heat capacity ii - temple universityfilin/heat capacity_fall2012.pdf · kundt’s tube chem 4396...

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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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)


Fall 2012

Heat Capacity


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


If we know the speed of sound, we can calculate g


CHEM 4396 (W237)


Fall 2012

Heat Capacity II


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]


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


CHEM 4396 (W237)


Fall 2012

Heat Capacity II




CHEM 4396 (W237)


Fall 2012

Experimental Setup

Generator

To oscilloscope

Speaker

Microphone

Kundt’s tube

Gas in

Gas out

Movable rod

Heat Capacity II



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.


Generator

Scope

Y1

t

Speaker Microphone

Y2 Kundt’s tube

CHEM 4396 (W237)


Fall 2012

Heat Capacity II


Generator

Scope

Y

Speaker Microphone

X


)(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


CHEM 4396 (W237)


Fall 2012

Node Antinode

Heat Capacity II


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)



CHEM 4396 (W237)


Spring 2012

Heat Capacity II


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



CHEM 4396 (W237)


Fall 2012

Heat Capacity II


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)



CHEM 4396 (W237)


Fall 2012

Heat Capacity II


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



CHEM 4396 (W237)


Fall 2012

Heat Capacity II


Generator

Scope

Y

Frequency is given

by generator

Speaker

X

How to measure the wavelength?

Microphone


CHEM 4396 (W237)


Fall 2012

Heat Capacity II


Generator

Scope

Y

Frequency is given

by generator

Speaker

X


Microphone

4


CHEM 4396 (W237)


Fall 2012

Heat Capacity II


Generator

Scope

Y

Frequency is given

by generator

Speaker

X


2

Microphone


CHEM 4396 (W237)


Fall 2012

Heat Capacity II


Generator

Scope

Y

Frequency is given

by generator

Speaker

X


43

Microphone


CHEM 4396 (W237)


Fall 2012

Heat Capacity II


Generator

Scope

Y

Frequency is given

by generator

Speaker

X


Microphone


CHEM 4396 (W237)


Fall 2012

Heat Capacity II



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)


Fall 2012

Heat Capacity II



CHEM 4396 (W237)


Fall 2012


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)

heat capacity ii - temple universityfilin/heat capacity_fall2012.pdf · kundt’s tube chem 4396...

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