48
2 Macroscopic Thermodynamics
expression that we have just obtained for the entropy of a photon gas, we can see
that constancy of the entropy during an adiabatic change in volume means that the
temperature must change as V
−
1
3 in order to compensate the volume change: this
result, taken together with the expression for the pressure, P (T ), for a photon gas
implies that the pressure changes as V
−
4
3 or, equivalently, that pressure and volume
are related by
P V
4
3 = const .
Although this result may be reminiscent of the condition P V γ = const., with γ =
C P /C V =
5
3 , that applies to adiabatic changes in a classical ideal monatomic gas,
the resemblance is purely superficial, as C P does not, in fact, exist for a photon gas,
since it is not possible to vary the temperature while simultaneously holding the
pressure fixed, as would be necessary for a determination of C P .
2.4 Expression for Heat Capacity Difference
The heat capacity of a thermodynamic system is defined as the rate of change with
temperature of the heat taken up by the system in a reversible process that is carried
out under specified conditions, such as at constant volume, V , or constant pressure,
P . Expressions for the heat capacities at constant volume and at constant pressure
have been given in Eqs. (2.2.15c) and (2.3.1c), respectively.
To obtain a general expression for the difference between C P and C V , we begin
with the thermodynamic defining relations for C P and C V . We may utilize the
formal definition (2.3.2) of the enthalpy to obtain an equivalent expression,
C P =
∂H
∂T
P
=
∂U
∂T
P
+ P
∂V
∂T
P
,
that will prove to be helpful in obtaining the desired result for the difference C P −
C V for a general thermodynamic system. Indeed, this expression for C P leads to
C P − C V being given as
C P − C V =
∂U
∂T
P
−
∂U
∂T
V
+ P
∂V
∂T
P
.
Now, by starting from the formal mathematical expression for the total differential
dU of the internal energy as a function of T and V , then treating V as a function of
T and P and forming the total differential dV for V (T , P ), dU can be obtained as
dU =
∂U
∂T
V
+
∂U
∂V
T
∂V
∂T
P
dT +
∂U
∂V
T
∂V
∂P
T
dP .
2 Macroscopic Thermodynamics
expression that we have just obtained for the entropy of a photon gas, we can see
that constancy of the entropy during an adiabatic change in volume means that the
temperature must change as V
−
1
3 in order to compensate the volume change: this
result, taken together with the expression for the pressure, P (T ), for a photon gas
implies that the pressure changes as V
−
4
3 or, equivalently, that pressure and volume
are related by
P V
4
3 = const .
Although this result may be reminiscent of the condition P V γ = const., with γ =
C P /C V =
5
3 , that applies to adiabatic changes in a classical ideal monatomic gas,
the resemblance is purely superficial, as C P does not, in fact, exist for a photon gas,
since it is not possible to vary the temperature while simultaneously holding the
pressure fixed, as would be necessary for a determination of C P .
2.4 Expression for Heat Capacity Difference
The heat capacity of a thermodynamic system is defined as the rate of change with
temperature of the heat taken up by the system in a reversible process that is carried
out under specified conditions, such as at constant volume, V , or constant pressure,
P . Expressions for the heat capacities at constant volume and at constant pressure
have been given in Eqs. (2.2.15c) and (2.3.1c), respectively.
To obtain a general expression for the difference between C P and C V , we begin
with the thermodynamic defining relations for C P and C V . We may utilize the
formal definition (2.3.2) of the enthalpy to obtain an equivalent expression,
C P =
∂H
∂T
P
=
∂U
∂T
P
+ P
∂V
∂T
P
,
that will prove to be helpful in obtaining the desired result for the difference C P −
C V for a general thermodynamic system. Indeed, this expression for C P leads to
C P − C V being given as
C P − C V =
∂U
∂T
P
−
∂U
∂T
V
+ P
∂V
∂T
P
.
Now, by starting from the formal mathematical expression for the total differential
dU of the internal energy as a function of T and V , then treating V as a function of
T and P and forming the total differential dV for V (T , P ), dU can be obtained as
dU =
∂U
∂T
V
+
∂U
∂V
T
∂V
∂T
P
dT +
∂U
∂V
T
∂V
∂P
T
dP .