chem 365 w2012 mt2

8/12/2019 Chem 365 W2012 MT2

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Statistical Thermodynamics Chemistry 365 W2012Midterm 2 March 27, 2011 6-9 PM OMC 112

Examiner Prof. Paul W. Wiseman

There are 5 questions on the exam and equation data at the end of the exam. Answer the questions in theexam answer booklet provided

1. [5 pts] Identify the molecule

The plot above depicts the heat capacity as a function of temperature for 1 mole of a mystery molecule in thegas state. Identify the mystery molecule from the choices below and provide a brief justification taking intoaccount the two labeled limits (lines) on the heat capacity plot and the data below.

A) H2O B) CO 2

C) SO 2 D) CO

Data for H 2O (g) vib 5360K, 5160K, 2290K rot 40.1K, 20.9K, 13.4K =2Data for CO 2 (g) vib 3360K, 954(2)K, 1890K rot 0.561 =2Data for SO 2 (g) vib 1660K, 750K, 1960K rot 2.92K, 0.495K, 0.422K =2Data for CO (g) vib 3103K, rot 2.77K =1

0 500 1000 1500 2000 2500 3000

1

2

3

4

5

6

7

Cv/ N ak

T (K)

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2. [15 pts] The partition function unlocks thermodynamic properties

A) i) Starting with the ensemble partition function Q, prove that the chemical potential of a nonlinear polyatomic ideal gas is given by the equation below if the translational, vibrational and rotational contributionsare taken into account (assumes T> rot, j )

63

1

iv,

CBA

3

2T

iv,2

12

3

e1ln2T

ln N

V

h

MkT2ln

kT

n

i

T

B) The partion function of a monatomic van der Waals gas is given by:

where a and b are the van der Waals constants.

Prove that the internal energy of a monatomic van der Waals gas is given by:

VkT

2aN23N

e Nb-VhmkT2

!1

T)V,Q(N, N2

N

VaN

NkT23

E2

8/12/2019 Chem 365 W2012 MT2


3. [10 pts.] Rotation/Nuclear Spin

Lithium ( 7Li) has a nuclear spin quantum number of 3/2. Consider the diatomic molecule of this isotope,Li2 and its thermodynamic properties at low temperatures in terms of its distinct ortho and para forms.

A) Write the expression for the rot./nuc. partition function for Li 2 that would be valid for low temperatures.Write a brief (2 lines) explanation of the meaning of each of the major terms in the partition function

B) Sketch a plot of the expected percentage of para Li 2 as a function of temperature in a system of Li 2 thatcontains both species. Explain your answer in terms of the expression you gave in A) and clearly explainwhat is happening at the molecular level as T is increased from 0K in order to justify your plot.

C) The plot below is from R. W. Richards J. Chem. Edu. V43, 644 (1966) showing a high resolutionvibration/rotation spectrum of acetylene C 2H2 . The fine structure peaks are due to coupling with rotationalenergy levels.

i) Explain the underlying reason for the non uniform amplitude of adjacent rotational peaks.ii) What do you expect for the amplitude ratio for the alternate peaks? Briefly Explain.iii) What can you say about the nature of the molecular energy levels corresponding to the larger amplitude

rotational peaks?

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4. [10 pts.] Boltzmann Distribution and Active Modes

The plots above show fractional occupancy for two different degrees of freedom for HCl(g) at 2000K startingwith the ground state levels (quantum number = 0 on the x axes)

Data for HCl(g) vib 4227K, rot 15.02K =1

A) Identify the dof A and B from the following possible dof's1. Translational2. Vibrational3. Rotational4. Nuclear energy dof

Briefly justify your matching choices based on the plots and the data

B) In a mixture of ideal gases, the species are independent so the partition function of a mixture is the

product of the individual partition functions. Thus for a mixture of gas species A and B in equilibrium

i) From your understanding of the partition function, explain why equilibrium occurs in chemical reactions.Build your answer around the simple reaction A B and the figure of the energy levels shown belowfor species A and B. State formulas for the arbitrary dof molecular partition functions q A and q B andexplain the equilibrium state for the reaction in the context of the partition functions and the energy levels.

ii) What is so special about the equilibrium state in light of the distribution of energy quanta in discretestates within the system (i.e. why do we observe reactions to go to equilibrium)?

(Maximum 1 page )

A B

00

T)V,,Q(NT)V,,Q(NT)V,, N,Q(N BABA

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5. [15 pts.] Calculation of an Equilibrium Constant

A) [10] Using partition functions and statistical thermodynamics, set up the expression for K p(T) for thefollowing reaction but do not calculate it (leave it in symbolic form but with each dof term shownwith the correct formula for the partition function):CO 2 (g) + H 2 (g) CO (g) + H 2O(g)

Note you may answer this in separate modules for each species and show where they fit in the finalexpression (due to limited page width space).

