1]); %Close all valves pause(20); end
[raw_data]= AC_Compression(n,sNI):
function [raw_data]= AC_Compression(n,sNI) %function [raw_data]= AC_Compression(n,sNI) % raw_data = returned matrix with raw sensor reagings (n rows, 3 % columns, V,T,p) %n = numberof ponts to be collected during compression %sNI = data acquisition object
f = mfilename(); disp([f ': Performing adiabatic compression']); raw_data = []; %zeros(n,3); sNI.outputSingleScan([0 1 1 1]); tic for i=1:n raw_data = [raw_data; sNI.inputSingleScan]; end toc pause(5); disp([f ': ' num2str(n) ' data points are collected']); sNI.outputSingleScan([1 1 1 1]);
[T, p, V] = AC_SensorCals(raw_data):
function [T, p, V] = AC_SensorCals(raw_data) %function [T, p, V] = AC_SensorCals(raw_data) %raw_data = matrix with n rows (number of points collecred), and 3 %columns: V, T, p %[T, p, V] = converted readings to respective units % % T = smooth(33.785 .* raw_data(:,2)+295.193,3); p = smooth(100000 .* (raw_data(:,3)+0.094),7); V = smooth(3.218e-5*raw_data(:,1)+1.25e-4,7); % p = -raw_data(:,3).*1.0100e5+1.4823e5; % V = smooth(2.8326e-5*raw_data(:,1)+9.191e-5,3); TempFunc = @(K,data)
1./(K(1)+K(2).*log(data)+K(3).*log(data).^2+K(4).*log(data).^3); K = [3.354E-03 2.5648E-04 2.388E-06 8.377E-8]; R = 2000; Vex = 5.08; Rt = raw_data(:,2)*R ./(Vex - raw_data(:,2)) ./10000; T = smooth(TempFunc(K,Rt),7);
end
AC_Run_Data_Analysis_KL_no_temp:
%Analyzes data collected thusly: [T1 p1 V1] = AC_SensorCals(data);
%Does not use temperature data
plot(V1,p1)
xlabel('V (m^3)')
ylabel('p (Pa)')
disp('===================================================================');
disp('Click on first point to be considered for fitting, then right click!');
disp('===================================================================');
[q1 q2] = getpts;
q = find(V1 > q1(1));
stop = q(end); %Index of first point considered for fitting
disp('===================================================================');
disp('Click on last point to be considered for fitting, then right click!');
disp('===================================================================');
[q1 q2] = getpts;
q = find(V1 > q1(1));
start = q(end); %Index of last point considered for fitting
%start and stop appear switched because V1 indexes from highest V to lowest
%truncate data between desired points
T=T1(start:stop);
p=p1(start:stop);
V=V1(start:stop);
%calculate logs
lnp=log(p);
lnV=log(V);
lnT=log(T);
%representation 1: -gamma*lnV + C = lnp
subplot(1,2,1)
plot(lnV, lnp,'.')
hold on
%perform linear fit
par1=polyfit(lnV,lnp,1);
lnp_calc=polyval(par1, lnV);
%calculate gamma from slope
gamma1=-par1(1);
plot(lnV,lnp_calc)
gtext(['\gamma = ' sprintf('%.2f',gamma1)])
xlabel('ln(V)')
ylabel('ln(p)')
hold off
%representation 1A: p=C*V^-gamma
subplot(1,2,2)
%initial guess for parameters C and gamma
C=5;
gamma=1.2;
p0=[C gamma];
%perform nonlinear fit:
f= @(p, V) p(1)*V.^(-p(2));
pf=nlinfit(V,p,f,p0);
%obtain gamma from fit parameters:
gamma_nonlin=pf(2);
plot(V, p,'.')
hold on
p_calc=f(pf,V);
plot(V,p_calc)
gtext(['\gamma = ' sprintf('%.2f',gamma_nonlin)])
xlabel('V (m^3)')
ylabel('p (Pa)')
hold off
%No equations using temperature, if temperature sensor is not working
%find and display average gamma
avg_gamma= mean([gamma1, gamma_nonlin])
%find and display Cv and Cp
R=8.314;
Cv=R/(avg_gamma-1)
Cp=Cv + R
AC_Run_Data_Analysis_KL:
%Analyzes data collected thusly: [T1 p1 V1] = AC_SensorCals(data);
plot(V1,p1)
xlabel('V (m^3)')
ylabel('p (Pa)')
disp('===================================================================');
disp('Click on first point to be considered for fitting, then right click!');
disp('===================================================================');
[q1 q2] = getpts;
q = find(V1 > q1(1));
stop = q(end); %Index of first point considered for fitting
disp('===================================================================');
disp('Click on last point to be considered for fitting, then right click!');
disp('===================================================================');
[q1 q2] = getpts;
q = find(V1 > q1(1));
start = q(end); %Index of last point considered for fitting
%start and stop appear switched because V1 indexes from highest V to lowest
%truncate data between desired points
T=T1(start:stop);
p=p1(start:stop);
V=V1(start:stop);
%calculate logs
lnp=log(p);
lnV=log(V);
lnT=log(T);
%representation 1: -gamma*lnV + C = lnp
subplot(2,2,1)
plot(lnV, lnp,'.')
hold on
%perform linear fit
par1=polyfit(lnV,lnp,1);
lnp_calc=polyval(par1, lnV);
%calculate gamma from slope
gamma1=-par1(1);
plot(lnV,lnp_calc)
gtext(['\gamma = ' sprintf('%.2f',gamma1)])
xlabel('ln(V)')
ylabel('ln(p)')
hold off
%representation 1A: p=C*V^-gamma
subplot(2,2,2)
%initial guess for parameters C and gamma
C=5;
gamma=1.2;
p0=[C gamma];
%perform nonlinear fit:
f= @(p, V) p(1)*V.^(-p(2));
pf=nlinfit(V,p,f,p0);
%obtain gamma from fit parameters:
gamma_nonlin=pf(2);
plot(V, p,'.')
hold on
p_calc=f(pf,V);
plot(V,p_calc)
gtext(['\gamma = ' sprintf('%.2f',gamma_nonlin)])
xlabel('V (m^3)')
ylabel('p (Pa)')
hold off
%representation 2: (gamma-1/gamma)lnp + C = lnT
subplot(2,2,3)
plot(lnp, lnT,'.')
hold on
%perform linear fit:
par2=polyfit(lnp,lnT,1);
lnT_calc=polyval(par2, lnp);
%calculate gamma from slope
gamma2=-1/(par2(1)-1);
plot(lnp,lnT_calc)
gtext(['\gamma = ' sprintf('%.2f',gamma2)])
xlabel('ln(p)')
ylabel('ln(T)')
hold off
%representation 3: (1-gamma)lnV + C = lnT
subplot(2,2,4)
plot(lnV, lnT,'.')
hold on
%perform linear fit:
par3=polyfit(lnV,lnT,1);
lnT_calc=polyval(par3, lnV);
%calculate gamma from slope
gamma3=-par3(1)+1;
plot(lnV,lnT_calc)
gtext(['\gamma = ' sprintf('%.2f',gamma3)])
xlabel('ln(V)')
ylabel('ln(T)')
hold off
%find and display average gamma
avg_gamma= mean([gamma1, gamma2, gamma3])
%find and display Cv and Cp
R=8.314;
Cv=R/(avg_gamma-1)
Cp=Cv + R
Sample data
Data file: AC_CO2_KL_5-26-2015.mat
Image/Figure file(s): AC_CO2_KL_5-26-2015.fig
Results
Experimental = 1.31
Literature13 = 1.29
Error 1.55%
13 http://webbook.nist.gov/cgi/cbook.cgi?ID=C124389&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
Data file: AC_Ar_KL_5-26-2015.mat
Image/Figure file(s): AC_Ar_KL_5-26-2015.fig
Results
Experimental = 1.56
Literature14 = 1.67
Error 6.59%
14
http://webbook.nist.gov/cgi/cbook.cgi?ID=C7440371&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
Data file: AC_He_KL_5-26-2015.mat
Image/Figure file(s): AC_He_KL_5-26-2015.fig
Results
Experimental = 1.45
Literature15 = 1.67
Error 13.2%
15
http://webbook.nist.gov/cgi/cbook.cgi?ID=C7440597&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
Data file: AC_N2_KL_5-26-2015.mat
Image/Figure file(s): AC_N2_KL_5-26-2015.fig
Results
Experimental = 1.41
Literature16 = 1.40
Error 0.709%
16
http://webbook.nist.gov/cgi/cbook.cgi?ID=C7727379&Units=SI&Mask=1&Type=JANAFG&Table=on#JANAFG
EXPERIMENTS 7: SOLVING AN ORDINARY DIFFERENTIAL
EQUATION (ODE) SYSTEM
Space holder
EXPERIMENTS 8: SOLVING A PARTIAL DIFFERENTIAL EQUATION
(PDE) SYSTEM
Space holder
References
1. Yung, R.M.G.Y.L., Atmospheric Radiation, in Atmospheric Radiation. Oxford University Press. p. 88.
2. McQuarrie, D.A., Second Virial Coefficient, in Statistical Mechanics. p. 233-237.
3. Reif, F., Fundamentals of statistical and thermal physics. 2009, Waveland Press. p. 422-424.
4. Lemmon, E.W., M.O. McLinden, and D.G. Friend. Thermophysical Properties of Fluid Systems, NIST
Chemistry WebBook. [cited 2014 02/23/2014]; Available from: http://webbook.nist.gov.
5. Clapham, D., Calcium signaling. Cell, 2007. 131(6): p. 1047-1058.
6. Cuthbertson, K. and T. Chay, Modelling receptor-controlled intracellular calcium oscillators. Cell
Calcium, 1991. 12(2-3): p. 97.
7. Chay, T.R., Y.S. Lee, and Y.S. Fan, Appearance of phase-locked wenckebach-like rhythms, devil’s
staircase and universality in intracellular calcium spikes in non-excitable cell models. Journal of
Theoretical Biology, 1995. 174(1): p. 21-44.
8. Borghans, J.A.M., G. Dupont, and A. Goldbeter, Complex intracellular calcium oscillations A theoretical
exploration of possible mechanisms. Biophysical Chemistry, 1997. 66(1): p. 25-41.
9. Dupont, G., Theoretical insights into the mechanism of spiral Ca2+ wave initiation in Xenopus oocytes.
1998, Am Physiological Soc. p. 317-322.
10. Goldbeter, A., G. Dupont, and M.J. Berridge, Minimal model for signal-induced Ca2+ oscillations and for
their frequency encoding through protein phosphorylation. Proceedings of the National Academy of
Sciences of the United States of America, 1990. 87(4): p. 1461.
