partition functions of ideal gases. we showed that if the # of available quantum states is >>...

Partition functions of ideal gases

We showed that if the # of available quantum states is >>

N

!

),(),,(

N

TVqTVNQ

N

The condition is valid when

18

2/32

Tmk

h

V

N

B

Examples gases at low densities (independent molecules)2

Monoatomic ideal gas

eleci

vibi

roti

transi

eleci

transi

general

MIG

)(),(),( TqTVqTVq electrans

3

Translational partition function MIG

2

2

2

2

2

22

8 c

n

b

n

a

n

m

h zyxnnn zyx

If cubic (a=b=c)

Particle in a parallelepipeda,b,c

222

2

2

8 zyxnnn nnnma

hzyx

nx, ny, nz =1,2,3…

1 1 1

222

2

2

1,,

/

8exp

),( ,,

x y z

zyx

Bznynxn

n n nzyx

nnn

Tk

trans

nnnma

h

eTVq

4

1 1 12

22

2

22

2

22

8exp

8exp

8exp),(

x y zn n n

zyxtrans ma

nh

ma

nh

ma

nhTVq

...8

exp2

2

2

2

2

2

8

9

8

4

8

12

22

ma

h

ma

h

ma

h

n

eeema

nh

3

12

22

8exp),(

ntrans ma

nhTVq

5

this sum cannot be expressed as a sum of a series but…

6

3

0

8 2

22

),(

dneTVq ma

nh

trans

2/1

0 4

2

dne n

Vh

TmkTVq B

trans

2/3

2

2),(

example: average translational energy MIG

7

Electronic partition function MIG

8

...)( 221 eggegTq l

llelec

the ground state energy is taken as the zero of energy

the other terms are negligible since they are typically in the order of 10-5 and smaller

exceptions: some of the halogens may have contributions from the first terms

Summary MIG

9

!

)(),(),,(

N

TqTVqTVNQ

Nelectrans

Vh

TmkTVq B

trans

2/3

2

2),(

221)( eggTqelec

elecB

VB

VNB q

eNgTk

T

qTNk

T

QTkE

2222

,

2

2

3lnln

Diatomic ideal gas

10

eleci

vibi

roti

transi

rot

qqqqTVq vibelectrans),(

!

),,(N

qqqqTVNQ

Nvibrotelectrans

Vh

TkmmTVq B

trans

2/3

221 )(2

),(

Zeros of energy

• rotational: J=0 (rotational energy =0)• vibrational: a) the bottom of the well or

b) the ground vibrational state; in (a) the ground vibrational state is h/2

• electronic: the energy is zero when the two atoms are completely separated

11

Dissociation energy (Do)and ground state electronic energy (De)

12

2

hDD eo

Anharmonic oscillator: HCl

vibrational partition function DIG

13

harmonic oscillator approximation

2

1vhv

0

/)2/1()(

Tkhvib

BeTq Tkh Bex /

T

T

vib vib

vib

e

eTq /

2/

1)(

Bvib k

h

h

h

vib e

exTq

1

)(2/

0

2

1

vibrational temperature:

average vibrational energy

14

12

ln/

2Tvibvib

Bvib

Bvib vibeNk

dT

qdTNkE

vibrational contribution to Cv

fraction of molecules in the jth vibrational state

15

vibq

ejyprobabilit

j

)(

hh

h

hhh

eee

eee

112/

2/

see problems 3.35; 336 (we solved them last class)

most molecules are in the ground vibrational state at room T

rotational partition function DIG

16

0

)2/()1(2

)12()(J

TIkJJrot

BeJTq (sum is over levels)

Brot Ik2

2rotational temperature

0

/)1()12()(J

TJJrot

roteJTq

17

rotational partition function DIG

18

0

/)1()12()(J

TJJrot

roteJTq

because the ratio rotational temperature/T is small for most molecules at ordinary Ts

dJeJTq TJJrot

rot

0

/)1()12()(

integrating

TT

dxeTq rotrot

Txrot

rot

for id val)(

0

/

much better at high T; is the high T limit

average rotational energy

19

TNkdT

qdTNkE B

rotBrot

ln2

total rotational contribution to Cv is R; R/2 per rotational degree of freedom for a diatomic

fraction of molecules in the Jth rotational state

)(

)12( )2/()1(2

Tq

eJf

rot

TIkJJ

J

B

see problem 3.37 that we solved last class

physical meaning of the rotational temperature

20

It gives us an estimate of the temperature at whichthe thermal energy (kT) equals the separation between rotational levels. At this T, the population of excited rotationalstates is significant.

