Statistical Mechanics I PHY 6536
Problem Set 1, due Jan. 30, 2008
Problem 1
Making use of the fact that the entropy S(N, V, E) is an extensive quantity,
show that
∂S
∂S
∂S
S = N
+V
+E
∂N V,E
∂V N,E
∂E V,N
Problem 2
Show that for an ideal gas composed of monoatomic molecules the entropy
change between any two temperatures, when the pressure is kept constant is
5/3 times the corresponding entropy change when the volume is kept constant.
Problem 3
Consider a nearly ideal gas for which an adiabatic reversible transformation of
an ideal gas satisfies
P V γ = cst .
Consider a mixture of two ideal gases with mole fractions f1 and f2 and respective exponents γ1 and γ2 . Show that the effective exponent γ for the mixture is
given by
1
f1
f2
=
+
γ−1
γ1 − 1 γ2 − 1
Problem 4
Establish thermodynamically the formulae
∂P
∂P
S = V
,
N = V
.
∂T µ
∂µ T
Express the pressure P of an ideal gas in terms of the variables µ and T , and
verify these formulae.
1
Problem 5
The equation of state of a gas is found to be
Np
,
T
where a is a constant. The heat capacity at constant pressure is given by
pV = N kT − a
5
N k + 2 a N f (T ) .
2
Find f (T ), and derive expressions for the heat capacity of the gas at constant
volume, entropy, internal energy, chemical potential, and Gibbs potential.
Cp =
Problem 6
A small hole has formed in the wall of a container enclosing a volume V, containing N molecules in the gaseous state.
1-) Show that the number of molecules per unit area per unit time which leak
through is given by
N <v>
,
4V
where < v > is the average speed of the molecules.
2-) Show that the rate at which energy escapes through the hole is 2 kT . Why
it is greater than the average energy of a molecule in the gas.
Problem 7
Consider a system composed of a liquid and its saturated vapor in thermal
equilibrium. Assuming the ideal gas equation of state for the vapor, show that
the vapor pressure dependence on the temperature T is given by
λM
,
RT
where λ is the heat of vaporization, and M the molecular weight.
ln p = ln p0 −
2
Problem Set 2, due February 6, 2008
Sethna: problems (1.6), (2.1),( 2.2), (2.6).
You will find the following link quite useful:
http://pages.physics.cornell.edu/sethna/StatMech/ComputerExercises.html
Problem Set 3, due February 13, 2008
Sethna: problems (3.5),(3.8),(3.9),(3.10),(3.11).
Problem Set 4, due February 20, 2008
Problem 1
Show that the phase space volume element
3N
Y
dqi dpi
i=1
is invariant under canonical transformations.
Problems 2, 3, 4: Sethna, (4.2), (5.4), (5.7)
3
Problem Set 5, due February 27, 2008
Problems 1, 2, 3, 4: Sethna, (5.13), (5.17), (6.3), (6.10)
Problem Set 6, due March 5, 2008
Problems 1, 2, 3: Sethna, (7.1), (7.3), (7.4)
Problem Set 7, due March 19, 2008
Problems 1, 2, 3: Sethna, (7.5), (7.6), (7.8)
4
Problem Set 8, due March 26, 2008
Problems 1, 2, 3: Sethna, (7.9), (7.11), (7.12)
Problem Set 9, due April 2, 2008
Problem 1: Sethna, (7.10)
Problem 2
Consider an ideal Bose gas confined to a region of area A in two dimensions.
Express the number of particles in the excited states, Ne , and the number of
particles in the ground state, N0 , in terms of z, T and A, and show that the
system does not exhibit Bose-Einstein condensation unless T → 0.
Refine your argument to show that, if the area A and the total number of
particles N are held fixed and we require both Ne and N0 to be of order N , we
do achieve condensation when
T ∼
1
h2
mkB l2 ln N
p
where l ∼ A/N is the mean interparticle distance in the system. If both A
and N → ∞, keeping l fixed, then T does go to zero.
Problem 3
Consider an n−dimensional Bose gas whose single particle energy spectrum
is given by ∼ ps , where s > 0. Discuss the onset of B-E condensation in
this system, especially its dependence on n and s. Study the thermodynamic
behavior of the system and show that quite generally,
P =
s E
,
n V
Cv (T → ∞) =
n
N kB ,
s
5
Cp (T → ∞) = (1 +
n
)N kB .
s
Problem Set 10, due April 9, 2008
Problem 1: Sethna, (7.16)
Problem 2
Show that for an ideal Fermi gas
∂ ln z
∂T
= −
P
5 f5/2 (z)
2T f3/2 (z)
and check that at low temperature, the adiabatic ratio
γ =
CP
π2
∼ 1+
CV
3
kB T
F
2
,
where F is the Fermi energy, defined to be the highest energy of the single
particle states at zero temperature, that is
Z F
N =
n() d ,
0
where n() is the density of states at energy .
Problem 3
Consider an ideal Fermi gas with energy spectrum ∼ ps , contained in a box of
(hyper) volume V in n dimensions. Show that
PV =
s
E,
n
f
n 2 f (z)
CV
n n
n/s+1 (z)
n/s
=
+1
−
,
N kB
s s
fn/s (z)
s
fn/s−1 (z)
CP − CV
=
N kB
sCV
nN kB
6
2
f(n/s)−1 (z)
fn/s (z)
.
Problem Set 11, due April 16, 2008
Problem 1
Show that the paramagnetic susceptibility of an ideal Fermi gas can be written
in the form
χ =
N gµ2B f1/2 (z)
V kB T f3/2 (z)
Problem 2
Show that the chemical potential and energy per particle of a Fermi gas at low
temperature is given by
π2 2 π4 4
x −
x + ···
µ = F 1 −
12
80
and
E
3
= F
N
5
5π 2 2 π 4 4
x −
x + ···
1+
12
16
0
respectively, where x = kB T /F .
Problem 3
Derive both the low and high temperature magnetic susceptibilities of an assembly of charged massive particles of spin 3/2, to the next to leading order in the
relevant variable.
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