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4 Mean Values and Thermodynamics
an ideal gas of N such particles. Show how the pressure and internal energy are
related.
9. Obtain expressions for the internal energy, U ur , the heat capacity at constant
volume, C ur
V , and the pressure, P ur , for a gas of N noninteracting ultrarelativistic (ur) particles, and obtain the relation between the pressure and the
internal energy for such an ultra-relativistic ideal gas. How does the internal
energy, heat capacity, and pressure for an ultra-relativistic gas compare with
those for a classical (non-relativistic) gas?
10. Starting from the defining relation (4.2.8) for N and (4.2.7) for the probabilities
p N , with γ ≡ −βμ, show that N is obtained from the grand potential
, V , μ) of Eq. (4.2.31) as
N = −
∂∂
∂μ
T ,V
.
11. Starting from the fundamental expression
S(T , V , μ) ≡ −k B
N,r N
p N,r N ln p N,r N
for the entropy in the grand ensemble, show that the grand potential , V , μ)
is given in terms of thermodynamic quantities as
, V , μ) = U(T , V , μ) − T S(T , V , μ) − μN(T , V , μ) .
12. Consider the lowest twelve energy levels for a particle of mass m in a box of
volume V box , namely,
n x n y n z =
h 2
8mV
2
3
box
(n
2
x + n
2
y + n
2
z ) ,
with n x , n y , n z each taking nonzero positive integer values such that n 2
x + n 2
y +
n 2
z ≤ 27. The ground level for this model thus has energy 0 ≡ 3h 2 /(8mV
2
3
box ):
given that the mass m and volume V box are such that 0 /k B = 100 K, find the
relative energies n x n y n z // 0 corresponding to n 2
x + n 2
y + n 2
z ≤ 27, evaluate the
partition function z(T ) for T = 200 K, and compute the fractional populations
for the 12 energy levels. Carry out the same calculations, still for temperature
T = 200 K, but with the box volume decreased so that (V
box )
2
3 =
1
2 (V box )
2
3 ,
obtained by isothermally compressing the system. What can you say about how
system populations behave when the system volume is changed isothermally?
13. Show that the defining relation U ≡ NE = N
r p r E r and relation
(1.2) both give the same result for the contribution U int (T ) to the total
(thermodynamic) internal energy U(T ) arising from the internal states of the
4 Mean Values and Thermodynamics
an ideal gas of N such particles. Show how the pressure and internal energy are
related.
9. Obtain expressions for the internal energy, U ur , the heat capacity at constant
volume, C ur
V , and the pressure, P ur , for a gas of N noninteracting ultrarelativistic (ur) particles, and obtain the relation between the pressure and the
internal energy for such an ultra-relativistic ideal gas. How does the internal
energy, heat capacity, and pressure for an ultra-relativistic gas compare with
those for a classical (non-relativistic) gas?
10. Starting from the defining relation (4.2.8) for N and (4.2.7) for the probabilities
p N , with γ ≡ −βμ, show that N is obtained from the grand potential
, V , μ) of Eq. (4.2.31) as
N = −
∂∂
∂μ
T ,V
.
11. Starting from the fundamental expression
S(T , V , μ) ≡ −k B
N,r N
p N,r N ln p N,r N
for the entropy in the grand ensemble, show that the grand potential , V , μ)
is given in terms of thermodynamic quantities as
, V , μ) = U(T , V , μ) − T S(T , V , μ) − μN(T , V , μ) .
12. Consider the lowest twelve energy levels for a particle of mass m in a box of
volume V box , namely,
n x n y n z =
h 2
8mV
2
3
box
(n
2
x + n
2
y + n
2
z ) ,
with n x , n y , n z each taking nonzero positive integer values such that n 2
x + n 2
y +
n 2
z ≤ 27. The ground level for this model thus has energy 0 ≡ 3h 2 /(8mV
2
3
box ):
given that the mass m and volume V box are such that 0 /k B = 100 K, find the
relative energies n x n y n z // 0 corresponding to n 2
x + n 2
y + n 2
z ≤ 27, evaluate the
partition function z(T ) for T = 200 K, and compute the fractional populations
for the 12 energy levels. Carry out the same calculations, still for temperature
T = 200 K, but with the box volume decreased so that (V
box )
2
3 =
1
2 (V box )
2
3 ,
obtained by isothermally compressing the system. What can you say about how
system populations behave when the system volume is changed isothermally?
13. Show that the defining relation U ≡ NE = N
r p r E r and relation
(1.2) both give the same result for the contribution U int (T ) to the total
(thermodynamic) internal energy U(T ) arising from the internal states of the