88 K for H2, 15.2 K for HCl and 0.561 K for CO2

21

most molecules are in the excited rotational levels at ordinary Ts

Symmetry effects

22

TT

Tq rotrot

rot

for id val)( is for heteronuclear DIG

For homonuclear DIG TT

Tq rotrot

rot

for id val2

)(

the factor of 2 comes from the symmetry of the homonuclear molecule; 2 indistinguishable orientations

TT

Tq rotrot

rot

for id val)(

23

molecular partition function DIG

elecvibtrans qqqqTVqrot

),(

TkDT

T

rot

B Be

vib

vib

ege

eTV

h

TMkTVq /

1/

2/2/3

2 1

2),(

restrictions:Trot

only the ground state electronic state is populatedzero (electronic) taken at the separated atomszero (vibrational) taken at the bottom of the potential well

average energy DIG

24

V

B T

qTNkE

ln2

average Cv DIG

25

Vibrational partition function of a polyatomic molecule

26

...2,1,0 )2

1(

1

jj

n

jjvib

vib

h

nvib is the number of vibrational degrees of freedom3n-5 for a linear molecule3n-6 for a nonlinear moleculenormal modes are independent

since the normal modes of a polyatomic molecule are

independent

27

vib

jvib

jvibn

jT

T

vibe

eq

1/

2/

,

,

1

vib

jvib

jvibn

jT

Tjvibjvib

Bvibe

eNkE

1/

/,,

,

,

12

vib

jvib

jvibn

jT

Tjvib

BvibV

e

e

TNkC

12/

/2

,,

,

,

1

B

jjvib k

h ,

28

29

Rotational partition function of a linear polyatomic molecule

30

linear )1(8 2

2

JJIJ

J = 0, 1, 2, …

12 Jg Jdegeneracy

n

jjjdmI

1

2

mj is the distance from nucleus j to the center of mass of the molecule

TT

Tq rotrot

rot

for id val)(

is 1 for nonsymmetrical molecules (N2O, COS) and 2 for symmetrical such as CO2

Importance of rotational motion

31

32

symmetry number

33

is 1 for nonsymmetrical molecules (N2O, COS) is 2 for symmetrical such as CO2

how about NH3?

symmetry number is the number of different waysin which a molecule can be rotated into a configuration indistinguishable from the original

For water, =2, successive 180o rotations about an axis through the O atom bisectingthe two H atoms result in two identical configurations

for CH4, for any axis through one of the four CH bonds there are 3 successive 120o rotationsthat result in identical configurations, therefore = 4x3 =12

Linear polyatomic moment of inertia

34

Example HCN

Non linear rigid polyatomic

35

rigid non-linear polyatomic

36

3 moments of inertia

AcI

hA

28

BcI

hB

28

CcI

hC

28

if the three are equal, spherical top

only two equal, symmetric top

three different, asymmetric top

37

38

examples of rotational symmetry

http://www.learner.org/courses/learningmath/geometry/session7/part_b/

examples

39

spherical top symmetrical top

Bjjrot kI2

2

,

J=A, B, C

3 rotational temperatures

Spherical top molecules

40

0

/)1(2)12()(J

TJJrot

roteJTq

dJeJTq TJJrot

rot

0

/)1(2)12(1

)(

)1(2

2

JJIJ

Allowed energies:

J = 0, 1, 2, …2)12( Jg J

Degeneracy:

Trot for

large JTrot

Spherical top molecules

41

dJeJTq TJJrot

rot

0

/)1(2)12(1

)(

large JTrot dJeJTq TJrot

rot

0

/2 2

41

)(

Can be solved analytically

Rotational partition functions

42

2/32/1

)(

rot

rot

TTq

Spherical top

2/1

,,

2/1

)(

CrotArotrot

TTTq

Asymmetric top2/1

,,,

32/1

)(

CrotBrotArotrot

TTq

Symmetric top

Average rotational energy (nonlinear polyatomic)

43

dT

TqdTNkE rot

Brot

)(ln2

Partition function ideal gas of linear

polyatomic molecule

44

elecvibtrans qqqqTVqrot

),(

TkDN

jT

T

rot

B Be

jvib

jvib

ege

eTV

h

TMkTVq /

1

53

1/

2/2/3

2 ,

,

1

2),(

Tk

D

e

T

TTNk

U

B

eN

jT

jvibjvib

Bjvib

53

1/

,,

1

/

22

2

2

3,

53

12/

/2

,

,

,

122

2

2

3 N

jT

Tjvib

B

v

jvib

jvib

e

e

TNk

C

45

Partition function ideal gas of nonlinear polyatomic molecule

TkDN

jT

T

CrotBrotArot

B

Be

jvib

jvib

ege

ex

TV

h

TMkTVq

/1

63

1/

2/

2/1

,,,

32/12/3

2

,

,

1

.2

),(

Obtain U and Cv

Comparison to experiments

46

47

Summary• Considering the molecules that constitute a

macroscopic material, we construct q, and from q we construct Q, and from Q any thermodynamic property.

• For example, U and Cv are not just numbers in tables. We have some new insights about why different materials have different thermodynamic properties

• Next, we will discuss the laws that govern the macroscopic thermodynamic properties.

48