PHYS 530: Problem Set 2
Due: 4:30 pm, 31 January 2013
If the answer is shown, all the marks will be given for the derivation not for writing down
the answer.
1. [5] Justify
Ω(1
,V,ε
)
≈
Vf
(
ε
);
(1)
that is, prove that for a single point-particle of given energy
ε
conta
inedinabox
,the
number of accessible states is approximately proportional to the box volume.
2. [7] Prove that the mixing entropy for two identical gases initially at the same temper-
ature is either zero (when the two containers have the same initial density) or positive
(when the initial densities are diFerent). In other words, prove that the mixing entropy
of identical gases at the same temperature cannot be negative.
3. [5] Solve problem 1.3 in Pathria’s book.
Two systems
A
and
B
, of identical composition, are brought together and allowed to
exchange both energy and particles, keeping volumes
V
A
and
V
B
constant. Show that
the minium value of the quantity
dE
A
/dN
A
is given by
μ
A
T
B
−
μ
B
T
A
T
B
−
T
A
,
(2)
where the
μ
’s and the
T
’s are respective chemical potentials and temperatures.
4. [8] Solve problem 1.4 in Pathria’s book.
In a classical gas of hard spheres (of diameter
D
) the spatial distribution of the particles
is no longer uncorrelated. Roughly speaking, the presence of
n
particles in the system
leaves only a volume (
V
−
nv
0
)ava
i
lab
leforthe(
n
+ 1)th particle; clearly
v
0
would be
proportional to
D
3
. Assuming that
Nv
0
±
V
, determine the dependence of Ω(
N,V,E
)
on
V
(compare to Ω(
N,E,V
)
∝
V
N
) and show that, as a result of this,
V
in the ideal-
gas law
PV
=
NkT
=
nRT
gets replaced by (
V
−
b
), where
b
is four times the actual
volume occupied by the particles.
5. [5] Solve problem 1.6 in Pathria’s book.
A cylindrical vessel 1 m long and 0.1 m in diameter is ±lled with a monatomic gas at
P
=1atmand
T
= 300 K. The gas is heated by an electric discharge, along the axis
of the vessel, which releases an energy of 10
4
J. What will the temperature of the gas
be immediately after the discharge?
6. [22] Solve problem 1.7 in Pathria’s book.
Study the statistical mechanics of an extreme relativistic gas characterised by the
single-particle energy states
ε
(
n
x
,n
y
,n
z
)=
hc
2
L
±
n
2
x
+
n
2
y
+
n
2
z
²
1
/
2
;
n
x
,n
y
,n
z
=1
,
2
,
3
,...,
(3)
1