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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2002
Problem Set 3
Due Thursday, March 21, 11 am. Hand in to the grader.
 Problem 1
Consider a classical gas of very weakly interacting molecules of different species.
There are N_{i} molecules of species number i, i = 1, 2, ..., m.
The molecules of
different species may have different masses and different internal degrees of
freedom (such as vibrational or rotational modes), however you may assume
that the molecules do not interact with each other. The molecules of a given species are indistiguishable from each other.
(a) Show that the canonical partition function for the gas has the form
Q = Q^{(1)}Q^{(2)}....Q^{(m)}
where
Q^{(i)} =
[Q_{1}^{(i)}]^{Ni}/N_{i}!
and Q_{1}^{(i)} is the single particle partition function
for molecules of species i. [4 points]
(b) Using the result in (a), show that the total pressure of
the gas is the sum of the pressures that each of the species of
molecule would have on its own (total pressure is the sum of
the "partial pressures"). Similarly show that the total entropy
of the gas is the sum of the entropies that each species would have
on its own. [2 points]
(c) By taking the appropriate derivative of the total Helmholtz
free energy of the gas, compute the chemical potential µ_{i}
of species i. Express your answer in terms of Q_{1}^{(i)}.
[2 points]
(d) Assume that the molecules are free (i.e. not in any external potential,
except for their confinement to a box of volume V). Show that
Q_{1}^{(i)} can be written as
Q_{1}^{(i)} = Vq_{1}^{(i)}(T) where
q_{1}^{(i)} depends only on temperature T. [2 points]
(e) Suppose that the species of molecules undergo the chemical reaction
a_{1}A_{1} + ... +
a_{j}A_{j} <> a_{j+1}A_{j+1} + ... +
a_{m}A_{m}
where a_{i} is the number of molecules of species A_{i}
(i = 1,...,j) that combine to create a_{k} molecules of species
A_{k} (k = j+1,...,m). As discussed in lecture, the equilibrium
number of molecules of each species will be determined by the condition
a_{1}µ_{1} + ... + a_{j}µ_{j}
= a_{j+1}µ_{j+1} + ... + a_{m}µ_{m}
(make sure you understand where this result comes from!).
Use the above, and the results of the previous parts,
to show that the equilibrium concentrations of the species
are given by
([n_{1}]^{a1}
[n_{2}]^{a2}...[n_{j}]^{aj
})/([n_{j+1}]^{aj+1}...[n_{m}]^{am
}) = K(T)
where n_{i} is the concentration of species i,
n_{i} = N_{i}/V, and K(T) is a
function of temperature only. Derive an expression for K(T) in terms
of the q_{1}^{(i)}(T). The above result is known as the
"law of mass action". [4 points]
(f) Consider the reaction
A + B <> C
Suppose that an energy E_{o} is released when A and B combine to
form C, i.e. E_{o} is the binding energy of the molecule C when
compared to its separated constituents A and B. Assume that A, B, and C
are free point particles (i.e. no internal degrees of freedom are excited).
By explicitly evaluating the single particle partition function of the three
species, determine the function K(T). Remember, you must properly include the
binding energy E_{o} of molecule C when you compute its partition
function. [4 points]
(g) Suppose that initially there are equal concentrations of A and B,
n_{A} = n_{B} = n_{o}, while the concentration of C
is initially n_{C} = 0. Find the resulting equilibrium concentrations
of A, B and C, in terms of n_{o} and the function K(T). [2 points]
 Problem 2
In the grand canonical ensemble, the probability to have a given state
with total energy E_{a} and total number of particles N_{a}
is,
P_{a} =
[e^{(EaµNa)/kBT}]/L
where
L = Sum_{a} [e^{(EaµNa)/kBT}]
is the grand canonical partition function.
(a) For a quantum ideal gas, with single particle states i of energy
e_{i}, many particle states are specified by the occupation
numbers {n_{i}} and have energy E = Sum_{i}
[e_{i}n_{i}]. Show that the probability for the state with
occupations {n_{i}} is given by
P({n_{i}}) = Product_{i} [p_{i}(n_{i})]
where p_{i}(n_{i}) is the probability that single particle
state i has occupation n_{i}
p_{i}(n_{i}) =
[e^{(eiµ)ni/kBT}]/w_{i}
where
w_{i} = Sum_{ni} [e^{(eiµ)ni/kBT}]
can be thought of as the partition function
for the single particle state i.
The above factorization says that the number of particles n_{i}
in state i, is independent of the number of particles
n_{j} in state j. [6 points]
(b) Using the above result, show that the Shannon definition of entropy can
be written as
S = k_{B} Sum_{{ni}} [P({n_{i}}) ln
P({n_{i}})] = k_{B} Sum_{i}
Sum_{ni} [p_{i}(n_{i}) ln
p_{i}(n_{i})]
[6 points]
(c) Using the above result, show that the following expressions apply for
the entropies of an ideal gas of bosons and fermions, respectively
bosons: 
S = k_{B} Sum_{i} [(1+<n_{i}>)
ln (1+<n_{i}>)  <n_{i}> ln
<n_{i}>] 
fermions: 
S = k_{B} Sum_{i} [(1<n_{i}>)
ln (1<n_{i}>)  <n_{i}> ln
<n_{i}>] 
where <n_{i}> = Sum_{ni}
[n_{i} p_{i}(n_{i})] is the average occupation
number of state i.
[8 points]
 Problem 3
Consider photons of a given energy e = hbar omega.
(a) If <n> is the average number of such photons in equilibrium
at temperature T, show that the fluctuation in the number of photons is
<n^{2}>  <n>^{2} =  (1/e) (d<n>/dbeta)
where beta = 1/k_{B}T
[5 points]
(b) Using the forumula for the equilibrium value of <n>,
apply the above result to determine the relative fluctuation in the number
of photons
[<n^{2}>  <n>^{2}]/<n>^{2}
Is this large or small? [5 points]
