Heat Capacity Theory and measurement of heat capacity ratio, g
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Heat capacity is the measurable physical quantity that characterizes the
amount of heat required to change a substance's temperature by a given
amount.
Units: In the International System of Units (SI), heat capacity is expressed
in units of joule (J) per Kelvin (K). [J × K-1]
Specific heat capacity is the heat capacity per unit mass of a material.
Units: specific heat capacity is expressed in units of joule (J) per
kilogram (kg) × Kelvin (K). [J × kg-1 × K-1]
In chemistry, heat capacity is often specified relative to one mole and is
called the molar heat capacity.
Units: molar heat capacity is expressed in units of joule (J) per mol (mol)
× Kelvin (K). [J × mol-1 × K-1]
Definition and units
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Thermodynamic definition
The internal energy dU of a closed system changes either by adding or
removing heat dQ to the system, or by the system performing work dW or
having work done on it.
dU = dQ – dW
If the process is performed at constant volume, then the second term of
this relation vanishes and one readily obtains
V
V V
U QC
T T
CV is the heat capacity at constant volume.
For work as a result of compression or expansion of the system volume dV:
dU = dQ – PdV
Heat capacity at constant volume
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
At constant pressure we have
Thermodynamic definition Heat capacity at constant pressure
Using dU = dQ – PdV it can be simplified to
dH = dQ + VdP
P
P P
H QC
T T
CP is the heat capacity at constant pressure.
The enthalpy H of the system is given by
dH = dU + d(PV)
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Heat Capacity Ratio g
P
V
C
Cg
By the definition:
Using the thermodynamics one can show, that for ideal gas
P VC C R
Or
1V
R
Cg
Where R = 8.31447 J × K-1 × mole-1 is the universal gas constant
P
P P
H QC
T T
V
V V
U QC
T T
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Degree of Freedom is the minimum number of coordinates required to
specify the position of a particle or system of particles
Equipartition Theorem The Equipartition Theorem states that energy is shared equally among
all energetically accessible degrees of freedom of a system.
1 atom: 3 degrees of freedom (x, y, z)
2 atoms: 3 × 2 = 6 degrees of freedom (x1, y1, z1, x2, y2, z2)
N atoms: 3 × N degrees of freedom (x1, y1, z1, x2, y2, z2 , … , xN, yN, zN)
Equipartition Theorem states that each quadratic degree of freedom will
possess an energy ½kbT , or ½RT per mole.
where kb = 1.38065 x 10-23 J × K-1 is Boltzmann’s constant
R = 8.31447 J × K-1 × mole-1 is Universal gas constant
A quadratic degree of freedom is one for which the energy depends on the
square of some property (for instance, velocity v or angular velocity ω).
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Equipartition Theorem
Consider the kinetic and potential energies associated with translational,
rotational and vibrational energy.
Instead of keeping an eye on the each atom, one can watch
i) translation of center of mass of the system
ii) rotation of the system (change of the orientation in space)
iii) vibration of the system (displacement of the atoms from their equilibrium)
Number of Degrees of Freedom DFs of system with N atoms
DFs = DFtrans + DFrot +DFvib = 3N
Internal energy U of system
trans rot vibU E E E
_ _trans rot vib kin vib potU E E E E
_ _vib vib kin vib potE E E
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Equipartition Theorem Monatomic gases
22 21 1 1
2 2 2trans x y zE m v m v m v
Internal energy can be expressed as
Ideal monatomic gas has only translation degree of freedom
transU E
Or, according to equipartition theorem, per 1 mole:
(number of degrees
of freedom times RT/2)
1 1 1 3
2 2 2 2transE RT RT RT RT
3 3
2 2
trans
V
V VV
EUC RT R
T T T
5
2P VC C R R
