5.7 Problems for This Chapter
255
25. Show, by substituting ln , P ; N) from expression (5.1.15) for the
isothermal–isobaric partition function for a gas of N structureless atoms into
the formal expression (4.3.19a) for the entropy, that the result is indeed the
Sackur–Tetrode equation (5.1.16a).
26. Employ Eq. (4.2.5) for the grand partition function T , V ) together with
Eq. (4.2.30b) for the entropy S to show that it is given explicitly in terms of the
single-particle canonical partition function z as
S = Nk B
1 + T
∂ ln z
∂T
V
+ ln
z
N
,
and that it reduces to the Sackur–Tetrode equation for the translational entropy
for an (ideal) gas of structureless atoms.
27. Consider an ideal gaseous binary mixture consisting of N 1 A atoms and N 2 B
atoms in thermal equilibrium at a temperature T in a container of volume V .
Employ Eq. (4.1.28) for the canonical partition function for this binary mixture
together with Eq. (5.1.1) to show that the internal energy, U , and the Helmholtz
energy, A, are extensive functions. Obtain an expression for the total pressure,
P mix , of the binary AB mixture, and show that P mix follows Dalton’s law of
partial pressures for an ideal gas mixture. Given the extensivity of U and A,
what can you say about the entropy, S, the enthalpy, H , and the Gibbs energy,
G?
28. An approximate canonical partition function for the Dieterici nonideal gas has
the form
Z(N, V , T ) =
1
N!
exp
N
V
0
e
−aN/(N 2
0 k B T y)
y − NB
dy
,
in which a and B are constants, and N 0 is the Avogadro number. Obtain
expressions for the Dieterici equation of state and the internal energy U for
this model nonideal gas. Your final expression for U will contain an integral for
which there is no closed-form solution.
29. Assume that a monatomic Einstein solid, in which each atom behaves as
a simple three-dimensional harmonic oscillator with the same fundamental
vibrational frequency, ν E , is in thermal equilibrium with its vapour, and that
an energy is required to take an atom from the solid phase into the vapour
phase. The canonical partition function for the N s atoms making up the solid
phase is, according to the Einstein model, given by
Z s (T , V s ; N s ) = e
βN s
2 sinh
E
2T
−3N s
,
while the canonical partition function for the N g atoms making up the vapour
phase is given by
255
25. Show, by substituting ln , P ; N) from expression (5.1.15) for the
isothermal–isobaric partition function for a gas of N structureless atoms into
the formal expression (4.3.19a) for the entropy, that the result is indeed the
Sackur–Tetrode equation (5.1.16a).
26. Employ Eq. (4.2.5) for the grand partition function T , V ) together with
Eq. (4.2.30b) for the entropy S to show that it is given explicitly in terms of the
single-particle canonical partition function z as
S = Nk B
1 + T
∂ ln z
∂T
V
+ ln
z
N
,
and that it reduces to the Sackur–Tetrode equation for the translational entropy
for an (ideal) gas of structureless atoms.
27. Consider an ideal gaseous binary mixture consisting of N 1 A atoms and N 2 B
atoms in thermal equilibrium at a temperature T in a container of volume V .
Employ Eq. (4.1.28) for the canonical partition function for this binary mixture
together with Eq. (5.1.1) to show that the internal energy, U , and the Helmholtz
energy, A, are extensive functions. Obtain an expression for the total pressure,
P mix , of the binary AB mixture, and show that P mix follows Dalton’s law of
partial pressures for an ideal gas mixture. Given the extensivity of U and A,
what can you say about the entropy, S, the enthalpy, H , and the Gibbs energy,
G?
28. An approximate canonical partition function for the Dieterici nonideal gas has
the form
Z(N, V , T ) =
1
N!
exp
N
V
0
e
−aN/(N 2
0 k B T y)
y − NB
dy
,
in which a and B are constants, and N 0 is the Avogadro number. Obtain
expressions for the Dieterici equation of state and the internal energy U for
this model nonideal gas. Your final expression for U will contain an integral for
which there is no closed-form solution.
29. Assume that a monatomic Einstein solid, in which each atom behaves as
a simple three-dimensional harmonic oscillator with the same fundamental
vibrational frequency, ν E , is in thermal equilibrium with its vapour, and that
an energy is required to take an atom from the solid phase into the vapour
phase. The canonical partition function for the N s atoms making up the solid
phase is, according to the Einstein model, given by
Z s (T , V s ; N s ) = e
βN s
2 sinh
E
2T
−3N s
,
while the canonical partition function for the N g atoms making up the vapour
phase is given by