TABLE OF CONTENT EXPERIMENT 1 .................................................................................................... 5
DETERMINATION OF THE GAS CONSTANT, R ............................................................................. 5 BACKGROUND ........................................................................................................ 5 EXPERIMENTAL PROCEDURE ................................................................................ 6 RESULTS, CALCULATIONS ..................................................................................... 8
EXPERIMENT 2 .................................................................................................. 11
DETERMINATION OF THE COMPOSITION OF AIR ....................................................................... 11 BACKGROUND ...................................................................................................... 11 EXPERIMENTAL PROCEDURE .............................................................................. 11 RESULTS, CALCULATIONS ................................................................................... 14
EXPERIMENT 3 .................................................................................................. 17
DETERMINATION OF MOLECULAR WEIGHT .............................................................................. 17 WITH GAS EFFUSION ...................................................................................................... 17 BACKGROUND ...................................................................................................... 17 EXPERIMENTAL PROCEDURE .............................................................................. 18 RESULTS, CALCULATIONS ................................................................................... 21
EXPERIMENT 4 .................................................................................................. 24
DETERMINATION OF HEAT OF COMBUSTION ............................................................................ 24 BACKGROUND ...................................................................................................... 24 EXPERIMENTAL PROCEDURE .............................................................................. 26 RESULTS, CALCULATIONS ................................................................................... 30
EXPERIMENT 5 .................................................................................................. 32
DETERMINATION OF HEAT OF VAPORIZATION USING THE CLAUSIUS CLAPEYRON EQUATION ................ 32
BACKGROUND ...................................................................................................... 32 EXPERIMENTAL PROCEDURE .............................................................................. 34 RESULTS AND CALCULATIONS ............................................................................ 34
EXPERIMENT 6 .................................................................................................. 37
DETERMINATION OF MOLECULAR WEIGHT BY THE FREEZING POINT DEPRESSION METHOD ..................... 37 BACKGROUND ...................................................................................................... 37 EXPERIMENTAL PROCEDURE .............................................................................. 37 RESULTS AND CALCULATIONS ............................................................................ 41
EXPERIMENT 7 .................................................................................................. 42
DETERMINATION OF DISSOCIATION CONSTANT WITH UV VISIBLE SPECTROSCOPY .......................... 42
BACKGROUND ...................................................................................................... 42 EXPERIMENTAL PROCEDURE .............................................................................. 43 CALCULATIONS .................................................................................................... 46 APPENDIX ............................................................................................................. 48
EXPERIMENT 8 .................................................................................................. 51
DETERMINATION OF DISSOCIATION CONSTANT OF A WEAK ACID OR WEAK BASE ................................. 51 BACKGROUND ...................................................................................................... 51 EXPERIMENTAL PROCEDURE .............................................................................. 53 RESULTS AND CALCULATIONS ............................................................................ 54
EXPERIMENT 9 .................................................................................................. 57
CONDUCTIVITY OF SOLUTIONS ............................................................................................ 57 BACKGROUND ...................................................................................................... 57 EXPERIMENTAL PROCEDURE .............................................................................. 59 RESULTS AND CALCULATIONS ............................................................................ 59
EXPERIMENT 10 ................................................................................................ 61
LIGHT ABSORPTION OF DYES ............................................................................................ 61 BACKGROUND ...................................................................................................... 61 EXPERIMENTAL PROCEDURE .............................................................................. 61 RESULTS AND CALCULATIONS ............................................................................ 62
EXPERIMENT 11 ................................................................................................ 64
THE THERMODYNAMICS OF THE DANIELL CELL ......................................................................... 64 BACKGROUND ...................................................................................................... 64 EXPERIMENTAL PROCEDURE .............................................................................. 66 RESULTS AND CALCULATIONS ............................................................................ 66
EXPERIMENT 12 ................................................................................................ 69
DETERMINATION OF RATE CONSTANT OF THE IODINE-ACETONE REACTION ........................................ 69 DETERMINATION OF SURFACE TENSION BY THE CAPILLARY RISE METHOD ........................................ 69 BACKGROUND ...................................................................................................... 69 EXPERIMENTAL PROCEDURE .............................................................................. 70 RESULTS AND CALCULATIONS ............................................................................ 71
EXPERIMENT 13 ................................................................................................ 73
KINETIC STUDIES WITH CONDUCTOMETRY- ............................................................................. 73 DETERMINATION OF ACTIVATION ENERGY .............................................................................. 73 BACKGROUND ...................................................................................................... 73 EXPERIMENTAL PROCEDURE .............................................................................. 75 RESULTS AND CALCULATIONS ............................................................................ 76
EXPERIMENT 14................................................................................................ 78
DETERMINATION OF RATE CONSTANT AND THE ACTIVATION ENERGY FOR THE DECOMPOSITION OF A DIAZONIUM
COMPOUND ................................................................................................................. 78 BACKGROUND ...................................................................................................... 78 EXPERIMENTAL PROCEDURE .............................................................................. 79 RESULTS AND CALCULATIONS ............................................................................ 80 APPENDIX ............................................................................................................. 83
EXPERIMENT 15 ................................................................................................ 86
MEASUREMENT OF REFRACTIVE INDEX .................................................................................. 86 BACKGROUND ...................................................................................................... 86
EXPERIMENTAL PROCEDURE .............................................................................. 86 RESULTS AND CALCULATIONS ............................................................................ 87
EXPERIMENT 16 ................................................................................................ 90
DETERMINATION OF HEAT CAPACITY OF GASES WITH RUCHHARDT’S METHOD ................................... 90 BACKGROUND ...................................................................................................... 90 EXPERIMENTAL PROCEDURE .............................................................................. 92 RESULTS AND CALCULATIONS ............................................................................ 93
EXPERIMENT 17 ................................................................................................ 96
REVERSIBLE WORK OF GASES ........................................................................................... 96 BACKGROUND ...................................................................................................... 96 EXPERIMENTAL PROCEDURE .............................................................................. 97 RESULTS AND CALCULATIONS ............................................................................ 99
EXPERIMENT 18 .............................................................................................. 104
DETERMINATION OF THE RATIO OF SPECIFIC HEATS OF GASES 2. ................................................ 104 BACKGROUND .................................................................................................... 104 EXPERIMENTAL PROCEDURE ............................................................................ 106 RESULTS AND CALCULATIONS .......................................................................... 107
EXPERIMENT 19 .............................................................................................. 111
INVERSION OF SUCROSE ................................................................................................. 111 BACKGROUND .................................................................................................... 111 EXPERIMENTAL PROCEDURE ............................................................................ 113 RESULTS AND CALCULATIONS .......................................................................... 115
EXPERIMENT 20 .............................................................................................. 118
ENZYME KINETICS ........................................................................................................ 118 BACKGROUND .................................................................................................... 118 EXPERIMENTAL PROCEDURE ............................................................................ 121 RESULTS, CALCULATIONS ................................................................................. 123
Experiment 1
Determination of the Gas
Constant, R
Background
The perfect gas equation of state characterizes gases that
behave like ‘ideal gases’. The equation is a limiting law
assuming that there is no interaction between molecules.
That assumption holds if the pressure of the gas
approaches zero.
The law is:
nRTpV ( 1.1)
If we solve ( 1.1) for R:
nT
pVR
( 1.2)
where p is the pressure, V is the volume, n is the chemical
amount, and T is the temperature in Kelvin. You are
going to put known amount of dry ice (solid carbon
dioxide) into a closed container. Dry ice vaporizes at
room temperature (sublimes). You are going to measure
the built pressure and the temperature from which the
value of R can be obtained.
Experimental Procedure
The experimental setup can be seen in Figure 1.1. There
is stainless steel pressure proof Gas Tank that will have a
pressure and temperature sensor connected.
The probes are connected to the interface box with DIN
connectors and the interface box is connected to the
computer.
Start the software (Logger Pro) and load the Exp_1.mbl
configuration file. The interface should look like Figure
1.2. The top left panel follows the pressure changes, the
lower left panel follows the temperature changes. On the
right hand side the recorded data points are in a table.
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Insert the temperature probe into the rubber cork (Figure
1.3) and insert it into the lower port of the gas tank
(Figure 1.4) air tight. Insert the pressure sensor into the
quick-release port (Figure 1.5) by pulling back the
cylinder and pushing in the nipple. Tight the blue bolts
(Figure 1.6) with the wrench very tight. Make sure that
the red ball valve is closed. Place the balance next to
the experimental setup to be able to weigh the dry ice
quickly.
Figure 1.5
Figure 1.6
Prepare some fresh pieces of dry ice (get instructions
from your instructor). Break a piece and weigh it to two
significant figures. It should be between 1.5 and 2.0 g.
Place the piece of dry ice into the Gas Tank through the
loading port and place the rubber cork back really
quick. Click on the ‘Collect’ button to start to collect
the pressure and temperature readings.
Once the pressure levels off, stop collecting data. If the
pressure reaches a maximum and starts to decline, your
system is leaking; you have to check for leaks and you
have to repeat the run. Remove the rubber stopper (it
will sound popping noise due to the built up pressure).
Save your data. If you are making more runs, just drop
a new piece of dry ice, place the stopper back and start
to collect data. If you are done, loosen the blue bolts
with the wrench.
Results, Calculations
Locate the final pressure value on the curve. Lookup the
corresponding temperature value. Calculate the number
of moles of carbon dioxide you put in the Gas Tank from
its weight and formula weight (44g mol-1). Prepare a table
in Excel and enter your data and necessary formulas (
). The volume of the Gas Tank is 1640 cm3. Calculate an
average from your trials. Calculate your error:
Do not handle dry ice with bare hand. It may cause burning injuries!
100exp
% xvalueltheoretica
valueerimentalvalueltheoreticaerror
(
1.3
)
Figure 1.7
Experiment 2
Determination of the
Composition of Air
Background
Air is a mixture of gases. The exact composition changes
all the time; it depends on the time, geographical location
etc. The oxygen content very likely would be much lower
in New York City than in Brazil. The carbon dioxide
content however would be much higher in New York.
Generally, air consists of nitrogen (around 78%), oxygen
(around 20%), argon (around 1%) and some other, gases
like carbon dioxide, sulfur dioxide, nitrogen oxides.
We are going to determine the oxygen content of the air
based on the fact that oxygen supports burning and the
other components don’t.
According to Dalton’s Law:
j
j
O
i
iatm pppp 2
( 1.4)
where i indicates all components, j indicates all
components except oxygen and carbon dioxide (the latter
one will be absorbed by KOH). From the atmospheric
pressure (patm) and the final pressure the partial pressure
of oxygen can be calculated.
Experimental Procedure
The experimental setup can be seen in Figure 1.8. There
is stainless steel pressure proof Gas Tank that will have a
pressure sensor connected. The top loading port is
plugged.
The probe is connected to the interface box with a DIN
connector and the interface box is connected to the
computer.
Start the software (Logger Pro) and load the Exp_2.mbl
configuration file. The interface should look like Figure
1.9. On the graph in the left panel we’ll follow the
pressure changes. On the right hand side the recorded
data points are in a table.
Figure 1.8
Figure 1.9
Figure 1.10
Figure 1.11
Insert the pressure sensor into the quick-release port by
pulling back the cylinder and pushing in the nipple. Insert
the KOH container into the middle of the bottom of the
tank. Tight the blue bolts with the wrench very tight.
Make sure that the red ball valve is closed. Insert the
candle in the hole of the rubber stopper (Figure 1.10)
firmly. Start collecting data with the computer. After
about 50s, lit the candle and insert the stopper into the
lower port (Figure 1.11). Due to the compression caused
by the stopper, the pressure initially will jump and then it
will dive as the oxygen will be used up (Figure 1.12).
Figure 1.12
Once the pressure levels off, stop collecting data.
Remove the rubber stopper (it will sound popping noise
due to the built up pressure). Save your data. If you are
making more runs, remove the stopper from the top port
and flush the tank with compressed air (flush the carbon
dioxide out). Replace the top stopper and repeat the
experiment. If you are done, loosen the blue bolts with
the wrench. Replace the KOH container into the glass
container and cover it carefully.
Results, Calculations
The pressure decrease is due to the consumption of
oxygen. According to Dalton’s Law:
atm
O
Op
px 2
2
( 1.5)
In terms of per cent of oxygen content:
100% 2 xp
poxygen
atm
O
( 1.6)
Experiment 3
Determination of Molecular
Weight With Gas Effusion
Background
For an ideal gas, the total energy of the gas is merely the
average kinetic energy, given as (the potential energy is
being zero):
2
2
1mvE
( 2.1)
where m is the molar mass of the gas and v is the root
mean square speed (RMS) of the molecules. There are
two gases, one with known and one with unknown molar
weight. Since the gases are at equal temperature and
pressure, the kinetic energy is the same for each gas,
which leads to the following equation:
1
2
2
1
1
2
2
2
2
1
2
2
2
1
21
2
22
2
11
2
1
2
1
2
1
2
1
m
m
v
v
m
m
v
v
mvmv
EE
mvEmvE
( 2.2)
Thus, the ratio of the RMS is inversely proportional to
the square roots of the molecular weights. From here, it
must be noted that the time it takes a certain volume of
gas to effuse through an orifice is inversely proportional
to the speed of the molecules. This leads to the working
equation for the experiment
1
2
1
22
1
2
1
2
1
2
2
1
1
2
mt
tm
m
m
t
t
m
m
v
v
t
t
( 2.3)
Therefore if the time it takes for the same amount of the
two gases to escape through the orifice is measured, the
molar weight of the unknown gas can be calculated.
Experimental Procedure
The experimental setup can be seen in Figure 2.1. A
stainless steel pressure proof Gas Tank will be used with
a pressure sensor connected to it. The probe is connected
to an interface box and the interface box is connected to a
computer.
Configure the software to display a pressure vs. time
graph and a digit display for pressure.
1. Close all three valves (Figure 2.2). Disconnect the
computer from the interface and move the tank near
the vacuum pump.
2. Connect the vacuum pump to the QuickConnect
port. Turn on the vacuum pump and follow the
pressure drop on the screen of the GLX. When the
pressure drops below 8kPa, disconnect the pump.
Stop the pump.
3. Connect the loading hose from to the gas cylinder
and to Valve #1 and tighten it with the wrench. Set
the pressure on the regulator to about 30PSI
(270kPa).
4. VERY SLOWLY open Valve #2 and the valve on
the regulator and let the gas in until the pressure
reaches 210kPa. Shut Valve #1.
5. Open Valve #2 to purge the tank. When the pressure
drops to 100 kPa, shut Valve #2.
6. Repeat steps 4 and 5 two more times.
7. Load the tank to 210 kPa.
8. Connect the GLX to the computer. Turn on
monitoring (Alt-M) and observe the digit display.
9. Allow some gas to leave by opening Valve #2 very
little until the pressure drops to 200kPa.
10. Stop monitoring.
11. Open Valve #3 to allow the gas to escape through
the orifice and in the same time start the data
collection.
12. Collect data for 20 min.
13. Repeat the run (steps 7-12).
14. Repeat the experiment (steps 1-12) with two other
gases.
Figure 2.1
Figure 2.2
Figure 2.3
Results, Calculations
The resulting data should look like Figure 2.3 (only two
gases are displayed for clarity). Determine the time it
took the pressure of each gas to drop from 200kPa to
150kPa (t1, t2 and t3). Since the volume of the vessel is
the same, initial and final pressures are the same, the
escaped volumes of gases are the same.
Using Excel, collect your data on a spreadsheet. Calculate
an average escape time for each gas from the three runs.
Using each gas, determine the molecular weight of the
other two unknown gases.
.
Experiment 4
Determination of Heat of
Combustion
Background
Thermochemistry is a powerful technique for
determining experimentally the enthalpy change, H, or
the energy change, U, accompanying a given isothermal
change in state of the system, usually one in which a
chemical reaction is the source of the energy. Since the
conditions are well controlled, any energy change that
occurs can be measured accurately, including the heat of
combustion.
Figure 3.1
The reaction is carried out in an oxygen bomb
calorimeter (Figure 3.1).
The parts of the calorimeter are shown in Figure 3.1. The
outside jacket (Figure 3.1,3) is insulated which prevents
heat exchange between the calorimeter and the
environment. There is pail inside the jacket with 2000ml
water (Figure 3.1, 4). An air pillow provides further
insulation between the pail and the jacket. The bomb
(Figure 3.1, 1) is sitting in the middle of the pail. A stirrer
(Figure 3.1, 2) operated by an electric motor (Figure 3.1,
5) makes sure that the water bath is homogeneous. The
temperature is followed by a high precision thermometer
(Figure 3.1, 8). Leds (Figure 3.1, 7) are connected to the
bomb for ignition.