552
3 3
2
P
V
RC
CR
g For ideal monatomic gas
1.67g
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Equipartition Theorem Diatomic gases
7 7
2 2V
V V
UC RT R
T T
9
2P VC C R R
9
7
P
V
C
Cg
For ideal diatomic gas 9
1.297
g
_ _trans rot vib kin vib potU E E E E
2
2
2
_ 12
2
_ 0
1v ;
2
1
2
1v ;
2
1
2
trans
rot
vib kin
vib pot
E
E I
E
E kr
1 1 1 1 73 2 1 1
2 2 2 2 2U RT RT RT RT RT
1 2
1 2
1 2
2
0
0
12
, - masses of atoms
- reduced mass of molecule
- moment of inertia,
- average distance between atoms
- angular velocity
v - velocity of center of mass
v - relative velocity of a
m m
m m
m m
I r
r
toms
- spring constantk
Internal energy of the system
m1 m2 r0
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Equipartition Theorem Polyatomic gases
Calculation of vibrational degrees of freedom for triatomic molecules
Linear molecule: CO2; N = 3 => DFs = 9; DFtranslation = 3; DFrotational = 2
DFvibrational = 9 – 3 – 2 = 4
Symmetric
stretch
Asymmetric
stretch
Bending(1) Bending(2)
Bending Symmetric
stretch Asymmetric
stretch
Noninear molecule: H2O; N = 3 => DFs = 9; DFtranslation = 3; DFrotational = 3
1 13 13 15
3 2 4 2 ; ; ;2 2 2
1.12
5V PU RT RT C R C R g
DFvibrational = DFs - DFtranslation - DFrotational
DFvibrational = 3
1 12
3 3 3 2 ;2 2
12 14; ;
21.1
27V P
U RT RT
C R C R g
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Adiabatic expansion Experiment
1. Fill the carboy with the N2 gas (P1>P2=Patm)
2. Wait to reach the T1=Troom (explain, why)
3. Measure the pressure P1 (initial pressure)
4. Release the gas by removing the stopper
5. Wait to reach the T3=Troom (explain, why)
6. Measure the pressure P3 (final pressure)
7. Repeat the measurements 3 times
Experimental setup
1 1 1, ,P V T2 2 2, ,P V T 3 3 3, ,P V T
I II
1 1 room, ,P V T atm 2 2, ,P V T 3 2 room, ,P V TI II
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Adiabatic expansion
2 2 2
1 1 1
T PV
T PV2 2
1 1
ln lnV
T VC R
T V
2 2
1 1
ln lnP
V
P C V
P C V
1
2
2
1
ln
ln
P
V
PPC
VCV
g
PV nRTdV
nRTV
– –dU dQ PdV PdV
V
V
UC
T
VdU C dT
dVnRT
V V
dT dVC R
T V (for 1 mole)
2 11 1 3 2
1 3
V PPV PV
V P
1
2
1
3
ln
ln
P
V
PPC
PCP
g
1 1 room, ,P V T atm 2 2, ,P V T 3 2 room, ,P V TI II
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity
Instructor: Dr. Aleksey I. Filin
Adiabatic expansion References
1. William M. Moore, The Adiabatic Expansion of Gases and the
Determination of Heat Capacity ratios, Journal of Chemical
Education 61, v. 12, p1119 (1984);
2. Gary L. Bertrand and H. O. McDonald, Heat Capacity Ratio of a
Gas by Adiabatic Expansion, Journal of Chemical Education 63,
v. 3, p252 (1986).
Sound Velocity vS can be expressed as:
Heat Capacity II
Sound Velocity Method
1
2vRT
S M
g
Where: P
V
C
Cg
R – universal gas constant
T – temperature
M – molecular weight
Instructor: Dr. Aleksey I. Filin
If we know the speed of sound, we can calculate g
Sound Velocity Method
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Vibrations transmitted through an elastic solid or a liquid or gas,
with frequencies in the approximate range of 20 Hz to 20 kHz
What the sound is?
Definition of sound:
Characteristics of sound:
f – frequency [number of vibrations per unit of time, Hz, or s-1]
– wavelength [m]
Instructor: Dr. Aleksey I. Filin
So, our goal is to measure the wavelength and the frequency of the sound
vS
f By definition Speed of sound can be found as vS f
Sound Velocity Method
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Instructor: Dr. Aleksey I. Filin
Sound Velocity Method
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Experimental Setup
Generator
To oscilloscope
Speaker
Microphone
Kundt’s tube
Gas in
Gas out
Movable rod
Heat Capacity II
Sound Velocity Method
Instructor: Dr. Aleksey I. Filin
Experimental Setup
If the Generator output is connected to Y1 input of the Scope and
the Microphone output is connected to Y2 input, the Scope
shows 2 similar sin waveforms.
