2.3 New Thermodynamic State Functions
47
The energy thus transferred will be seen either as an increase (for > 0) or as a
decrease (for V < 0) in the number of photons making up the photon gas. This is
a consequence of N not being an independent variable for a photon gas.
Because a slow (quasistatic) isothermal volume change is a reversible change,
the entropy change is given by Q/T , giving S as 5
S =
4σ
3c
T
3 ,
,V ≡ V f − V i .
We may now employ this result, with V i = 0, N = 0, corresponding to the internal
piston face being flush with the back of the cylinder, to create the photon gas by
moving the piston quasistatically to a position at which V = V f , with N photons in
the cylinder. As the internal energy U equals zero for V = 0, it thus follows from
the First Law of Thermodynamics that Q = 0, whence S ≡ Q/T = 0.
The above result for S implies that the entropy of a gas of N photons occupying
a container of volume V at temperature T will be [8, 9]
S(T , V ) =
4σ
3c
V T
3 .
The total energy required to carry out such an isothermal expansion of a photon gas
from volume 0 to volume V is given by Q exp = T S. Because the pressure of a
photon gas depends only upon T , the expansion occurs at constant pressure, so that
Q exp ≡ Q P corresponds to the change H exp in the enthalpy H = U + P V . We
may thus deduce that the enthalpy of a photon gas in equilibrium at temperature T
is given by
H (T , V ) =
4σ
3c
V T
4 .
This clearly shows that the enthalpy H represents the energy required to form a
photon gas at temperature T plus that needed to carry out the pressure–volume work
necessary to generate the space that the photon gas occupies.
It may also be interesting to see how an adiabatic change is characterized for
a quantum ideal gas. We first recall that an adiabatic change is characterized by
allowing no energy to be exchanged between the thermodynamic system and its
surroundings. In the context of a photon gas, this means explicitly that no photons
may be absorbed by or emitted from the container walls during the process: this is
equivalent to saying that the container walls must be perfectly reflecting mirrors.
Hence, an adiabatic volume change for a photon gas requires that the number of
photons in the gas remain constant, so that both the entropy, S(T , V ), of the photon
gas and the number, N(T , V ), of photons in the gas must remain constant. From the
5 Note that this entropy change differs from the logarithmic volume dependence of the entropy
change for an isothermal volume change for the classical ideal gas, given by Eq. (2.3.12b).
47
The energy thus transferred will be seen either as an increase (for > 0) or as a
decrease (for V < 0) in the number of photons making up the photon gas. This is
a consequence of N not being an independent variable for a photon gas.
Because a slow (quasistatic) isothermal volume change is a reversible change,
the entropy change is given by Q/T , giving S as 5
S =
4σ
3c
T
3 ,
,V ≡ V f − V i .
We may now employ this result, with V i = 0, N = 0, corresponding to the internal
piston face being flush with the back of the cylinder, to create the photon gas by
moving the piston quasistatically to a position at which V = V f , with N photons in
the cylinder. As the internal energy U equals zero for V = 0, it thus follows from
the First Law of Thermodynamics that Q = 0, whence S ≡ Q/T = 0.
The above result for S implies that the entropy of a gas of N photons occupying
a container of volume V at temperature T will be [8, 9]
S(T , V ) =
4σ
3c
V T
3 .
The total energy required to carry out such an isothermal expansion of a photon gas
from volume 0 to volume V is given by Q exp = T S. Because the pressure of a
photon gas depends only upon T , the expansion occurs at constant pressure, so that
Q exp ≡ Q P corresponds to the change H exp in the enthalpy H = U + P V . We
may thus deduce that the enthalpy of a photon gas in equilibrium at temperature T
is given by
H (T , V ) =
4σ
3c
V T
4 .
This clearly shows that the enthalpy H represents the energy required to form a
photon gas at temperature T plus that needed to carry out the pressure–volume work
necessary to generate the space that the photon gas occupies.
It may also be interesting to see how an adiabatic change is characterized for
a quantum ideal gas. We first recall that an adiabatic change is characterized by
allowing no energy to be exchanged between the thermodynamic system and its
surroundings. In the context of a photon gas, this means explicitly that no photons
may be absorbed by or emitted from the container walls during the process: this is
equivalent to saying that the container walls must be perfectly reflecting mirrors.
Hence, an adiabatic volume change for a photon gas requires that the number of
photons in the gas remain constant, so that both the entropy, S(T , V ), of the photon
gas and the number, N(T , V ), of photons in the gas must remain constant. From the
5 Note that this entropy change differs from the logarithmic volume dependence of the entropy
change for an isothermal volume change for the classical ideal gas, given by Eq. (2.3.12b).