The sample is loaded into the bomb with 30atm oxygen.
A wire is fused through the sample that can be heated
with electric current. At the moment of ignition, the wire
is heated up and the combustion occurs. The released heat
is absorbed by the wall of the bomb that eventually
transfers the heat to the surrounding water. The raise of
the temperature of the water bath is monitored. The
released amount of heat can be obtained from ( 3.1).
TnCQ p ( 3.1)
where Q is the released heat, n is the chemical amount
of the burnt material, Cp is the heat capacity of the
calorimeter and T is the temperature increase due to the
reaction. In order to determine the calorimeter constant, a
material with known heat of combustion is burnt.
Experimental Procedure
There are three basic steps involved in carrying out the
experiment: (1) Preparation of the sample (pellet) and (2)
Burning the standard sample to determine the calorimeter
constant, (3) Burning the unknown sample.
To prepare pellets of the calibrating material, benzoic
acid, weigh about 1.3g material into a weighting dish.
Place the base plate of the press on a solid surface. Place
the mold on the top of it and fill the material into the
mold (Figure 3.2, A). Place the piston into the mold and
pound it down a little bit with a hammer (Figure 3.2, B).
Place the press (the mold and the piston) into the vise
press it as much as you can (Figure 3.2, C). Loosen the
vise and place the space holder such a way that the pellet
could be pushed out freely by the piston. Place the press
and the space holder into the vise and tighten it very
slowly (Figure 3.2, D). The piston will push the pellet into
the space holder. Loosen the vise. Take about a 2” long
wire, weigh it (with four significant figures) and mount it
on the fusing leds Figure 3.2, E). Turn up the controller
knob half way to heat up the wire and fuse the wire into
the pellet (Figure 3.2, F). The wire should be fused in half
way.
Figure 3.2
Fix the bomb in the filling bracket with the bolt in it.
Weigh the pellet with four significant figures and
subtract the weight of the wire to get the weight of the
pellet. Mount the pellet in the head of the bomb (Figure
3.3, A) to the ignition electrodes very carefully. Wind
the extra wire around the electrode so it wouldn’t cause
short circuit. Place a pan under the sample (Figure 3.3,
B). Insert the head into the bomb and tighten the screw
as much as you can without a wrench (Figure 3.3, C).
Place the oxygen hose on the filling port (Figure 3.3, D).
Make sure that the valve on the regulator of the oxygen
cylinder is closed. Open the valve on the oxygen
cylinder. Slowly open the valve on the regulator and set
the pressure to 25atm (Figure 3.3, E). That will fill up
the bomb with 25atm oxygen. Never exceed 30atm
pressure! Relese the gas from the hose by opening the
butterfly valve on the regulator (Figure 3.3, F). Release
the oxygen from the bomb very slowly by opening the
needle valve on the bomb to flush out the nitrogen from
the bomb that was originally present (Figure 3.3, G).
Close the needle valve. Fill the bomb with oxygen again.
Place the pail into the jacket on the three mounting
points. Make sure that the pail is sitting securely on the
mounting points. Place the bomb into the pail (Figure
3.3, H). Connect the ignition leds (Figure 3.3, I). Fill a
2L volumetric flask to the mark with water (you may
want to run the tap water for a while to get as cold water
as possible) and pour the water very slowly into the pail.
Make sure you transfer all the water and the water
Figure 3.3
doesn’t splash. The accuracy of the measurement greatly
depends on the fact, that the amount of water is always
the same in the calorimeter since the water is the main
medium that adsorbs the heat therefore the main
contributor to the heat capacity of the calorimeter. Place
the lid on the calorimeter (Figure 3.3, J). Place the belt
on the pulley and the shaft of the motor (Figure 3.3, K).
Place the thermometer into the hole opposite from the
stirrer. Make sure the thermometer is sitting securely.
Handle the thermometer very gently since the mercury
line can break very easily. Turn on the motor. Start to
take temperature readings once in every 2min until you
establish a steady rate of climbing temperature (Figure
3.4). When the base line is established, push the ignition
button (Figure 3.3, L) and start to take readings in every
minute. The temperature will start to rise after about 30s.
Take readings until the temperature levels off and you
get the climbing r.ate as before the ignition.
Take the calorimeter apart. Release the oxygen through
the needle valve. Never attempt to remove the top of the
bomb under pressure! When the bomb is depressurized,
remove the top. Make sure all the sample is burnt. If
there is any sample left, you have to
Figure 3.4
repeat the experiment because you wouldn’t know how
much of the sample actually burnt. If you want to have
more runs, wipe out the water from the bomb that is a
product of the combustion and fill the bomb as
described before. Drain the water out from the pail and
wipe it dry. It may seem unnecessary to wipe it dry
since you will fill it with water, but once again, the
amount of water in the pail has to be known precisely
and has to be the same to maintain constant heat
capacity. Have at least two benzoic acid runs and one
run for each unknown.
When you are done, wipe all the parts dry.
Close the valve on the oxygen cylinder.
Results, Calculations
The heat of combustion of benzoic acid is
H(BA)=3224 kJ/mol. The amount of heat released by
the benzoic acid pellet is
)()(
)( BAHBAM
mBAHnQ
w
( 3.2)
where m is the weight of the pellet, Mw(BA) is the
molecular weight of benzoic acid. The amount of heat
released by the sample is Q. That heat was absorbed by
the calorimeter (all the parts and the water):
TcQ ( 3.3)
where c is the calorimeter constant and T is the
temperature increase by the reaction. Combination of (
3.2) and ( 3.3)
)()(
BAHBAMT
mc
w
( 3.4)
From ( 3.4) the calorimeter constant can be obtained.
To determine the temperature jump from the
Temperature-Time graph, draw a tangent to the part of
the graph that was recorded before the ignition and an
other one to the part that was recorded after the
temperature leveled off after the ignition (Figure 3.4).
Determine the distance between the two lines in terms
of temperature. These two lines should be close to
parallel. Calculate an average of calorimeter constant
from the 2 benzoic acid runs.
To determine the heat of combustion for the
unknown, combine ( 3.3) and ( 3.4) a different way:
m
XMTcXH w )(
)(
( 3.5)
where m now is the weight of the pellet made from the
unknown, H(X) is the heat of combustion of the
unknown and T is the temperature jump caused by the
combustion of the unknown.
Experiment 5
Determination of Heat of
Vaporization Using the
Clausius Clapeyron Equation
Background
Vapor-liquid phase equilibrium can be characterized
with the Clausius-Clapeyron equation which provides a
relationship between the vapor pressure and
temperature:
dp
dT
H
RT
v
2
( 4.1)
where Hv is the enthalpy change accompanying the
phase transition from liquid to vapor, i.e. the heat of
vaporization. Integration of ( 4.1) can be performed
assuming that Hv is independent of the temperature (in
most cases this holds):
dp
p
H
RT dTv
2 ( 4.2 )
ln pH
R TCv
1 ( 4.3 )
where C is the integration constant. According to ( 4.3
), by measuring the dependence of the vapor pressure
over a liquid on the temperature, the heat of
vaporization can be determined; A plot of ln p vs. 1/T
has a slope of -Hv/R from which Hv can be
determined.
Experimental Procedure
The experimental setup can be seen in Figure 4.1.
Please spend some time to locate the parts of the
apparatus. You should receive instructions from your
T.A. regarding the temperature range for the
experiment. Open Exp_4.mbl configuration file in
Logger Pro. Conect the pressure and temperature
sensors to the interface box. Connect the vacuum pump
hose to the lower arm of the stopcock (Figure 4.1, F).
Connect the pressure sensor (Figure 4.1, E) and turn on
the vacuum pump. Turn the stopcock (Figure 4.1, F)
such that it would connect the vessel with the vacuum
pump and start to evacuate the reaction vessel). Once
the pressure gauge reading is less than 5kPa, turn the
stopcock (Figure 4.1, F) to closed position. Check the
pressure readings for 10 min. If the pressure changes
more than 2 kPa, that indicates a leak in the vacuum
system. If the sealing is good, place 15 ml of your
unknown into the sample holder. Open the Stopcock on
the sample holder (Figure 4.1, C) very slowly and make
sure no air seeps into the reaction vessel! Close the
stopcock before all of the sample enters the reaction
vessel. If any air enters, you must repeat the
preparations. Turn on the heating elements (Figure 4.1,
D). Start to record your data. Continue to collect data
until the pressure reaches the atmospheric pressure.
After stopping the data collection, turn the heating
elements off.
Results and Calculations
Highlight all your data in KLogger Pro. Export the
highlighted values into an Excel file. Open the saved
data in Excel. The three columns are the measured time,
pressure and temperature. Create a column for ln p and
1/T (please remember to convert Co to K). Create a plot
of ln p – 1/T. Fit a straight line on the points (Figure
4.2). The slope of the line is related to the heat of
vaporization according to ( 4.3 )
Do NOT knock anything into the reaction vessel! Any fracture may result in an explosion because of the vacuum!
Some of the unknowns may cause skin irritation upon contact! Be aware, most of the unknowns is highly flammable!
Figure 4.1
0.0020
0.0022
0.0024
0.0026
0.0028
0.0030
0.0032
0.0034
0.0036
0.0038
0.0040
7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00
ln p
1/T
Methanol
Ethanol
Acetone
1-Propanol
2-Propanol
Benzene
Ethyl acetate
n-Butyl alcohol
s-Butyl alcohol
i-Butyl alcohol
Figure 4.2
Experiment 6 Determination of Molecular
Weight by the Freezing Point
Depression Method
Background
The freezing point of any particular solvent is lowered as
a result of vapor pressure lowering of a volatile solvent
when a nonvolatile solute is added. By measuring the
extent of the freezing point depression (T) of a known
quantity of pure solvent when a known amount of
nonvolatile pure solute is added, the molecular weight of
the solute can be determined.
The freezing point depression is by the following
equation:
mKT ff ( 5.1)
where Tf is the freezing point depression, Kf is the
cryoscopic constant of the solvent (Kf = 6.8 K kg mol-1
for naphthalene) and m is the molality.
Experimental Procedure
1. Connect the stirrer rod and connect the
temperature sensor to the digital thermometer as
Probe A. Connect the Recorder output of the
thermometer to the lab interface. The thermometer
provides a signal of 10mV for each oC (i.e. 25oC
would be 0.250 V) that we will measure and
record with the data acquisition station. Make sure
that the thermometer probe selector is set to Probe
A (Figure 5.2,1), the reading is set to oC (Figure
5.2,3) and the range selector switch is set to 0-
100oC (Figure 5.2,2). Start Logger Pro and load
the Exp_6.mbl configuration file.
2. Weigh approximately 30g of naphtalene to the
nearest 0.01g. Grind it to fine powder in a mortar.
Assemble the experimental setup as shown Figure
5.1.
3. Place the stirrer into the test tube and mount the
test tube. Place the temperature sensor inside the
loops of the stirrer and mount it so that the tip of
the temperature sensor is as close to the bottom of
the test tube as possible. Tip of the probe senses
the temperature. Add the naphtaline in the test
tube. You may need to turn on the stirrer to make
the powder to fall to
Figure 5.1
Figure 5.2
the bottom of the test tube.
4. Once the powder is in place, place a beaker of
water under the test tube and start to heat it. When
the water temperature reaches 80 oC, the
naphtalene starts to melt. Turn on the stirrer.
5. When all the solid particles disappeared, very
carefully replace the beaker with an empty
Erlenmeyer flask (Figure 5.3). The flask slows
down and makes the cooling process more even.
Start to record the temperature.
6. Initially the temperature will drop quickly and
then it remains constant for a while (Figure 5.4)
and the initially tranparent liquid will have
flurries. The constant temperature is the freezing
point which won’t change until there is any liquid
left. Make sure that the stirring is appropriate.
Figure 5.3
Figure 5.4
7. Weigh approximately 1.00g to the nearest 0.001g.
8. After about 20 min recording, stop the data
collection and store the data set.
9. Replace the Erlenmeyer flask with the beaker of
water and melt the naphthalene again. Carefully
The naphthalene and naphthalene with unknown solutions should be placed in the appropriate waste container
remove the water and remove the test tube. Make
sure all the drips of naphthalene will be collected
in the test tube as you remove the test tube.
10. Put the unknown into the test tube. Make sure all
the crystals fall into the melted naphthalene.
11. Replace the test tube and put the hot water back
until all the crystals are dissolved.
12. Replace the beaker with the empty Erlenmeyer
flask and start to record the cooling curve as
before.
13. When done, melt the mixture again, add again
about 1.000g (that you weighed to the nearest
0.001g) and record the cooling curve again. You
will notice that as you add the unknown, the
freezing point will drop for each run.
Results and Calculations
The exact value of the freezing point of the pure
naphthalene and of the solutions is obtained graphically.
Export the data from Logger Pro to Excel and place all
three curves on one graph. Print out the graph. Use a
straight edge to extrapolate the portion of the cooling
curve before the freezing. Extrapolate the nearly
horizontal portion of the curve to the temperature axis.
Use your graph to calculate Tf (the difference in
freezing point of the solution from the solvent), the
molecular weight of the unknown from each run using the
freezing point depression values (Tf), and the known
freezing point depression constant for naphthalene.