Sound Velocity Method
Generator
Scope
Y1
t
Speaker Microphone
Y2 Kundt’s tube
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Generator
Scope
Y
Speaker Microphone
X
Instructor: Dr. Aleksey I. Filin
)(xyy
If the Generator output is connected to Y1 input of the Scope and
the Microphone output is connected to X input, the Scope shows
so called Lissajous pattern .
A Lissajous pattern is a graph of one frequency plotted on the y axis combined
with a second frequency plotted on the x axis. In our case both f are the same
Experimental Setup
Sound Velocity Method
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Node Antinode
Heat Capacity II
Sound Velocity Method
Mathematical description of the simplest Lissajous pattern
)(xyy
Scope shows y as a function of x
We know the time dependence both y and x
)sin()(
)sin()(
0
0
ftXtx
ftYtyWhere is the phase difference
between y(t) and x(t)
x(t) y(t)
Instructor: Dr. Aleksey I. Filin
Sound Velocity Method
CHEM 4396 (W237)
Physical Chemistry Laboratory
Spring 2012
Heat Capacity II
Sound Velocity Method
Assuming X0=Y0
)sin()(
)sin()(
0
0
ftXtx
ftYty
y(x)=x
y(x)
x
If = p/2
(a quarter
of period) )cos()2
sin()(
)sin()(
ftfttx
ftty
p
y2(t)+x2(t)=1
y(x)
x
If = 0 )sin()(
)sin()(
fttx
ftty
1)(cos)(sin)()( 2222 ftfttxty
Instructor: Dr. Aleksey I. Filin
Mathematical description of the simplest Lissajous pattern
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Assuming X0=Y0
)sin()(
)sin()(
0
0
ftXtx
ftYty
y(x)=-x
y(x)
x
If = 3p/2
(three quarters
of period) )cos()
23sin()(
)sin()(
ftfttx
ftty
p
1)(cos)(sin)()( 2222 ftfttxty
y2(t)+x2(t)=1
y(x)
x
)sin()sin()(
)sin()(
ftfttx
ftty
p
If = p
(a half
of period)
Instructor: Dr. Aleksey I. Filin
Mathematical description of the simplest Lissajous pattern
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Assuming X0=Y0
)sin()(
)sin()(
0
0
ftXtx
ftYty
y(x)=x
y(x)
x )sin()(
)sin()(
fttx
ftty
If = 2p
(full period)
Note: if X0 = Y0 the angle will be not 450 and the circles become the ellipses
Full period corresponds to
Instructor: Dr. Aleksey I. Filin
Mathematical description of the simplest Lissajous pattern
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Generator
Scope
Y
Frequency is given
by generator
Speaker
X
How to measure the wavelength?
Microphone
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Generator
Scope
Y
Frequency is given
by generator
Speaker
X
How to measure the wavelength?
Microphone
4
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Generator
Scope
Y
Frequency is given
by generator
Speaker
X
How to measure the wavelength?
2
Microphone
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Generator
Scope
Y
Frequency is given
by generator
Speaker
X
How to measure the wavelength?
43
Microphone
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Generator
Scope
Y
Frequency is given
by generator
Speaker
X
How to measure the wavelength?
Microphone
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Instructor: Dr. Aleksey I. Filin
Experiment
1. Measure the shape of Lissajous pattern as a function of
microphone position
2. Distance between two positions corresponding to straight
450 line is equal to the wavelength of the sound in given gas
at given sound frequency
3. Calculate the speed of sound in given gas at given frequency
using formula
vS f
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Heat Capacity II
Sound Velocity Method
Instructor: Dr. Aleksey I. Filin
CHEM 4396 (W237)
Physical Chemistry Laboratory
Fall 2012
Sound Velocity Method
References
1. C. A. ten Seldam and S. N. Biswas, Determination of the
equation of state and heat capacity of argon at high pressures
from speed-of-sound data. Journal of Chemical Physics, 94
Issue 3, p2130 (1991)
2. J. Clayton Baum , R. N. Compton and Charles S. Feigerle, Laser
Measurement of the Speed of Sound in Gases: A Novel
Approach to Determining Heat Capacity Ratios and Gas
Composition, J. Chem. Educ., 85 (11), p1565 (2008);
3. Arthur M. Halpern and Allen Liu, Gas Nonideality at One
Atmosphere Revealed Through Speed of Sound Measurements
and Heat Capacity Determinations, J. Chem. Educ., 85 (11),
p1568 (2008)