2.3 New Thermodynamic State Functions
45
with S 0 (N) the integration constant. This expression for the absolute entropy,
however, is not extensive: for example, a simple calculation for a doubling of both
the number of particles in and the volume occupied by an ideal classical gas does
not result in a doubling of the absolute entropy of that thermodynamic system.
We shall employ a mathematical artifice in order to arrive at a correct form for
the absolute entropy: we note that Eq. (2.3.12a) is the sum of two terms, but only
the first term has the normal format in which products of thermodynamic variables
occur as products between extensive and intensive variables, as exemplified, for
example, by P V , T S, μN . We shall therefore rewrite Eq. (2.3.12a) as
dS ideal gas =
C V (T )
T
dT +
Nk B
(V /N )
d
V
N
(2.3.13a)
which, upon integration, gives the absolute entropy as
S ideal gas = S 0 + C V ln T + Nk B ln
V
N
.
(2.3.13b)
We shall see later (in Chap. 5) that the integration constant, S 0 (N), for an N-particle
ideal classical gas is obtained as
(S 0 ) ideal gas =
5
2
+
3
2
ln
2πmk B
h 2
Nk B ≡ Ns 0 .
(2.3.14)
Note that as S 0 (N) is proportional to N, it is an extensive quantity. Upon carrying
out a doubling of both the number of particles and the volume occupied by an ideal
classical gas, it will be clear that the absolute entropy of that gas is also doubled, so
that extensivity of the absolute entropy is maintained by Eq. (2.3.13b).
Either Eq. (2.3.12a) or Eq. (2.3.13a) may be utilized to obtain a correct
expression for the entropy change S between an initial (reference) state (T 0 , V 0 )
and a final state (T , V ) for an ideal gas of classical structureless particles, namely,
((S) ideal gas = Nk B ln
V
V 0
+ C V ln
T
T 0
,
(2.3.15)
with C V =
3
2 Nk B .
Example 2.3 Thermodynamic functions for a classical ideal gas.
We have seen that the internal energy, U(T , V ), for a thermodynamic system can
be obtained in principle via Eq. (2.3.8), while the enthalpy, H , is given in terms
of U(T , V ) by Eq. (2.3.2), and the Helmholtz and Gibbs energies may then be
determined from Eqs. (2.3.5a) and (2.3.6c), respectively.
We begin by recalling that an ideal gas is defined by its equation of state, P V =
Nk B T . Because we see from Eq. (2.3.9) that the internal energy U for an ideal gas
45
with S 0 (N) the integration constant. This expression for the absolute entropy,
however, is not extensive: for example, a simple calculation for a doubling of both
the number of particles in and the volume occupied by an ideal classical gas does
not result in a doubling of the absolute entropy of that thermodynamic system.
We shall employ a mathematical artifice in order to arrive at a correct form for
the absolute entropy: we note that Eq. (2.3.12a) is the sum of two terms, but only
the first term has the normal format in which products of thermodynamic variables
occur as products between extensive and intensive variables, as exemplified, for
example, by P V , T S, μN . We shall therefore rewrite Eq. (2.3.12a) as
dS ideal gas =
C V (T )
T
dT +
Nk B
(V /N )
d
V
N
(2.3.13a)
which, upon integration, gives the absolute entropy as
S ideal gas = S 0 + C V ln T + Nk B ln
V
N
.
(2.3.13b)
We shall see later (in Chap. 5) that the integration constant, S 0 (N), for an N-particle
ideal classical gas is obtained as
(S 0 ) ideal gas =
5
2
+
3
2
ln
2πmk B
h 2
Nk B ≡ Ns 0 .
(2.3.14)
Note that as S 0 (N) is proportional to N, it is an extensive quantity. Upon carrying
out a doubling of both the number of particles and the volume occupied by an ideal
classical gas, it will be clear that the absolute entropy of that gas is also doubled, so
that extensivity of the absolute entropy is maintained by Eq. (2.3.13b).
Either Eq. (2.3.12a) or Eq. (2.3.13a) may be utilized to obtain a correct
expression for the entropy change S between an initial (reference) state (T 0 , V 0 )
and a final state (T , V ) for an ideal gas of classical structureless particles, namely,
((S) ideal gas = Nk B ln
V
V 0
+ C V ln
T
T 0
,
(2.3.15)
with C V =
3
2 Nk B .
Example 2.3 Thermodynamic functions for a classical ideal gas.
We have seen that the internal energy, U(T , V ), for a thermodynamic system can
be obtained in principle via Eq. (2.3.8), while the enthalpy, H , is given in terms
of U(T , V ) by Eq. (2.3.2), and the Helmholtz and Gibbs energies may then be
determined from Eqs. (2.3.5a) and (2.3.6c), respectively.
We begin by recalling that an ideal gas is defined by its equation of state, P V =
Nk B T . Because we see from Eq. (2.3.9) that the internal energy U for an ideal gas