## Transcribed Text

1. Distributions
(A) Calculate the number of ways to divide 17 distinguishable objects into 5 groups as
follows:
group A has 5 objects
group B has 0 objects
group C has 7 objects
group D has 2 objects
group E has 3 objects
(B) Suppose you can’t evaluate factorials (x!) directly. Show me how you can use the
simple version of Stirling’s approximation to calculate 17!. Does it work well?
2. Boltzmann distribution
Consider a system of exactly 8 moles of independent but distinguishable particles. Each
particle has just three levels available The lowest level has an energy of 0 and is not
degenerate. The second level has an energy of and is two-fold degenerate. The third
level has an energy of 3 and is not degenerate.
(A) What is the molecular partition function at absolute zero temperature?
What are the occupation numbers (N0, N1, N2, N3) of the four states?
What is the average energy of a particle?
(B) What is the molecular partition function at infinite temperature?
What are the occupation numbers (N0, N1, N2, N3) of the four states?
What is the average energy of a particle?
(C) What is the molecular partition function at a temperature of 2/k?
What are the occupation numbers (N0, N1, N2, N3) of the four states?
What is the average energy of a particle?
(D) Is N0 = 3.0 mol, N1 = 2.5 mol, N2 = 1.5 mol, N3 = 1.0 mol a Boltzmann distribution?
Explain.
3. Entropy OR equilibrium
The following problems use related information at P = 1.000 bar and T = 1000. K.
(A) Calculate the standard molar entropy, Sm
o
, of I2(g) using statistical mechanics.*
(B) Calculate the equilibrium constant for I2(g) 2I(g) using statistical mechanics.**
The following information will be helpful:
I: q
T
/N = 1.165109
g0
E = 4
I2: D0 = 12,366 cm1 g0
E = 1 B᷉= 0.0374 cm1 v᷉= 214.5 cm1
To make your life (and mine) easier, you might want to first calculate and report some of
the following quantities:
I: q/N
I2: , R
,
V
, q
T
/N, q
R
, q
V
, q
E
* By comparing your answer to (A) with the appendix value (not at 1000 K) you might
find confidence or an error in your answer.
** By comparing properties of I2 on this exam with those of Na2 from the lecture, you
should be able to judge whether your calculations are reasonable for (B).
(C) : You shouldn’t understand why g0
E = 4 for an iodine atom (it has to do with the
spin-orbit effect which we did not cover), but you should be able to propose a different
value for g0
E
and justify it well.
4. Internal energy and constant-volume heat capacity
Let’s consider I2 again, but this time at the more reasonable temperature of 400 K. Both
the molar internal energy (Um) and the molar constant-volume heat capacity (CV,m) can be
expressed in the following manner where (i) we reasonably ignore any electronic
contributions, (ii) x1 and x2 are just numbers representing the amount of RT or R that is
needed, and (iii) v1 and v2 represent the vibrational contributions.
Um = x1RT + v1 CV,m = x2R + v2
(A) What is x1? What is x2?
(B) Accurately determine v1 in units of J mol1
.
(C) Accurately determine v2 in units of J K1 mol1
.
Equation 15E.6 (page 638) has a typo and is just the high-temperature limit, so you’ll
want to think hard and refer to your notes. Also, remember that the “V” in CV,m refers to
“constant volume” not “vibrational mode”
Answer
(A) or (B)
5. Equilibrium
The diagram below shows the energy levels (assume nondegenerate) for the reactants and
products of a 1:1 ideal gas chemical reaction R(g) P(g). I want to know how the
equilibrium constant behaves with temperature. The four possibilities are listed below.
(A) Is this reaction exothermic or endothermic, i.e. is rH
o > 0 or rH
o < 0?
(B) Does entropy increase or decrease in the forward direction, i.e. is rS
o > 0 or rS
o <
0?
(C) Explain which possibility (1-4) for the temperature dependence of the equilibrium
constant is observed by considering the extreme cases: T = 0 and T = . You may
consider the occupation numbers (state populations) graphically by replicating the
diagram above – or you may consider the contributions to the expression of the
equilibrium constant given above.

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