Experiment 7
Determination of Dissociation
Constant with UV Visible
Spectroscopy
Background
Let us consider a weak acid, HA with the dissociation
process
HA H A ( 6.1 )
and dissociation constant, K:
KH A
HA
[ ][ ]
[ ] ( 6.2 )
If the weak acid is suitably chosen, the HA and A- species
have different colors, i.e. different absorption spectra.
Acid-base indicators often have this usually like that.
Considering a wavelength where the two forms have
significantly different absorption (max ), the ratio of their
concentrations can be expressed as
[ ]
[ ]
max
max
A
HA
A A
A A
HA
A
( 6.3 )
where A is the absorption at max, AAmax is the absorption
at max when all the weak acid is in A- form, and AHAmax is
the absorption at max when all the weak acid is in HA
form (for detailed derivation refer to the Appendix). The
value of K can be calculated using this equation and the
value of pH.
Experimental Procedure
At the beginning of the lab session you will be assigned
pH values for six buffer solutions as well as your
unknown. Clean eight 25 ml volumetric flasks and
number them. In six you should prepare buffer solutions
with the assigned pH, in one you should prepare a buffer
solution with pH=1 (i.e. where all the acid is in the HA
form) and one with pH=9 (i.e. where all the unknown is
in the A- form). Each of your solutions will consist of a
combination of two solutions that determine the pH (i.e.
they form a buffer with the desired pH), the unknown and
5 ml of 1 M KCl to set the ionic strength constant. For
more details and an example refer to the table below.
Pipet these solutions into all of your volumetric flasks
(again six with the assigned pH values, one with pH=1
and one with pH=9), and fill the volumetric flasks to the
mark and shake them.
Example:
Assigned buffers: pH=4.4, 4.6, 4.8, 5.0, 5.2, 5.4
Unknown to be used: 1.00 ml/25ml
From the table of buffers (Figure 6.1)
Flask # Desired
pH
Buffers
1 4.4 3.50ml I + 1.95ml J
2 4.6 2.55ml I + 2.45ml J
3 4.8 2.00ml I + 3.00ml J
4 5.0 5.00ml D + 2.39 F
5 5.2 5.00ml D + 3.00 F
6 5.4 5.00ml D + 3.55 F
7 1.0 5.00ml A + 9.7ml B
8 9.0 5.00mlA + 0.88ml F
Add 1.00ml of unknown and 5.00ml 1M KCl to each flask
and fill them to the mark.
Turn on the photometer. For operation of the
photometer refer to the handout. First you have to record
the base line (between 730-330nm). To do so, place a cell
(filled 2/3 with distilled water) in the cell compartment.
Take extra caution keeping the transparent walls of the
cells clean (i.e. do not touch them and wipe them dry with
Kimwipes). Record the base line. Rinse the sample cell
with a small portion of the first solution and fill it up to
2/3. Wipe it dry and record the spectrum. Wash the
sample cell, rinse it with a small portion of the second
solution and fill it 2/3rds. Record the spectrum. Make
sure you do the recording in overlay mode (see Figure
6.2). Repeat the same procedure with all your solutions.
Do not dispose of your solutions until you have
completed your calculations because you may have to
repeat some of the scanning.
Make sure that the
cells are clean and
dry outside. Never
touch the
transparent walls of
the cells!
Mlliliters of buffer solution
desired pH A B D F I J O P
1.00 5.00 9.70
2.00 5.00 1.06
3.00 5.00 2.03
3.20 1.47 5.00
3.40 0.99 5.00
3.60 0.60 5.00
3.80 4.40 0.60
4.00 4.10 0.90
4.20 3.68 1.32
4.40 3.05 1.95
4.60 2.55 2.45
4.80 2.00 3.00
5.00 5.00 2.39
5.20 5.00 3.00
5.40 5.00 3.55
5.60 5.00 3.98
5.80 5.00 4.30
6.00 5.00 4.55
6.20 8.15 1.85
6.40 7.35 2.65
6.60 6.25 3.75
6.80 5.10 4.90
7.00 3.90 6.10
7.20 2.80 7.20
7.40 1.90 8.10
7.60 1.30 8.70
7.80 0.85 9.15
8.00 0.53 9.47
9.00 5.00 8.80
Figure 6.1
Figure 6.2
Calculations
On the spectrum pick a wavelength where the two form
have significantly different absorption (max ) and read
the absorption for each scan at that particular wavelength.
Record the corresponding pH absorption values in a
spreadsheet :
pH A(520) [A]/[HA] K
520
1.00 0.652 N/A N/A
3.40 0.470 0.512676 2.04E-04
3.60 0.415 0.79 1.98E-04
3.80 0.330 1.497674 2.37E-04
4.00 0.285 2.158824 2.16E-04
4.20 0.245 3.130769 1.98E-04
4.40 0.210 4.652632 1.85E-04
9.00 0.115 N/A N/A
Calculate the [A-]/[HA] ratio from Equation ( 6.3 ), where
AAmax
is the absorbance of the solution at pH=9 (i.e. when all
the molecules are in A- form), AHAmax is the absorbance at
pH=1 (when all the molecules are in HA form). Enter
these ratios into your spreadsheet. From the pH value you
can obtain [H+] for each solutions. Substituting these [H+]
and [A]/[HA] values into Equation ( 6.3 ) ) you can
obtain K for each solutions. Note, that at pH=1 and pH=9
you cannot evaluate K. Calculate an average value for K.
If any of the points is way off, try to repeat the scan with
that sample. If it is still off, ignore it when calculating the
mean value for K.
Appendix
The following derivation holds at any wavelength, however, we
consider the wavelength where one species has absorption maximum
(Ímax) and the other species has substantially lower absorption. The
correlation between absorbance and concentration is given by the
Beer-Lambert law:
A bc
where A is the absorbance, c is the concentration of absorbing species,
is the molar absorbance and b is the cell thickness.
The total absorbance is:
A A AHA A
where AHA is the contribution from the HA form and AA is the
contribution from the A form to the total absorption, A. The
absorbance of the solution that has only HA (i.e. in the presence of
strong acid) will be AHAmax (AA=0) and the absorbance of the solution
that has only A- (i.e. in the presence of strong base) will be AAmax.
(AHA=0):
max
max
maxmax
max
max
[ ] [ ]
[ ] [ ]
AA A T A
T
HAHA HA T HA
T
HAHA HA
T
AA A
T
AA bc b
c
AA bc b
c
AA b HA HA
c
AA b A A
c
where cT is the total concentration of the weak acid:
c A HAT [ ] [ ]
At any intermediate pH, where both forms are present, the absorption
will be:
max max max max
max max max max max max
max
max max
[ ] [ ] [ ] [ ]
[ ] [ ] [ ]
[ ]
THA A HA A HA A
T T T T
HA HA A HA A HA
T T T
HA
T A HA
HA A c A AA A A A A A A
c c c c
A A AA A A A A A
c c c
A A A
c A A
and max max max max
max max
max max
max max
max
max
[ ] [ ][ ] [ ] [ ] [ ]
[ ] [ ]
([ ] [ ]) [ ] [ ]
[ ] [ ] [ ] [ ]
[ ]( ) [ ]( )
[ ]
[ ]
HA A HA AHA A HA A
T T
HA A
HA A
A HA
HA
A
A A A HA A AA A A b HA b A HA A
c c HA A
A HA A A HA A A
A HA A A A HA A A
A A A HA A A
A A A
HA A A
Experiment 8
Determination of Dissociation
Constant of a Weak Acid or
Weak Base
Background
Acids or bases that do not dissociate completely in water
are called weak acids or weak bases. Consider a weak
acid:
AHHA ( 7.1)
The corresponding equilibrium constant would be:
][
]][[
HA
AHK
( 7.2 )
The reason for the limited dissociation is that the
corresponding conjugated base (i.e. A-) is a strong base.
There are two processes in such aquaeous solutions
competing for the H+: one is ( 7.1)and the second is the
reaction of OH- with H+ forming water. Since A- is a
stronger base than OH- (i.e. it has stronger affinity for H+
), it will remove some H+ from water (this process is
frequently referred to as “hydrolysis”):
OHHAOHA 2 ( 7.3 )
The corresponding equilibrium constant would be:
][
]][[
A
HAOHK H ( 7.4 )
It can be shown, that the relationship between KH and K
is Kw = KH K, where Kw is the water ion product (about
10-14). Let us take a solution of such acid and start to add
a solution of a strong base, such as NaOH (strong base
means, again, that it completely dissociates into ions, Na+
and OH- in aqueous solution). The OH- ions react with
the weak acid and form a salt:
AOHHAOH 2 ( 7.5 )
Please note, that the process is quantitative, not an
equilibrium. The concentration of the added base (cb )
and the total concentration of the weak acid (ca) can be
calculated :
][][][
][][][
AAOHc
HAcHAAc
b
ba ( 7.6 )
Substituting Eqn.( 7.6 ) into ( 7.2 ) leads to:
ba
b
cc
cHK
][
( 7.7 )
We will use Eqn. ( 7.7) to calculate the dissociation
constant along the second part of a titration curve (see
Figure 7.1 part 2) from pH measurements.
At the equivalence point (Figure 7.1), point 3) all the weak
acid is converted to salt, and the pH is determined by
Eqn.( 7.3 ). From Eqn. ( 7.3 ) one can see, that the
solution will be somewhat alkaline due to the formation
of OH- ions.
If more NaOH solution is added, the pH will be
determined by the excess OH- concentration (Figure 7.1),
part 4).
Figure 7.1 Typical titration curve
Experimental Procedure
1. Check the condition of the glass electrode. It
should not have any salt precipitate on it and it
should be stored in a KCl solution. Take the
electrode out from the KCl solution and rinse it
thoroughly with distilled water.
Please note that the combination electrode is very fragile, handle it with
care!
2. Locate the three buffer solutions (pH=4, 7, and
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0.00 10.00 20.00 30.00 40.00
Volume of NaOH /ml
pH
1.
2. 4.
3.
10). Pour about 25ml of each into three 50ml
beakers. Immerse the electrode in the first one,
stir the buffer gently and wait until you can get a
stable reading). The reading should match the
nominal value of the buffer solution. Take the
electrode out, rinse it with water and wipe it
gently. Repeat the procedure with the other two
solutions. If any of the readings deviate more
than 0.03 units, the apparatus has to be
calibrated. For the details of the calibration
procedure refer to the manual of the pH meter.
3. Pipet 20.00ml of your unknown into a clean
100ml beaker with a volumetric pipet. Take a
note that the concentration of your unknown is
0.01M. Place a stirring bar in the beaker and
place the beaker on a magnetic stirrer.
Stir the solution gently and slowly, otherwise air bubbles will make the reading very
unstable and inaccurate!
4. Fill the buret with 0.01M NaOH solution. Set the
zero point on the buret.Immerse the electrode in
your solution.
5. Once the reading is stable, start to add the
titrating solution from the buret in 0.05ml
aliquots. Make sure the pH is stable before you
add the next portion of solution. Record the pH
and volume data pairs in your notebook. When
you are getting close to the equivalence point, it
will take longer for the pH to stabilize.
7. Once you reached the equivalence point, add
about 10 more ml in 0.05ml aliquots.
8. When you are done, rinse the electrode with
distilled water, wipe it off with kimwipes and
place it into the storage solution.
Please note that only the bulb part of the electrode should be in solution!
Results and Calculations
Prepare your titration curve: plot pH as a function of
volume of added solution. The dissociation constant can
be calculated based on Eqn.( 7.7 ) along part 2 of the
titration curve. Prepare a table with your data:
The total concentration of the added base (cb) and the
unreacted acid (ca) is:
0
00
00
0 )(
)(,
)(
)(
vNaOHv
cNaOHvcvcc
vNaOHv
NaOHvc ba
abb
( 7.8)
where cb0 is the initial concentration of the NaOH
solution (i.e. 0.01M), ca0 is the initial concentration of the
weak acid (i.e. 0.01M), v0 is the initial volume of the
weak acid solution. The calculated K values may vary in
the beginning and in the end since the concentration of
one of the components is very low compared to the other
and Eqn. ( 7.3 ) does not hold. Omit those values and
choose the ones that are approximately the same and
calculate an average value. That is the value you will
have to turn in. To verify that determined the right value,
take the volume of NaOH solution that was necessary to
reach the equivalence point, divide it by two and look up
the pH value on your curve that corresponds to the half
volume. The pH value should approximately match the –
log K value.
Notes
Experiment 9
Conductivity of Solutions
Background
Electrolyte solutions follow Ohm’s law in the sense that
due to U potential, I current is passing through the
solution and the relationship between U and I is R=U/I.
For practical reason, however, in chemistry we use
conductivity () instead of resistance (R) :
R
k
A
l
R
1
( 8.1)
where l is the distance between the electrodes and A is the
surface area of the electrodes. k is called the cell constant
indicating that it depends only on the geometric
parameters of the cell.
The conductivity of a solution depends on the
concentration of the ions in the solution since ions carry
the charges; the higher the ion concentration, the higher
the conductivity. In diluted solutions where the
interaction between the ions can be neglected, the
conductivity is directly proportional to the ion
concentration. In order to normalize the conductivity to 1
mol substance, the molar conductivity (m) was
introduced:
i
mc
( 8.2 )
where ci is the ion concentration (anion or cation; for
simplicity, we consider ions with equal, but opposite
charges).
