2.5 Thermodynamic Engines
61
Fig. 2.3 Carnot cycle in the
P V -plane for a quantum
ideal gas (a gas of photons).
Reproduced from [9] with the
permission of the American
Association of Physics
Teachers
adiabat
isotherm (T=T expn )
isotherm (T=T comp )
adiabat
1
2
3
4
Carnot cycle for a quantum ideal gas
V
P
u(T expn )V 2 −
1
3 u(T m ))V − u(T comp )(V 2 + ) = 0 ,
in which T m is a temperature such that T comp < T m < T expn and satisfies the
intermediate-value theorem. For T expn − T comp T comp and V V , we may
approximate V by the differential dV and, upon approximating the difference
between the internal energy densities for the isothermal segments of the Carnot
cycle by du, we may replace u(T comp ) by
u(T comp ) = u(T m ) + [u(T comp ) − u(T m )] ≤ u(T m ) + du ,
so that to first order in differentials, U Carnot = 0 is reasonably approximated by
the differential equation
V du −
4
3 u(T ) dV 0 .
Upon utilizing the condition ((S) cycle = 0 for the entropy as well, we may
replace u(T ) dV by
u(T ) dV = V du −
uV
T
dT = 0 ,
whence the Carnot cycle for a photon gas requires that the internal energy density
u(T ) satisfy the differential equation
du
u
− 4
dT
T
= 0 .
Integration of this differential equation gives u(T ) as