Problem Set 3. Due in class: Tuesday, Feb 24 2015
Reading: lecture notes 9, 10, and 11.
1. (Debye model.) Use the Debye approximation to find the following thermodynamic func-
tions of a solid as a function of temperature T . Express your answers in terms of the function
D(y) =   3
and the Debye temperature ΘD  = hωD/kB  = (hv/kB)(6π2N/L3)1/3.
(a) Show that the partition function of the Debye model is given by
Z Debye = n
1− e−βhωn .   (1)
In the problem, we drop the contribution of the zero-point energy for the quantum
harmonic oscillator.
(b) Show that the Helmholtz free energy F  = −kBT  log Z  is given by
F  = 3NK BT  log[1− exp(−ΘD/T )] −NkBTD(ΘD/T ).   (2)
Hint: use  log
iBi =
i logBi  and integration by parts    x2 log(1−e−x)dx =  x3/3log(1−
e−x)−   x3/(ex − 1)dx.
(c) Show that the mean energy E   is given by
E  = 3NkBTD(ΘD/T ).   (3)
S  =  NkB[−3 log(1− exp(−ΘD/T )) + 4D(ΘD/T )].   (4)
(e) Evaluate D(y) in the limit y   1 and y   1.
(f) Use these results to find F ,  E , and S  when T   ΘD  and T   ΘD. What conclusions
can you draw? Do they agree with what we found in class?
2. (Gas in gravitational field.) Consider a column of atoms each of mass  M  at temperature
T   in a uniform gravitational field  g. Find the thermal average potential energy per atom.
The thermal average kinetic energy density is independent of height. The total heat capacity
is the sum of contributions from the kinetic energy and from the potential energy. Take the
zero of the gravitational energy at the bottom h = 0 of the column. Integrate from h = 0 to
h = ∞. Show that the heat capacity is 5kB/2.
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3. (Plant cell.) The concentration of potassium K+ ions in the internal sap of a plant cell of 
a fresh water alga may exceed the concentration of potassium K + ions in the pond water in
which the alga is growing in by a factor 104. The chemical potential of the K+ ions is higher
in the sap because their concentration n is higher there. Estimate the difference in chemical
potential at 300K and show that it is equivalent to a voltage of 0.24V across the cell wall, i.e.
0.24 eV (electron volt). Assume the system of the ions as an ideal gas.
4. (Grand partition function.)
(a) Consider a system that may be unoccupied with energy zero or occupied by one particle
in either of 2 states, 0 and . Show that the grand partition function (also known as the
Gibbs sum) for this system is
Z  = 1 + λ + λ exp(−β),   (5)
where λ  = exp(βµ). We exclude the possibility of one particle in each state at the same
time.
(b) Show that the thermal average occupancy of the system is
N  =  λ + λ exp(−β)
Z    (6)
(c) Show that the thermal average occupancy of the state at energy   is
N () =  λ exp(−β)
Z    (7)
(d) Find an expression for the thermal average energy of the system.
(e) Allow the possibility that the orbital at 0 and   may be occupied each by one particle
at the same time. In this case, show that
Z  = 1 + λ + λ exp(−β) + λ2 exp(−β) = [1 + λ][1 + λ exp(−β)].   (8)
Since Z  can be factorized as shown, we have in effect, two independent system.
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