In this experiment we will study a weak electrolyte,
acetic acid. The conductivity in acetic acid is due to the
H+ and OAc- ions. The concentration of these ions is:
cOAcHci ][][ ( 8.3 )
where c is the initial acetic acid concentration and is
the degree of ionization. The corresponding equilibrium
constant would be:
22
22
1
)1(
))((
][
]][[
cc
c
c
cc
cc
HOAc
OAcHK a
( 8.4 )
By arranging ( 8.4 ) one can get:
aK
c
1
1
( 8.5 )
The conductivity of a weak electrolyte depends on the ion
concentration which depends on the degree of ionization
as shown in ( 8.4 ):
0
mm ( 8.6 )
where m0 is the limiting molar conductivity. The limiting
molar conductivity is the conductivity of the electrolyte
that corresponds to the conductivity at zero concentration.
The combination of ( 8.5 ) and ( 8.6 ) leads to:
c
Km
ammm
200
111
( 8.7 )
According to ( 8.7 ), from the plot of 1/m vs. mc, m0
and Ka can be determined.
Experimental Procedure
Assemble the conductivity setup: connect the connectivity
probe to the interface box and start the acqusition
program.
Prepare 1L of 0.05M acetic acid solution from Glacial
Acetic Acid in a volumetric flask (Solution #1).
Pipet 25.00ml of the 0.05M acetic acid into a 100ml
volumetric flask and fill the flask to the mark
(Solution #2).
1. Pipet 25.00ml of Solution #2 into a 100ml
volumetric flask and fill the flask to the mark
(Solution #3)
2. Pipet 25.00ml of Solution #3 into a 100ml
volumetric flask and fill the flask to the mark
(Solution #4).
3. Measure the conductance of Solutions #1-4. The
cell constant, k is 1cm-1.
Results and Calculations
Make a spreadsheet with your data:
c (M) G (mS) (S/cm) (Scm2mol-1) 1/m mc
0.05
0.0125
0.003125
0.000781
Make a plot of of 1/m vs. mc. Calculate of m0 from
the intercept and Ka from the slope.
Experiment 10
Light Absorption of Dyes
Background
By measuring the amount of electromagnetic radiation
absorbed by matter, information about the properties of
matter may be obtained. The spectrum (light absorbed as
a function of wavelength) of a sample of matter is
characteristic of that matter and follows the Beer-Lambert
Law (A=bc).
Experimental Procedure
Familiarize yourself with the CHEM-2000
spectrophotometer before starting the experiment. Start
the program that controls the spectrophotometer (see
Figure 9.1). Obtain the dark current background by
blocking the path of the light and click on the button.
This background calibrates the electronics of the
spectrophotometer. Place your cell with distilled water in
the path and obtain the reference background by clicking
on the button. This background serves as a reference
point at each wavelength; it has everything in it except
the absorbing species. Once the background is taken, the
buttons become available indicating that you
are ready. Select the first button to put the
spectrophotometer into Absorbance mode. Click on the
button and set the Absorbance scale to 1.0.
1. Prepare the 5 standard solutions in accordance
with the instructions given in Table I. Calculate
the concentration (in mg/liter) of these solutions.
Make sure each solution is well mixed in a
beaker before taking absorbance measurements.
2. Take the spectrum of each solution. Start with
the highest concentration. Select the peak on the
spectrum in the visible region with the
buttons. Take all the
absorbance measurements at this wavelength.
Absorbance can be read from the bottom of the
spectrum. Save the sample files.
6. Fill the volumetric flask with the unknown to the
mark and shake it well. Measure the absorbance
of the unknown at the previous wavelength.
Results and Calculations
Calculate the concentration in mg/liter of each of the
solutions prepared in step 1 above. Make a linear plot of
absorbance versus concentration in mg/liter for each of
the known solutions. Read the concentrations in mg/l
from the plot of the unknown solutions and the measured
absorbance of the known solutions. Calculate the
concentrations of the unknown in moles/liter and in parts
per million (molecular weight of methyl orange = 327.34
gm/mol; 1 mg/L = ppm). Include the following items
with your report: The spectrum (graph of absorbance
versus wavelength) for the cuvette and for the dye you
used, and the graph of absorbance versus concentration at
the maximum wavelength of absorbance (use the whole
sheet of paper (8½ x 11) in making your graphs and draw
curves carefully).
TABLE I. Directions for preparation of standards.
Volume of Stock
Solution Added
Total Volume
Methyl Orange can
be washed down
the sink with large
amounts of water.
1 ml
100 ml
2 ml
100 ml
4 ml
100 ml
6 ml
100 ml
8 ml
100 ml
10 ml
100 ml
Figure 9.1
Experiment 11
The Thermodynamics of the
Daniell Cell
Background
The electromotive-force (EMF) data from a reversible
chemical cell can be used to obtain the values of reaction
Gibbs-energy, G, the reaction enthalpy, H and the
reaction entropy, S for the cell reaction. When a cell
operates reversibly at constant pressure and temperature
with only electrical work done, G can be obtained:
nFEG ( 10.1 )
from which S can be readily derived:
pp T
EnF
T
GS
( 10.2 )
According to ( 10.2 ) S can be calculated from the
dependence of EMF on the temperature. Also H can be
calculated knowing, that:
STHG
( 10.3 )
We are going to study the Daniell cell:
ZnZnCuCu aqaq |||| 22
( 10.4 )
Experimental Procedure
Prepare 500ml of 1.0M ZnSO4-solution and 1.0M
CuSO4-solution. Fill the ceramic cylinder 4/5th with the
CuSO4-solution and place it into the plastic cup. Fill the
cup half an inch below the edge of the ceramic cylinder
with ZnSO4-solution (Figure 10.1). Mount the Cu
electrode in the ceramic cylinder and mount the Zn
electrode in the ZnSO4-solution (Figure 10.2 and Figure
10.3). Mount the small stirring motor next to the Zn-
electrode and turn it on. Be careful, the stirring propeller
is made of glass and it is very fragile. Connect the
electrodes to the interface box with the right polarity (i.e.
the potential reading should be about 1.1V). Place plenty
of ice into the water bath with some distilled water.
Mount the plastic cup into the water bath (Figure 10.4).
Turn on the circulator pump. Make sure that the plastic
cup is immersed into the water as much as possible but
there is no water stirred into the cup. Place the
temperature probe into the ZnSO4-solution (Figure 10.5).
Set the temperature of the water bath to 90 oC and start
collecting data (one point in every 10s). Once the
temperature in the plastic cup reached about 80 oC, stop
the data acquisition and disassemble the apparatus. Stop
the circulating pump. Place the ceramic cylinder into
large volume of hot water (take it from the water bath) to
clean off the cupper residue. Clean the electrodes.
Results and Calculations
Make a spreadsheet of your T,E data (the time values
won’t be necessary). Make a plot (don’t forget to convert
the temperature readings to K). From ( 10.2 ) and the plot
calculate S. From ( 10.1 ) and S obtain G and from (
10.3 ) obtain H. Interpret your results!
Figure 10.1
Figure 10.2
Figure 10.3
Figure 10.4
Figure 10.5
Experiment 12
Determination of Rate
Constant of the Iodine-
Acetone Reaction
Background
Acetone is known to react with iodine in two steps: the
first is the enolization of acetone and the second is the
reaction of the enol form with molecular iodine.
CH CO CH H CH COH CH CH COH CH H
CH COH CH I CH CO CH I H I
3 3 3 3 3 2
3 2 2 3 2
The first reaction is slow and the second is fast. Therefore
the rate limiting step is the enolization. This process is
catalyzed by acids, so the rate depends on the acid
concentration. Since the enolization a second order
reaction, the rate constant of the form:
XAcetoneH
XHAcetone
HAcetonetk
00
00
00 ][][
][][ln
][][
1
( 11.1)
where X is the concentration of acetone enolized, that is,
the amount reacted up to time t.
XAcetoneH
XHAcetone
HAcetonekt
00
00
00 ][][
][][ln
][][
1
( 11.2 )
According to Eqn ( 11.2 ) of as a function of time will
give a straight line with a slope of k. Please note that
every equivalent of reacted acetone will generate one
equivalent of acid so the acid concentration will increase
during the course of the reaction. X can be determined by
following the [I2] spectrophotometrically. The convenient
wavelength is 470nm where the I2/I3- system has an
isosbestic point. Therefore the I3- formation does not
disturb our measurements (i.e. both I2 and I3- have the
same absorption coefficient so it does not matter in what
form the iodine exist).
l
AAIIX t
t
0
202 ][][ ( 11.3
)
where [I2]0 is the initial iodine concentration, [I2]t is the
iodine concentration at time t, A0 is the absorbance at t=0,
At is the absorbance at time t, is the molar absorption
coefficient at 470nm for the iodine (225 M-1cm-1) and l is
the path (1 cm).
Experimental Procedure
Turn on the thermostat and set it to 25 oC. Turn the
photometer on and let it warm up for 30 min.
Turn on the cell thermostating unit and set the
desired temperature. Start the control
program.Configure the spectrophotometer for
Time Acquisition @ 460nm once every second.
Please note that it is essential, that the temperature is constant since the
rate constant strongly depends on the temperature!
Pipet 25.00ml of saturated iodine solution into a 50ml
volumetric flask. Calculate the volume of acid
solution you will need to add as catalyst:
500
acid
acidacid
c
cv ( 11.4 )
where vacid is the volume of acid solution to be added,
cacid is the desired final concentration of acid (that you
will be assigned), c0acid is the concentration of the stock
acid solution. Add that amount of acid to the volumetric
flask. Use a graduated pipet and not a graduated cylinder.
Calculate the volume of the acetone to be added:
acetone
acetoneacetoneacetone
cMwv
500
( 11.5 )
where Mwacetone is the formula weight, c0acetone is the
desired acetone concentration (that you will be assigned)
and acetone is the specific gravity of acetone.
Prepare a clean, 1cm cell. Make sure that the transparent
windows of the cell are free of any
contamination. Obtain the dark current spectrum.
Fill the cell 2/3 with distilled water, place it into
the spectrophotometer and obtain the background
spectrum. Make sure the transparent walls are in
the optical path.
Add the calculated amount of acetone to the volumetric
flask, quickly fill the flask to the mark with
distilled water and shake it well. Rinse the cell
with a small portion of the solution. Fill the cell
2/3 with the solution and place the cell into the
photometer. Make sure that the transparent
windows are in the optical path. Start to collect
absorbance versus time data for about half an
hour or while the absorbance hits zero.
Turn the lamp off but do not leave the program before
saving your data (read ‘the Data processing and
report filing’ section)!
Results and Calculations
Iodine solution may cause slight skin irritation upon contact.
Notice, that Eqn.( 11.2 ) and ( 11.3 ) have the [H+] and
not cacid . To find out what [H+] is in cacid use Eqn. ( 11.6 )
,
acidacid cKH *][ ( 11.6 )
where Kacid=1.75x10-5, the dissociation constant for
acetic acid.
Prepare a table with the following data:
Time /s At X
Make a plot of as a function of time and carry out a
linear regression to determine the slope and so the rate
constant. Turn in the rate constant, and the plot.
References
Atkins, P.W., "Physical Chemistry", 3rd ed., Freeman: New York,
1986.
Note
Experiment 13
Kinetic Studies with Conductometry-
Determination of Activation Energy
Background
Conductometry is a simple yet very accurate analytical technique to estimate
ion concentration. Because of its simplicity it is widely used in standard
testing procedures in industry.
Benzoyl-chloride is known to hydrolyze in water according to the following
equation:
C H COCl H O C H COOH H Cl6 5 2 6 5
This reaction is very important in many organic syntheses and it has been
studied very extensively. We are going to study it in an acetone-water
mixture in which the rate constant strongly depends on the concentration of
water. Under these circumstances, the reaction follows first order kinetics.
Since it produces ions, it can be followed by conductometry. The reactants
(i.e. benzoyl-chloride and water) are neutral species, therefore they do not
contribute to the conductance. However, the products are charged and they
will contribute to the conductance. The rate constant can be determined by
Eqn. ( 12.1),
ttk
0ln1
( 12.1)
where 0 is the initial conductance, t is the conductance at time t and is
the conductance that corresponds to the solution after the reaction has gone
to completion.
However, sometimes it takes too long to reach completion of the reaction
(like in this case). To solve this problem, we are going to use the
Guggenheim-Method. By using this method, not only does the reaction not
have to reach completion, but we do not need to know the exact initial
concentrations either. The only requirements are that the reaction must be
first order (which is true) and the readings have to be done in equal time
steps.
Let us assume that the conductance is directly proportional to the ion
concentration, therefore it is directly proportional to the hydrolyzed
benzoyl-chloride concentration:
ktkt
t ececctcc 1)( 0000 ( 12.2 )
where c0 is the initial bezoil-chloride concentration, ct is the benzoyl-
chloride concentration at time t and is a proportionality factor. According
to Eqn. ( 12.2 ), the conductance at time (t+t) would be:
)(
0
)(
000 1)( ttkttk
tt ececcttcc
( 12.3 )
When we subtract Eqn. ( 12.3 ) from Eqn. ( 12.2 ) we find:
1
)()(
000
)(
0000
tkkttkktkt
ttkkt
ttt
eeceecec
eccecc
( 12.4 )
Taking the logarithm of Eqn. ( 12.4 ) we have:
ktec tk
ttt
1lnln 0 ( 12.5 )
According to Eqn. ( 12.5 ) if one plots the logarithm of the difference
between two consecutive conductance readings, ln|t-t+t| vs. t, the slope
will give the rate constant.
It is well known that rates of reaction strongly depend on the temperature.
The temperature dependence of the rate constant of many reactions can be
approximated with the Arrhenius equation:
TR
EAk
Aek
a
RTEa
1lnln
/
( 12.6 )
where Ea is the activation energy, A is the pre-exponential factor, and R is
the gas constant (8.314 J mol-1K-1). According to Eqn. ( 12.6 ), a plot of the
logarithm of the rate constant measured at various temperatures (ln k) as a
function of the reciprocal temperature gives a straight line with a slope of -
Ea/R.
