208
4 Mean Values and Thermodynamics
to the Helmholtz free energy of an imperfect crystal in thermal equilibrium at
temperature T will attain its minimum for n equal to n (the equilibrium number
of Frenkel defects), minimize A(n) with respect to n, and show that w is given
in terms of N, M, and n by
w = k B T ln
(N − n)(M − n)
n
2
,
and show that for a given value of w the equilibrium number of Frenkel defects
can be approximated as
n
√
MN e
−w/(2k B T ) .
19. By utilizing expressions (4.1.32–34) for the thermodynamic functions U mix ,
P mix , and A mix , respectively, together with the thermodynamic defining relations for the enthalpy, H , entropy, S, and Gibbs energy, G, show that H mix ,
S mix , and G mix for a multicomponent mixture occupying the common volume
V mix at the same temperature T are additive.
20. Consider two ideal monatomic gases A and B and show that if the N!
factor is not present in the definition of Z(T , V , N), i.e., if Z(T , V , N) for
a pure monatomic gas is defined simply as Z (T , V , N) ≡ z N (T , V ), the
Helmholtz energy A defined as A = −k B T ln Z (T , V , N) is not an extensive
thermodynamic quantity, and hence also the entropy S will not be extensive.
We know from experiment that if we double the number of atoms in an ideal gas
and at the same time also double the volume occupied by the gas, the entropy
doubles. By how much is S in error when we change N to 2N and V to 2V ?
Show explicitly that the incorporation of N! into the definition of Z(T , V , N)
fixes this problem for the Helmholtz energy, and hence also for the entropy.
Why does the internal energy not show a similar problem?
21. The molar entropy of Ar at temperature T = 298 K under a pressure P = 1 bar
is 154.72 J K −1 . Estimate the number of states available to Ar atoms under
these conditions.
22. From the total differential for the characteristic function for the grand ensemble,
Eq. (4.2.28), show that the differential of the chemical potential μ can be
obtained as
dμ = v dP − s dT ,
with v and s the specific (i.e., per particle) volume and entropy, respectively.
23. By considering μ to be a function of V and T , rather than of P and T , show
that
(a)
∂μ
∂v
T
= v
∂P
∂v
T
; (b)
∂μ
∂T
v
= v
∂P
∂T
v
− s .
4 Mean Values and Thermodynamics
to the Helmholtz free energy of an imperfect crystal in thermal equilibrium at
temperature T will attain its minimum for n equal to n (the equilibrium number
of Frenkel defects), minimize A(n) with respect to n, and show that w is given
in terms of N, M, and n by
w = k B T ln
(N − n)(M − n)
n
2
,
and show that for a given value of w the equilibrium number of Frenkel defects
can be approximated as
n
√
MN e
−w/(2k B T ) .
19. By utilizing expressions (4.1.32–34) for the thermodynamic functions U mix ,
P mix , and A mix , respectively, together with the thermodynamic defining relations for the enthalpy, H , entropy, S, and Gibbs energy, G, show that H mix ,
S mix , and G mix for a multicomponent mixture occupying the common volume
V mix at the same temperature T are additive.
20. Consider two ideal monatomic gases A and B and show that if the N!
factor is not present in the definition of Z(T , V , N), i.e., if Z(T , V , N) for
a pure monatomic gas is defined simply as Z (T , V , N) ≡ z N (T , V ), the
Helmholtz energy A defined as A = −k B T ln Z (T , V , N) is not an extensive
thermodynamic quantity, and hence also the entropy S will not be extensive.
We know from experiment that if we double the number of atoms in an ideal gas
and at the same time also double the volume occupied by the gas, the entropy
doubles. By how much is S in error when we change N to 2N and V to 2V ?
Show explicitly that the incorporation of N! into the definition of Z(T , V , N)
fixes this problem for the Helmholtz energy, and hence also for the entropy.
Why does the internal energy not show a similar problem?
21. The molar entropy of Ar at temperature T = 298 K under a pressure P = 1 bar
is 154.72 J K −1 . Estimate the number of states available to Ar atoms under
these conditions.
22. From the total differential for the characteristic function for the grand ensemble,
Eq. (4.2.28), show that the differential of the chemical potential μ can be
obtained as
dμ = v dP − s dT ,
with v and s the specific (i.e., per particle) volume and entropy, respectively.
23. By considering μ to be a function of V and T , rather than of P and T , show
that
(a)
∂μ
∂v
T
= v
∂P
∂v
T
; (b)
∂μ
∂T
v
= v
∂P
∂T
v
− s .