Statistical Mechanics - Physics at Oregon State University

2.9. PROBLEMS FOR CHAPTER 2 39
If the temperature is small, only the first term is important. If the temperature
is large, the second term is dominant. This is true in general. At low temperatures
pressure is mainly related to changes in the internal energy, while at high
temperature it is mainly due to changes in the entropy.
Finally, note that in our example U = −pV . The Gibbs-Duhem relation is
U = T S − pV , and these two results are inconsistent. This is not surprising.
The Gibbs-Duhem relation is based on the assumption that the energy is an
extensive quantity. In our example this is not true. Neither the entropy nor
the energy double their value when the volume is increased by a factor of two.
This indicates that one has to be careful in introducing extensive parameters in
a quantum-mechanical formalism. Our example is too simple, again!
2.9 Problems for chapter 2
Problem 1.
The quantum states of a system depend on a single quantum number n, with
n = 0, 1, 2, · · · , ∞. The energy of state n is ɛn = nɛ (with ɛ > 0 ). The system
is at a temperature T.
(1) Calculate the partition function Z(T )
(2) Calculate the energy U(T) and the entropy S(T)
(3) Calculate T(U)
(4) Calculate S(U) and check that 1
T =
∂S
∂U
Problem 2.
The partition function of a system with quantum states n = 0, 1, 2, · · · , ∞
and energies f(n) is given by Zf (T ) and for a system with energies g(n) by Zg(T ).
The corresponding internal energies are Uf (T ) and Ug(T ); the corresponding
entropies are Sf (T ) and Sg(T ). The quantum states of a composite system
depend on two quantum numbers n and m, with n, m = 0, 1, 2, · · · , ∞. The
energy of a state n,m is ɛn,m = f(n) + g(m).
(1) Find the partition function Z(T ) for the composite system in terms of
Zf (T ) and Zg(T )
(2) Do the same for the energy and the entropy
Problem 3.