In order to determine the activation energy, you have to determine the rate
constant at three different temperatures. Then plot the logarithm of the rate
constant as a function of the reciprocal of the temperature where the rate
constant was determined. From the slope, Ea can be obtained as discussed
above.
Experimental Procedure
1. Make sure the conductometer is assembled properly and the cell is
connected to the interface box. Set the range to 2000S with the switch on
the probe AND in the interface configuaration window. Set the data
collection to 1 sample per second and 1400 s collection time.
2. Make sure there is sufficient water in the thermostating container. If not,
fill it approximately 80% full with distilled water. Set the temperature to
25 oC.
If the thermostat does not have sufficient amount of water, it
overheats and damages the circulator!
3. Put 80 ml of the acetone-water mixture in the reactor (Figure 12.1). Start
stirring slowly (Figure 12.2).
Please note that acetone is more volatile than water, therefore if you
leave the solvent container open during the experiment, the
composition will change
Please note that if the stirring is too fast, air bubbles get into the cell
resulting in unstable readings!
Benzoyl-chloride if exposed to air may cause slight eye irritation. Keep the container with benzoyl-chloride and the reactor closed all the time
4. Add 2 ml of benzoyl chloride to the mixture very quickly (Figure 12.3)
and replace the teflon top with the cell. Start the data collection.
5. When the data collection is completed, suck out the reaction mixture with
the aspirator into the waste bottle (Figure 12.4). Rinse the cell and the
reactor a few times with acetone and suck that out too.
6. Repeat the experiment on 35 oC and 45 oC too.
7. When done, shut off the thermosating bath and the magnetic stirrer.
Results and Calculations
Prepare the following table in your notebook for each temperature:
Time /s /Ms ln|t-t+t|
Plot ln|t-t+t| vs. t for each temperature. You should get a straight line.
When you calculate the ln|t-t+t| values, pick points about 150 readings
apart, but keep the difference constant (i.e. t has to be constant!).
The initial points may not fall on a line since it took some time until the
solution was well mixed. You may ignore those data points. Also, at the end
of the reaction, the concentration of benzoyl-chloride was very low and
there was no significant change between two consecutive readings; ignore
those points too. Carry out a linear regression on the points that follow the
straight line. The slope will give the rate constant.
Repeat the same process at every temperature. Prepare a table such as the
one below:
t /oC T /K T-1 /K-1 k ln k
Make a plot of ln k as a function of T-1. Determine the slope and the
activation energy. Turn in this table, the plot and the value of the activation
energy. Report the exact temperature values at which you carried out the
experiment.
Figure 12.1
Figure 12.2
Figure 12.3
Figure 12.4
Notes
Experiment 14
Determination of Rate Constant and the
Activation Energy for the Decomposition
of a Diazonium Compound
Background
Diazonium ion forms after mixing the hydrochloric salt of an amine with
nitrite solution, according the following scheme:
The diazonium ions are only stable at low temperatures (generally lower
than 70 0C), so at room temperature they begin to decompose according to
the following scheme:
X
Y Z
N N+ ]
Cl-
ZY
X NH3
+
Cl
]-
+ NO2
-
The process, as one can see from the equation above, follows first order
kinetics. The reaction can be monitored spectrophotometrically because
most of the diazonium ions absorb in the UV region and the change in their
concentrations can be followed. However, in this experiment we are going
to monitor the progress of the reaction by trapping the released nitrogen gas.
To evaluate the experimental data, the Guggenheim-method will be used. In
order to use the Guggenheim-method, the reaction must have a physical
property that can be monitored and it is directly proportional to the
concentration of the reactant and/or one of the products. In this case, that
will be the volume of nitrogen gas which is proportional to the number of
moles of diazonium decomposed. Additionally the process must follow first
order kinetics. A further criterion is that the data must be collected in equal
time intervals, i.e. the time between any two data points should be constants.
In this case the following equation can be used:
ln ( ) ln V t e ktN
k t
21
( 13.1)
where V tN2( ) is the volume increment of nitrogen gas as a function of
time (collected with constant t) and is a constant (for a detailed
derivation of equation (1) refer to the Appendix). According to (1), the plot
of ln{ V tN2( ) } vs. t will yield a straight line with the slope of -k, which is
the rate constant.
Experimental Procedure
X
Y Z
N N +
] Cl
-
X
Y Z
OH + N 2
1. Place 10 ml 1M NaNO2, 10ml of your unknown in 25 ml Erlenmeyer
flasks and 50 ml of 1M HCl in a 100ml Erlenmeyer flask for 30 min on
crushed ice. The experimental apparatus is shown in Figure 13.1. You will
perform the experiment at three different temperatures between 10 oC 50 oC.
While the actual values of the temperatures is not important, make sure they
are spaced out. Start the thermostat and set the first desired temperature.
2. Test the setup for air tightness. Assemble the data acquisition setup
(probe, interface, computer). Setup the program to capture readings at every
30 s. Have a graph monitoring the pressure.
3. Once the temperature is constant, wait 10 more min. FIRST put the
NaNO2 solution and THEN the unknown solution from the ice into the 100
ml Erlenmeyer flask with the HCl in it slowly and while you are swirling the
HCl solution. Then after mixing, pour the solution into the reactor and close
very tightly with the rubber stopper. You should get a colorful, but yet
transparent solution. If a colorful precipitate forms, call your T.A.; you will
have to repeat the preliminary steps.
4. Immediately after you added the unknown, start the data collection.
Continue to collect data until the pressure levels off.
5. Once you are done running the experiment at three different
temperatures, turn off the thermostat and dispose of the contents of the
reactor into the appropriate waste bottle.
Results and Calculations
Create a spreadsheet with your experimental and calculated data (see Figure
13.2). Calculate the amount of nitrogen gas that was formed under
atmospheric pressure using the following equation:
0
0
1)(
)( Vp
tptV
( 13.2)
where V(t) is amount of nitrogen formed if the pressure was atmospheric
pressure by time t, p(t) is the total pressure at time t, p0 is the atmospheric
pressure, p(t) is the total pressure at time t, and V0 is the pressure of the
reactor. Calculate the V values at time t by using the equation
Appearance of a color precipitate upon adding the unknown indicates an undesired side reaction. If that happens, call your T.A.
Some of the diazonium compounds decompose violently! Do not spill any of the contents of the reactor! Upon drying the spill may explode!
V t V t V t t( ) ( ) ( )
where V(t) is the volume difference at time t, V(t) is the reading at time t
and V(t-t) is the reading at time t-t (i.e. previous reading). In the
following example,
Time /s V/ml dV
120 12
150 13.5 1.5
180 14.6 1.1
the V value at t=150 s equals 13.5-12.0=1.5 ml and at t=180 s 14.6-
13.5=1.1ml. Plot ln(V) vs. t. You should get similar results to those
obtained in Figure 13.3. Locate the portion of the plot where the points are
more or less on one line (i.e. ignore the initial part) and carry out a linear
regression. The points where you had to reset the buret will be missing.
Determine the slope (and the value of k). With the obtained slope and
intercept calculate the expected ln(V) values at every time point and list
them in your table. Plot the calculated values and connect them with a line
(see Figure 13.3).
Diazonium compounds are suspected to cause cancer. In case of skin contact, wash it off with plenty of water AND call your T.A.
Figure 13.1
t /s p /Pa V(t) /mL ΔV /mL ln(ΔV)
Figure 13.2 Table for Experimental and Calculated Data
Appendix
Derivation of equ.(1):
n t n n t
n t
Vc c t
n t
Vc c e
n t
Vc e
N D D
N
D D
N
D D
kt
N
D
kt
2
2
2
2
0
0
0 0
0 1
( ) ( )
( )( )
( )
( )
where n tN2( ) is the number of moles of nitrogen, and n tD ( ) is the number
of moles of diazonium ion at time t and nD
0 is the initial diazonium ion
concentration. Concentrations are indicated the same way. V is the volume
of the solution after mixing. The number of moles of nitrogen can be
substituted from the ideal gas law:
pV t n t RT
n tpV t
RT
N N
N
N
2 2
2
2
( ) ( )
( )( )
Figure 13.3 Typical plot of ln(V) vs. t (Experimental data)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 2 4 6 8 10 12 14 16 18 20
time /min
ln(d
V)
and after substitution:
V t p
VRTc e
V tVRTc
pe e
N
D
kt
ND kt kt
2
2
0
0
1
1 1
( )
( )
where V tN2( ) is the volume of released nitrogen at time t.
The next reading is to be taken in t and follow the same derivation as
above:
n t t n n t t
n t t
Vc c t t
n t t
Vc c e c c e e
n t t
Vc e e
N D D
N
D D
N
D D
k t t
D D
kt k t
N
D
kt k t
2
2
2
2
0
0
0 0 0 0
0 1
( ) ( )
( )( )
( )
( )
( )
V t t p
VRTc e e
V t tVRTc
pe e e e
N
D
kt k t
ND kt k t kt k t
2
2
0
0
1
1 1
( )
( )
The difference in volume of nitrogen between the two readings can be
evaluated:
V t V t t V t e e e
e e e e e
N N N
kt k t kt
kt kt k t kt k t
2 2 21 1
1
( ) ( ) ( )
Taking the logarithm of the obtained expression:
ln ( ) ln
ln ( ) ln
V t e e
V t e kt
N
kt k t
N
k t
2
2
1
1
Derivation of equ.(2):
By the time t, the amount of gas in the reactor is:
2)( Nnntn
where n0 is the number of moles of air in the reactor initially, n(t) is the total
number of moles of gas by the time t, and nN2 is the number of moles of N2
formed by the time t. Using the ideal gas law:
)(1)(
)()(
)()(
)()(
)()(
)()(
0
0
0
00
000
0000
0000
0000
tVVp
tp
tVp
Vptp
tVpVptp
tVpVpVtp
tVpVpVtp
RT
tVp
RT
Vp
RT
Vtp
where V(t) is the volume of N2 formed by the time t under atmospheric
conditions.
Experiment 15
Measurement of Refractive Index
Background
Although the use of refractive index for identification of unknowns has been
largely supplanted by spectroscopic methods, refractive index measurements
still provide a useful analytical tool for determining the purity of substances
or composition of mixtures (e.g. in distillation) and for following the course
of reactions or purification processes.
Experimental Procedure
Your unknown consists of two solvents. You have to determine the composition of
your unknown. Obtain the specific density of both components. You will have to
prepare four standard solutions with known composition. Make a table with the
necessary data (see Table 1). Prepare 0.1 mole of the standard solutions. From the
specific density of the components, calculate the mass and the volume to be used
in each standard solution according to the following formulas:
m n Fw
Vm
*
where m is the mass in g, n is the desired number of moles, Fw is the
formula weight, V is the volume in ml, is the specific density. Use two
Unknown, standard solutions and the pure components must be disposed into the organic waste bottle. Do not put the pure components back into the original container to avoid contamination!
burets to prepare the standard solutions. Measure the solvents into vials and
close them quickly to avoid evaporation.
You have to measure the refractive index of your unknown with the Abbe
refractometer (see Figure 14.1), the standard solutions and the pure
components. To measure refractive index, follow the following steps:
Turn on the cooling water at the tap and turn on the circulating pump, which
passes water through the prisms of the instrument. Check the
thermometer and adjust the thermostat so that the temperature in
the bath is at 25 ± 0.1C.
Clean prisms of the refractometer (labeled 2 and 5 in Figure 14.1) with a soft
cloth or cleaning tissue (Kimwipe) and acetone (USE CARE).
Place a few drops of distilled water on the prism using a medicine dropper or
pasteur pipette. Be careful not to scratch the soft glass prism!
Gently close the prism assembly. It may be necessary to add a few more
drops of water from time to time because of evaporation.
Turn on the main switch of the instrument, which is a rocker switch located
on the power cord of the instrument.
Look through the eyepiece slowly and move the arm bearing the light bulb
until you see a field divided into dark and light portions (see Figure
14.1). The light bulb is now at the proper position for most liquids.
Before taking the final reading, sharpen the line of demarcation between the
light and dark fields by rotating the fine adjustment knob on the
right hand side of the instrument.
The refractive index is then read through the eyepiece by pressing the button
marked "Light" on the left hand side of the instrument. The
refractive index of pure water at 25 is 1.3324. If you do not get this
reading within 0.001 consult the instructor.
Clean the prisms as in Step 2 (USE CARE) and add a few drops of the
unknown liquid to the prisms. More liquid may need to be added if
evaporation occurs.
Turn the knob on the right hand side of the instrument until the divided field
appears in the eyepiece (see Figure 14.1). Sharpen this line of
demarcation and make the final reading in the same manner as with
water.
Clean the prisms carefully (use alcohol or acetone if necessary) of all
substances and leave the prisms slightly ajar to dry.
Results and Calculations
Fill the entry in the table. Plot the refractive index as a function of
composition for the standards and pure components (see Figure 14.1). From
the graph, based on the refractive index of your unknown, determine the
composition of the unknown.
Turn in the plot and the result sheet.
