# Second hour exam

```Chem. 444, Fall 2006
Instructor: Nancy Makri
Name: ____________________________
SECOND HOUR EXAM
November 3, 2006
1. State whether the temperature T of the system, the entropy Ssys of the system and the entropy S tot
of the universe (i.e., that of the system and its surroundings) increase, decrease or remain constant
during each of the processes described below. In each case justify your answer in a single sentence.
(a) Reversible isothermal expansion of an ideal gas.
(b) Free expansion of an ideal gas.
(c) Reversible adiabatic compression of an ideal gas.
(20 points)
2. A fluid (not necessarily an ideal gas) is expanded and heated, such that its temperature is changed
from T1 to T2 and its volume is changed from V1 to V2 . Express an infinitesimal change of the
entropy in terms of infinitesimal changes in the temperature and volume. You can evaluate the
change in entropy by integrating this differential form. However, derivatives of the entropy are not
measurable directly. Therefore, you should first transform this relation to bring it in a form that
involves measurable quantities, such as pressure-temperature relations and/or heat capacities. For
this purpose you may find it useful to write the first law and to use one of Maxwell's relations.
(You get 5 points of extra credit for deriving Maxwell's relation.) Now you are ready to integrate
the differential form. Show that the change in the entropy of the fluid is given by the relation
S  
T2
T1
V2  P 
cv (T )
dT   
 dV .
V1
T
 T V
(20 points + up to 5 points of extra credit)
3. Consider a gas (which may consist of polyatomic molecules and which does not necessarily exhibit
ideal behavior). Use statistical mechanical arguments for the following questions:
(a) Show that the canonical partition function increases upon (i) increasing the temperature at
constant volume and (ii) expanding the gas at constant temperature.
(b) Argue whether the internal energy and the entropy of the gas will increase or decrease upon
increasing the temperature at constant volume. Does your conclusion agree with what you
expect from thermodynamics?
(20 points)
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No calculators on this test.
You must turn in this sheet with your solution.
```