218
5 Atomic Systems
If we choose to separate out the pressure dependence, we may write μ tr (P , T ) in
the conventional manner of chemical thermodynamics as
μ tr (T , P ) = μ
◦
tr (T ) + k B T ln P ,
(5.1.11)
wherein μ ◦
tr (T ) is given explicitly by
μ
◦
tr (T ) = − k B T ln
k B T
3
.
(5.1.12)
Note that not only have we arrived at the thermodynamic form taken by the chemical
potential for a pure substance in the gaseous state, but we have also obtained a
formula that allows us to calculate the actual value of the translational contribution
to μ ◦ (T ) for particles of a given mass m at a specified temperature T .
5.1.2 The Isothermal–Isobaric Partition Function
Expression (5.1.1) for the canonical partition function Z N (T , V ) enables us both to
extend the defining relation (3.4.9) for (T , ζ ) to an isothermal–isobaric ensemble
of N structureless atoms possessing only translational states as
, P ; N) =
∞
0
Z N (T , V )e
−βP V dV ,
(5.1.13)
with the translational states treated classically as continuum states and to obtain an
explicit closed-form expression for it. The closed-form expression is obtained upon
substituting Eq. (5.1.1) into the extended definition (5.1.13) to give
, P ; N) =
1
N! 3N (T )
∞
0
V
N e
−βP V dV ,
(5.1.14)
followed by evaluation of the definite integral over volume. For N 1, we thereby
obtain , P ; N) as
, P ; N) =
k B T
3 P
.
(5.1.15)
By substituting this result for , P ; N) into Eq. (4.3.19a) for the entropy, we
obtain the expression
S tr (T , P ; N) = Nk B ln
k B T e
5
2
3 P
,
(5.1.16a)
5 Atomic Systems
If we choose to separate out the pressure dependence, we may write μ tr (P , T ) in
the conventional manner of chemical thermodynamics as
μ tr (T , P ) = μ
◦
tr (T ) + k B T ln P ,
(5.1.11)
wherein μ ◦
tr (T ) is given explicitly by
μ
◦
tr (T ) = − k B T ln
k B T
3
.
(5.1.12)
Note that not only have we arrived at the thermodynamic form taken by the chemical
potential for a pure substance in the gaseous state, but we have also obtained a
formula that allows us to calculate the actual value of the translational contribution
to μ ◦ (T ) for particles of a given mass m at a specified temperature T .
5.1.2 The Isothermal–Isobaric Partition Function
Expression (5.1.1) for the canonical partition function Z N (T , V ) enables us both to
extend the defining relation (3.4.9) for (T , ζ ) to an isothermal–isobaric ensemble
of N structureless atoms possessing only translational states as
, P ; N) =
∞
0
Z N (T , V )e
−βP V dV ,
(5.1.13)
with the translational states treated classically as continuum states and to obtain an
explicit closed-form expression for it. The closed-form expression is obtained upon
substituting Eq. (5.1.1) into the extended definition (5.1.13) to give
, P ; N) =
1
N! 3N (T )
∞
0
V
N e
−βP V dV ,
(5.1.14)
followed by evaluation of the definite integral over volume. For N 1, we thereby
obtain , P ; N) as
, P ; N) =
k B T
3 P
.
(5.1.15)
By substituting this result for , P ; N) into Eq. (4.3.19a) for the entropy, we
obtain the expression
S tr (T , P ; N) = Nk B ln
k B T e
5
2
3 P
,
(5.1.16a)