Figure 14.1
# xA xB NA nB mA MB VA VB nD
1 2 3 Average
Pure B 0.00 1.00 0.00 0.10
1 0.20 0.80 0.02 0.08
2 0.40 0.60 0.04 0.06
3 0.60 0.40 0.06 0.04
4 0.80 0.20 0.08 0.02
Pure A 1.00 0.00 0.10 0.00
Unknown N/A N/A N/A N/A
Table 15.1
Necessary Data for Refrective Index Measurements
(A and B are the components, x is the mole fraction, n is the number of moles, m is the mass in g, V is the volume in ml)
Experiment 16
Determination of Heat Capacity of Gases
With Ruchhardt’s Method
Background
The Ruchhardt experiment is a classic method to measure γ for a gas; i.e.,
the ratio of heat capacity at constant pressure to that at constant volume.
The original apparatus consisted of a steel ball which frictionlessly can
move in a glass tube attached to a gas reservoir (see Figure 15.1). The piston
of mass m is displaced from equilibrium and released to oscillate in damped
simple harmonic motion. The frequency of oscillation is measured and this
measurement is combined with the known physical parameters of the system
to estimate γ.
Figure 15.1
The pressure inside the reservoir is:
0
mgp p
A ( 15.1)
When the ball is removed from its equilibrium, the force returning the ball to its equilibrium
is given by:
2
2
d xA dp m
dt ( 15.2)
where A is the surface of the ball, dp is the pressure change due to the displacement of the
ball and m is the mass of the ball. Since the oscillation of the ball has rather high frequency,
the compression-expansion of the gas in the reservoir can be considered for practical
purposes adiabatic:
.pV const ( 15.3)
Taking the derivative of both sides yields:
( 1)
( 1)
0V dp p V dV
p V pdp dV dV
V V
( 15.4)
The displacement of the ball (x) will change the volume of the gas with dV:
dV Ax ( 15.5)
Substituting ( 15.5) into ( 15.4) yields:
22
2
2
2
p d xA x m
V dt
d xkx m
dt
( 15.6)
This equation is Hook’s Law which characterizes a simple harmonic motion, where k is the
Hook constant. The Hook constant is related to the angular frequency () of the oscillation.
The angular frequency is also related to the period (T) of the oscillation:
2
2
22
k p A
m mV
mVT
p A
( 15.7)
From ( 15.7) one can obtain:
2
2 24
mV
pT A ( 15.8)
According to ( 15.8) by measuring the period of the oscillation, one can calculate the ratio
of the heat capacities.
Experimental Procedure
1. Assemble the experimental setup. Connect the pressure/temperature probe
to the GLX and the GLX to the computer. Make sure that the pressure is
measured by Pascals. Also, configure the experiment for “Delayed start”
(when pressure exceeds 160,000 Pa) and “Delayed stop” (collect data for
2s).
2. Connect the pressure probe to Port A on the piston and release the
corresponding clamp (see Figure 15.2). Open Port B by releasing the
corresponding clamp.
3. Pull the piston all the way up and close Port B with the clamp. The piston
should stay.
4. Prepare a table in Excel with the following headings:
m (g) T (s) x (mm) V (m3) p (Pa)
where m is the mass of the piston, T is the period of the oscillation, x is the
initial position of the piston, V is the volume of the gas in the cylinder, p is
the initial pressure, and is the ratio of the heat capacities. The diameter of
the piston is 32.5 mm and the mass of the piston is 35g. For V and enter
the right formulae.
5. Record the initial position of the piston, and the initial pressure by
switching on data monitoring in DataStudio. Stop data monitoring.
6. Start the data collection. Push down the piston with your thumbs by the
edge of the plastic plate. Make sure you apply equal pressure by the two
thumbs otherwise you will break off the plastic plate.
7. When the pressure exceeds 160,000 Pa, the data collection starts. Release
the plate suddenly. The data collection will continue for 2s.
8. Perform the same experiment with a 200g and then with two 200g weights
taped securely to the plastic plate.
9. Repeat the experiment with CO2 loaded into the piston through Port B
with all three weight combinations.
10. Record your data into the Excel spreadsheet.
Results and Calculations
From the graph, determine the period of the oscillation for each of the
measurements by using the smart tool. Alternatively, if there is at least a few
oscillations, highlight the oscillations and fit a sine function on them. Then
read the period of the oscillationfrom the parameter window.
Calculate average and standard deviation for the heat capacity ratio values
for both, the air and for CO2.
Figure 15.2
Experiment 17
Reversible Work of Gases
Background
A perfect gas can do the most expansion work under isothermal reversible
conditions. It is because every step the way the internal pressure is matched
with the external pressure therefore none of the pushing power of the gas
goes wasted. Since the pressure inside and outside are always matched, at
any given time with an infinitesimal change the expansion process can
reverse (or push forward), hence, the process is reversible. We will simulate
that situation by allowing water to drain slowly from a container on the
piston. Initially, the pressure imposed by the weight of the water will be
equal to the pressure inside of the system. Once a few drops of water are
drained (“infinitesimal” change of the pressure), the internal pressure will be
slightly higher, the piston will move up until the pressure will equalize
again. The external pressure will gradually decrease, always a slightly lower
than the internal pressure resembling reversible conditions.
The expansion work done by a gas is given by
f
i
V
V
w pdV ( 16.1)
Since the pressure inside the container (and since the process is reversible
also on the outside) changes with every step of the volume change:
ln
f
i
V
f
iV
VdVw nRT nRT
V V ( 16.2)
In this experiment you will measure the expansion work done by an ideal
gas sample by measuring the integral of the p = f(V) function in the range
where the gas expands. You will also measure portion of the expansion
work that the gas uses to perform mechanical work and you will determine
the associated efficiency.
Experimental Procedure
1. Assemble the experimental setup. Connect the pressure/temperature probe
and the rotary motion sensor to the GLX and the GLX to the computer.
Make sure that the pressure is measured by Pascals. Also, connect the
balance to a USB port of the computer (see Figure 16.1).
2. You need to configure DataStudio as follows:
Add the balance as a new instrument in the Setup menu (Setup
button/Add sensor or instrument button/Instruments from the drop
down list/OHaus Balance)
Configure the rotary motion sensor (Setup button/Select the Rotary
motion sensor/Measurement tab/Position has to be checked
only/Rotary motion sensor tab/Large Pulley in the Linear Scale
drop down box)
Configure the Pressure/Temperature sensor for measuring the pressure
in Pa and the temperature in K (Setup button/Select the
Pressure/Temperature sensor/N/m^2 has to be selected for the
pressure and K for the temperature)
Create two graphs: pressure vs. position and mass vs. position.
Create two digital displays: position and mass.
Configure DataStudio for delayed start (Experiment/Set sampling
options/Delayed start tab/Time 5s).
3. Close the clamp on the tubing. Raise the piston to the maximum while you
allow air to flow into the piston by disconnecting the hose from the
pressure probe. Reconnect the tubing to the pressure probe.
4. Pour about 60g of water into the bottle on the piston slowly.
5. Setup the rotary motion sensor by putting a thread on the large pulley,
connecting the thread to the bottle on the piston and to the counterweight.
Make sure that when the piston moves up, the displacement will be
positive (turn on data monitoring by Alt-M and move the piston up a little
and observe on the screen if the displacement is positive. If not, take off
the pulley, turn it around and mount the pulley on the other side).
6. Zero the balance with the empty container on it. Record the original
position of the syringe (on the scale of the syringe).
7. Click on the Start button. You have 5s to open the clamp before the data
collection starts. Open the clamp just enough so that the water will start to
flow but the clamp wouldn’t fall off (otherwise it will fall on the balance
and falsifies the mass measurements).
8. Continue to collect data until there is water in the container. Record the
final position of the syringe (on the scale of the syringe).
Figure 16.1
Results and Calculations
Once you competed the data collection, you need to create two calculated
columns in DataStudio. Click on the Calculate button on the main toolbar
and create the following two datasets:
A dataset that shows the mass on the piston at every point calculated
from the mass reading from the balance (see Figure 16.2). To select
the reading from the balance for x click on the down-pointing
triangle button, and select Data Measurement and select the Mass
readings (see Figure 16.3).
Figure 16.2
Figure 16.3
A dataset that shows the volume of the gas at every point calculated
from the original volume of the gas (5.100L) and from the position
and the diameter of the piston (see Figure 16.4). Select the Position
measurements for x the same way as you selected the mass
measurements in the previous step).
Figure 16.4
On the Pressure vs. Position change the independent variable to the
calculated volume dataset by clicking on the label and selecting the volume
from the list (see Figure 16.5).
Figure 16.5
On the Mass vs. Position change the dependent variable to the calculated
mass dataset by clicking on the label and selecting the volume from the list
(see Figure 16.6).
Figure 16.6
Calculate (in Excel):
The expansion work in SI units from the p vs. V. area
The expansion work from the integrated formula
The mechanical work done by the gas from the area under the m vs.
Position graph (m is the mass on the piston!)
The yield (how many percent of the work done by the gas was used
for mechanical work).
Parameters:
Volume of gas container: 5100L
Diameter of syringe: 32.0 mm
Mass of syringe: 35.00 g
Mass of empty bottle on syringe: 44.70 g
Calibration of syringe: 0.83 mL/mm
Counter weight: 10.2 g
(Reversible Expansion work2.xls, Reversible expansion work2.ds)
Experiment 18
Determination of the Ratio of Specific
Heats of Gases 2.
Background
For ideal gases the heat capacity is defined as:
V
V
UC
T
( 17.1)
The First Law of Thermodynamics is:
dU dq dw ( 17.2)
For an adiabatic process there is no heat exchange between the system and
its environment.
0 V
V
dU pdV C dT
pdV C dT
( 17.3)
Assuming that the gas obeys the Ideal Gas Law, the following derivation
holds:
1
pV nRT
pdV Vdp nRdT
pdV Vdp dTnR
( 17.4)
Substituting of ( 17.4) into ( 17.3) yields (using the Cp-CV = nR relationship):
( 1)
0
v
v
p v
p v
v
CpdV Vdp pdV
nR
CpdV Vdp pdV
C C
C CpdV Vdp pdV
C
pdV Vdp pdV
Vdp pdV
dV dp
V p
( 17.5)
where = Cp/Cv, the ratio of the specific heats at constant pressure and
constant volume, respectively. Integrating ( 17.5) yields:
ln lnV p k
pV k
( 17.6)
Eliminating pressure from ( 17.4) would yield a different form of ( 17.6):
1TV k ( 17.7)
Considering a gas that changes from T1, p1, V1 to T2, p2, V2 these equations
would have the form of:
1 1 2 2
1 1
1 1 2 2
pV p V
TV T V
( 17.8)
The ratio of specific heats depends on the nature of the gases. According to
the equipartition theorem, each degree of freedom contributes with R/2 to
the value of CV. Therefore a mono-atomic gas (that has three translational
degrees of freedom) has the value of CV=3/2R with Cp = 5/2R and = 5/3. ,
The same way a gas with di-atomic molecules (with three translational
degrees of freedom and two rotational degrees of freedom around the axes)
would have the value of CV=5/2R with Cp = 7/2R and = 7/5.
The expansion work done by the gas is given by (using also ( 17.6) and (
17.8)):
22 2
1
1
1 1
1 1 2 2
1 1
2 2 2 1 1 1 2 2 1 1
( )1
1 1
VV V
adiabatic
V V V
adiabatic
dV Vw pdV k pV
V
k pV p V
p V V pV V p V pVw
( 17.9)
Experimental Procedure
1. Assemble the experimental setup. Connect the pressure, temperature, and
position ports of the Adiabatic Gas Apparatus (AGA) to a GLX (see
Figure 17.1). The output from the measured quantities are given by
voltage signals and each signal has to be calibrated accordingly.
Figure 17.1
2. In DataStudio, add the three channels of input as “User defined sensor”
in the Setup section.
3. Configure DataStudio for data collection; create a graph with all three
measured quantities and three digital displays.
4. Put DataStudio in data monitoring mode and read the signal from the
position monitor at the two extrema of the piston.
5. Configure the data collection for delayed data collection (Experiment/Set
Sampling Options/Delayed Start) by setting the position channel to “Is
below” and a slightly below the value that you obtained when the piston
was all the way up (e.g. if the maximum value was 4.4V, set it to 4.35).
This way when the lever is moved down slightly, the data collection starts
automatically. Also set the “Automatic Stop” to 15 s.
6. Open one stopcock, move the piston all the way up, and close the
stopcock.
7. Start the data collection and with a sudden move the lever all the way
down and hold it down until the data collection is going (15s).
8. Repeat the experiment with adiabatic expansion; Configure the data
collection (Experiment/Set Sampling Options/Delayed Start) by setting
the position channel to “Is above” and a slightly above the value that you
obtained when the piston was all the way down (e.g. if the minimum value
was 0.17V, set it to 0.2). This way when the lever is moved up slightly, the
data collection starts automatically. Also set the “Automatic Stop” to 15
s.
9. Open one stopcock, move the piston all the way down, and close the
stopcock.
10. Start the data collection and with a sudden move the lever all the way up
and hold it up until the data collection is going (15s).
11. Run He and CO2 as well. To fill the piston with a particular gas connect
the gas line to one of the stopcocks, open it and allow the gas to fill the
piston. Then close the stopcock, open the other stopcock and move the
piston down to flush the cylinder. Repeat this step 4-5 times to eliminate
all the air.
12. Repeat steps 5-10 to collect data for the particular gas.
Results and Calculations
1. Once you competed the data collection, you need to create three calculated
columns in DataStudio to convert the voltage signals to the respective
physical quantities. Click on the Calculate button on the main toolbar and
create the following two datasets:
A dataset that converts the position readings to volume (see Figure
17.2). To select the reading from the position input for x click on the
Fill the cyliner with gases very slowly! Too fast gas flow will damage the sensors in the cyliner!
down-pointing triangle button, and select Data Measurement and
select the appropriate channel.