B) [5] Calculate the value of V2O H q

(only) from the expression for K p for a temperature of 900 K

Data for CO 2 (g) vib 3360K, 954(2)K, 1890K rot 0.561 =2Data for H 2 (g) vib 6332K, rot 85.3 =2Data for CO (g) vib 3103K, rot 2.77 =1

Data for H 2O (g) vib 5360K, 5160K, 2290K rot 40.1, 20.9, 13.4 =2Do=917.6 kJ/mol e1=1MW 18.015 g/mol k=1.38x10 -23 J/K h=6.626x10 -34 Js N a=6.022x10 23 mol -1

C) [5] The equilibrium constant at two temperatures is K p(900K) =0.43 and K p(1200K) =1.37 for thisreaction. Based on this temperature trend, explain whether the forward reaction is favoredentropically or enthalpically. Make reference to an appropriate equation to aid in your explanation.

8/12/2019 Chem 365 W2012 MT2


EquationsMicrocanonical ensemble (N, V, E) Canonical ensemble Q(N, V, T)

EV,

E N,

V N,

Nln

-kT

Vln

kT p

E

ln

kT

1

dN-dVT p dE

T1

dS

klnS

T

Grand canonical ensemble (V, T, ) Isothermal-isobaric ensemble (N, T, p)

!n!

(n)

j j

t

N'

V N,

2

TV,

T N,

V N,

j

kT

E

TlnQ

kTE

NlnQ

kT-

VlnQ

kT p

TlnQ

kTlnQkS

dNdV p-SdT-dF

kTlnQ-F

eT)V,Q(N, j

-

Vln

kT V

lnkT p

lnkT N

Tln

kTlnkS

dV p Nd - SdT pVd

lnkT pV

eT)V,Q(N,)T,(V,

T,

TV,

V,

N

kT N

pT,

T N,

p N,

V

kT pV

Nln

kT-

plnkT-V

Tln

kTlnkS

dNdpV dTS-dG

lnkT-G

eT)V,Q(N, p)T,(N,

N N N N ln!ln

j j j nE j

BBAA

DDCC

BA

DC

Vq

Vq

V

q

V

q

BA

DCc TK

TK kT p p p p

TK cBA

DC p

BADC

BA

DC

S T E F

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levels

states j

kTE

kT jE

eeT)V,Q(N,l

l

l

NT)q(V,T)V,Q(N, N

N!T)q(V,

T)V,Q(N,

...q q q q q q nuclear electroniclvibrationarotationalnaltranslatiomolecular

mkT2h

whereVhmkT2

V

TV,q 22

3

23trans

levelaforenergyMolecular

stateaforenergyMolecular

...)al,vibrationonal,translati(i.e.

Freedomof DegreeDOF

eeq levels

kT

states j

kT

DOF

DOF l

DOF j

ll

DOF l

DOF j

where

i

kT ieelect

i

eq

2i

kT ie1eelect

i1

eq

levelsi

kT

kT

DOF

kT

k

e

e

q

e f DOF

i

DOF k

DOF k

DOF i

DOF k

DOF k

T

2T

kTh

2kTh

vib v

v

e-1

e

e-1

e (T)q

bersin wavenumEk

hc

k h

v

where

6-3nor5-3n

1 j T

2T

6-3nor5-3n

1 j kT

h

2kT

h

vib j

j

j

j

e-1

e

e-1

e (T)q

bersin wavenumEk

hc

k

h

j

vj

j

j j

where

Ic8

h B

k

Bhc

e1J21

(T)q

2r

0Jrot

T

1)J(Jr

where

T T

(T)q r r

rot for

rhomonuclea2earheteronucl1

...T15

1

T31

1T

(T)q 2

r r

r rot

odd J

evenJnucrot,

T1JJr

T1JJr

e12J12II

e12J12I1I(T)q

Integer Spin

odd J

evenJnucrot,

r

T1JJr

e12J12I1I

e12J12II(T)q

Half- Integer Spin

eeeq

2hDD

D-

hkT1

kTD

1ekTD

1eelect

oe

ee1

eo

6-3nor5-3n

1 T

vib vj

e-1

1 (T)q j

8/12/2019 Chem 365 W2012 MT2


2/3

r

2/3

2

2

rot

Th

IkT8(T)q

CBA III

CBA III 2

1

CrArrot

TT(T)q

CBA III2/1

rCrBrA

3

rotT

(T)q

Spherical Top

Symmetric Top

Asymmetric Top

5)NkT-(3n NkT NkTE 22

23Linear Polyatomic

6)NkT-(3n NkT NkTE 23

23

Nonlinear Polyatomic

Equipartition of Energy

e1

e n

e1

e n N N

e wheree1

Tk

Tk

k

k k Tk

Tk

k

Tk

1

k

Tk

b

k

b

k

b

k

b

k

b b

k

FD

BE

FD

BE

FD BE

k

Tk

b

k k Tk

Tk

k k k

b

k

b

k

b

k

e1lnTk pV

e1

e n NE

FD

BE

FD

BE

i

kT

kT

kT

k i

kT

e

e

q

eP eq

i

k k

i

i

k k i

i

kT

kT

kT

k

e

e N

q e

Nni

k k

i

k k

kT

kT

kT

k

je

q e

N

q

e N

n

n k j

k

j

k

j

k

j

Boltzmann Distribution

chem 365 w2012 mt2

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