Figure 17.2
A dataset that converts the signal input to pressure (see Figure 17.3).
Select the appropriate channel for x.
Figure 17.3
A dataset that converts the signal input to temperature (see Figure
17.4. Select the appropriate channel for x.
Figure 17.4
2. Create a table in Excel for all of your data. Create the (p,V,T) vs. time
and p vs. V graphs.
3. With the Smart Tool read the initial and final values for p, V, and T and
calculate the values.
4. Determine the expansion work from the p vs. V graph with the Area
Tool and calculate it with ( 17.8).
5. Repeat the same calculations for each gas.
6. Compare your results to the theoretical values.
(adiabatic.ds, adiabatic.xls)
Experiment 19
Inversion of Sucrose
Background
Molecules with asymmetric carbon atom(s) are known to rotate the plane of
polarized light. Enantiomer molecules rotate the plane equal extent,
however, to the opposite directions. The extent of rotation is characteristic to
the particular molecule, however, it also depends on the path length that the
light has to pass, the wave length of the incident light, and the concentration
of the solution. The rotation is characterized with the specific rotation which
is defined as:
cl
][ ( 18.1)
where is the wavelength of the incident light, is the temperature, is
the rotation, c is the mass-concentration (kg/m3), and l is the length of the
cell with the solution. Usually Na-lamp is used (= 589 nm) and the
temperature is 20 oC and there are standard length cells (10 cm, 20 cm, etc.).
The extent of the rotation is measured with a polarimeter (see Figure 18.1).
Polarized light is mdade with a Nicol prism (called polarizer) which
eliminates all but the vertically oriented electromagnetic waves (see Figure
18.2). Then the vertically polarized light is transmitted through the optically
active solution which rotates the vertical plane. Then the light passes an
other Nicol prism (called the analyzer) which has to be rotated to align it
with the plane of rotation. The extent of the adjustment is the optical
rotation.
Figure 18.1
Figure 18.2
In this lab you will study the decomposition of sucrose (called “inversion”)
to fructose and glucose. You will monitor the extent of the reaction by
monitoring the optical rotation of the reaction mixture. Sucrose and glucose
rotates the plane of polarized light clockwise. Fructose, however, rotates the
plane counter-clockwise. Since the specific rotation of fructose is larger than
the specific rotation of glucose, in the end of the reaction the solution will
rotate the plane of polarized light counter-clockwise.
FructoseGlucoseSucrose
612661262112212
HOHCOHCHOHOHC
The rate of inversion is:
pnm HOHOHCkdt
OHCd][][][
][2112212
112212 ( 18.2)
Since water is in a great excess and the H+-ions are only used as catalyst (ie.
their concentration is constant):
pn
m
HOHkk
OHCkdt
OHCd
][]['
]['][
2
112212112212
( 18.3)
Assuming that the reaction is first order in sucrose (m = 1), the integrated
rate equation will be:
tkeOHCOHC '
0112212112212 ][][ ( 18.4)
We will monitor the changes of concentrations with monitoring the rotation,
α. Initially only the sucrose is responsible for the rotation (α0). Once the
reaction came to completion, both products will be responsible for the
rotation of the plane of the polarized light. The rotation of the solution then
is . If the rotation of the solution during the reaction is α, then the
following equation holds (for derivation see Appendix):
tk 'ln0
( 18.5)
0ln'ln tk ( 18.6)
Experimental Procedure
1. Turn on the Na-lamp of the polarimeter (see Figure 18.3). Allow the lamp
to warm up for 30 min. Turn on the thermostating bath and set the
temperature to 20 oC.
Figure 18.3
Figure 18.4
2. Prepare 100 mL 20 g/100mL sucrose solution and 250 mL 4M HCl
solutions.
3. Transfer 37.5 mL 20g/100 mL sucrose solution into a 125 mL Erlenmeyer
flask and mount it into the water bath.
4. Transfer 12.5 mL 4 M HClsolution into a 125 mL Erlenmeyer flask and
mount it into the water bath.
5. Hold the polarimeter cell vertically so that the wide neck is upward (see
Figure 18.4). Remove the top cap. Fill the cell with water until only about a
mm is left. Securely close the cell with the cap. Place the cell into the
water bath.
Please handle the window pieces of the cell carefully! They have
multiple pieces and they may fall apart when you empty the cell. Do not
empty the cell over the sink because if the window piece does fall apart,
the pieces might fall into the sink
6. After 30 min, remove the cell from the bath.
7. Wipe the side of the cell dry with paper towel. Wipe the windows of the
cell with Kim Wipes dry. Tilt the cell carefully so that the small bubble
would be trapped in the wide neck. Place the cell into the bed in the
polarimeter (see Figure 18.5). Make sure that the bubble remains in the
wide neck. Close the cover.
Figure 18.5
Figure 18.6
8. With the eyepiece adjust the focus (see Figure 18.6). Turn the adjusting
knob until the contrast between the dark and illuminated areas is the best.
This will require turning the knob back and forth very little at a time. Read
the scale in degrees. Please note, there is a caliper scale as well to read the
first decimal. Record your reading as “Blank”. Empty the cell.
9. Combine the sugar and HCl solutions, mix the solution, rinse the cell with
a little of the solution. Fill the cell the same way as you did with water.
Dry the wall of the cell with paper towel and the windows with Kim
Wipes. Place the cell into the bed so that the small bubble is trapped in
the wide neck. Close the cover.
10. Start the stop watch.
11. Turn the adjusting knob until the contrast between the black and
illuminated areas is the best. Take a reading. This is your initial rotation,
α0.
12. Place the cell back into the water bath.
13. Take readings of the rotation in every 10 min until there is change in the
value of the rotation. Your last reading (which won’t change anymore)
will be .
14. Once done, set the temperature first to 30 oC and repeat the experiment.
15. Finally, set the temperature to 40 oC and repeat the experiment.
16. When done, carefully clean the cell with water and turn off the
thermostating bath.
Results and Calculations
1. Arrange your data into a table in Excel. Subtract the blank from your
readings.
2. Since the logarithmic operation skews the error in the experimental data,
fit a
caey bt
type of equation onto your data to obtain a, b, and c witch translate to
)( 0 , k’, and - respectively (see Equation ( 18.6) ) using the
Solver tool in Excel.
3. Calculate the value of k at each temperature.
4. Perform and Arrhenius analysis to calculate EA and A.
5. Calculate the specific rotation for sucrose and for the mixture of fructose
and glucose. Calculate the error of your measurements if the literature
values for specific rotation are:
020][ D
(deg cm3/g dm)
Sucrose +66.54
Fructose -93.78
Glucose +52.74
Experiment 20
Enzyme Kinetics
Background
A living cell has great many reactions occurring simultaneously. These
reactions are coordinated like a very precise machinery and each is
responsible for one step and part of one or more cellular function. Most of
these reactions could not happen under the thermodynamical circumstances
of the living cell (pressure, temperature). The presence of special catalyst
proteins make the occurrence of these reactions possible. Each of these
proteins, called enzymes, catalyze on specific reaction. The word “enzyme”
was coined by Kühn in 1878 from Greek words meaning “in yeast”,
reflecting the popularity of yeast research in the late 19th century. Most
enzymes can be identified by the suffix “-ase” and a prefix that usually
indicates the type of reaction the enzyme catalyzes.
A catalyst lower the activation energy by forming a complex with the
reactant(s). Enzymes are catalysts of biological chemical reactions and they
lower the energy of activation in a manner similar to catalysts of inorganic
and organic (but non-living) chemical reactions. Enzymes are generally
much more efficient than chemical catalysts and are capable to acting within
a limited range of temperature and at one atmosphere of pressure.
The reactant(s) of an enzymatic reaction is called the substrate. When a
given substrate (S) is converted to a single product (P) in a reaction
catalyzed by an enzyme (E), the reaction occurs by the formation of an
enzyme-substrate complex (ES) at the active site. The product is freed from
the active site as the catalysis is completed. This reaction sequence is is
based on the formulation of L. Michaelis and M.L. Menten (1913):
b
aa
kEPES
kkESSE
', ( 19.1)
The rate of product formation is
][ESkv b ( 19.2)
Assuming the steady state approximation for the intermediate complex:
][
]][[
]][[]][[][
0][][]][[][
'
'
'
ES
SE
k
kkK
K
SESE
kk
kES
ESkESkSEkdt
ESd
a
baM
Mba
a
baa
(
1
9
.
3
)
Where KM is the Michelis constant. Usually [S]0 >> [E]0 therefore if we
study initial rates, we can assume that the substrate concentration does not
change during the initial period of time. Furthermore considering that [E]0 =
[E]+[ES]:
0
0
0
0
0
0
][1
][][
][
][][][
][
][][
][][][
S
K
EES
S
ESKESE
S
ESKE
ESEE
M
M
M
( 19.4)
Substituting [ES] into Equation ( 19.2):
0
0
][1
][
S
K
Ekv
Mb
( 19.5)
If [S]0 << KM :
00 ][][ ESK
kv
M
b ( 19.6)
If [S]0 >> KM the rate is maximum and independent from [S]0:
C
o 0max ][Ekvv b ( 19.7)
mbining ( 19.5) and ( 19.7) yields:
0
max
][1
S
K
vv
M
( 19.8)
Transformation of ( 19.8) that is suitable for linear analysis yields:
0maxmax ][
111
Sv
K
vv
M
( 19.9)
The plot made with is called the Lineweaver-Burk plot from which we can
get KM and kb if [E]0 is known.
The turnover frequency characterizes the effectiveness of an enzyme. It is
the frequency at which the active site of the enzyme makes a product
molecule therefore it is equal with kb:
a
a
a
An other way to characterize the effectiveness of an enzyme is the catalytic
efficiency:
a
During this lab you will study the Trypsin-catalyzed hydrolysis of DL-
arginine p-nitroanilide hydrochloride (D,L-BAPA):
The reaction will be monitored by following the production of p-nitro-
aniline at 410 nm with a spectrophotometer. The molar absorption
coefficient of p-nitro-aniline is 8,800 (Mcm)-1.
0
max
][E
vkk bcat ( 19.10)
M
cat
K
k ( 19.11)
Experimental Procedure
1. The experimental setup is shown in Figure 19.1.
Figure 19.1
Figure 19.2
2. Start the lab with:
Turn on the thermostat and set the temperature to 37 oC. The cell
holder for the spectrophotometer is thermostated also since enzyme
reactions are very sensitive to the temperature. Place a clean cell into
the holder (see Figure 19.2). Try to keep the cell in the holder as
much as possible.
Pipet 1.00 mL 0.001M HCl into three Eppendorf tubes and set them
on ice.
3. Prepare the following stock solutions:
a. 500 mL 40 mM TRIS buffer solution with 50 mM CaCl2. Set the pH
of the solution between 7.0 - 7.2 with 4M HCl solution added drop
wise.
b. Weigh 43.5 mg of the substrate (FW. 434 g/mol) in an Eppendorf tube.
Add 1.00 mL DMSO. Vortex the tube to make a homogenous solution.
Make two identical tubes. Set them in the “floater” in the 37 oC water
bath (see Figure 19.1).
c. Put a Trypsin tablet into each of the Eppendorf tubes on ice. Sonicate
the tubes for a minute and vortex them after long enough to get a
homogeneous solution. Try to minimize the time that the tubes are off
the ice as the enzyme tends to hydrolyze at higher temperature. Keep
the tubes on ice during the experiment.
4. Prepare sample tubes for the substrate concentrations listed in the table below.
You will need 5.00 mL sample in each tube. Calculate the volume of the
substrate solution that you need for each sample. The sample with the highest
substrate concentration will have only the substrate and TRIS solution;
calculate how much TRIS solution you would need. The other samples will
have the same amount of TRIS solution but less substrate solution. Make up
for the difference with pure DMSO.
[S] (mM)
v(S) (mL)
v(DMSO) (mL) v(TRIS)
(mL) v (total)
(mL)
4.00
5.00
3.50
3.00
2.50
2.00
1.50
1.00
0.75
0.50
0.40
0.30
0.20
0.10
5. Pipet the calculated amount of TRIS solution and DMSO into each sample
tube. Prepare a blank as well without any substrate solution. Set the sample
tubes into the 37 oC water bath in a test tube rack.
6. Use the blank solution to run the blank in the RedTide spectrometer. Place the
cell in the holder and save the dark scan. Turn on the light source with the
switch on the box (see Figure 19.2) Note, the light is not controlled on the
interface as usual. Save the reference scan. Configure the RedTide to take
kinetic measurements at 410 nm.
7. Take the first sample solution and add the calculated amount of substrate
solution. Replace the cap and mix the solution.
8. Pipet 0.90 mL of the solution from the sample tube into a clean Eppendorf
tube. Pipet 0.100 mL of the enzyme solution to it. Close the tube and mix
well.
9. Transfer the solution into the clean spectrophotometer cell. Take the
absorption spectrum at 410 nm for 200 s.
10. Repeat the same process with all of the other samples.
11. When done, turn off the light source.
Results, Calculations
1. Transfer the absorption traces into DataStudio. Create a
calculated dataset for each run with smoothing over 10 points.
2. Transfer your data to Excel. Transfer the absorption traces to
concentration traces knowing that the molar absorption
coefficient of the product is 8,800 (Mcm)-1. Calculate the initial
rates for each run. Consider the data collected between 0-120 s.
3. Make a plot showing the concentration of the product as a
function of time.
4. Make the Lineweaver-Burk plot. Calculate KM, kcat . The
formula weight of Trypsin is 2.33x104 g/mol.