2.5 Thermodynamic Engines
63
W 2→3 (gas) = −
V 3
V 2
K
V
1
3
dV
or
W 2→3 (gas) = 3(P 3 V 3 − P 2 V 2 ) .
During the third step of the Carnot cycle, which is a reversible isothermal
compression at temperature T comp from (V 3 , P comp ) to (V 1 , P comp ), we find that the
work done on the photon gas is
W 3→4 (gas) = −
V 4
V 3
P comp dV = −P comp (V 4 − V 3 ) ,
which is positive, as V 3 > V 4 .
During the final step of the cycle, which is a reversible adiabatic compression
from (V 4 , P comp ) at T comp to (V 1 , P comp ) at T expn , the work done by the gas has the
same form as that obtained from step 2, namely,
W 4→1 (gas) = 3(P expn V 1 − P comp V 4 ) .
The total work associated with the photon gas for the Carnot cycle is thus
W (cycle) = W 1→2 (gas) + W 2→3 (gas) + W 3→4 (gas) + W 4→1 (gas)
= −P expn (V 2 − V 1 ) + 3(P comp V 3 − P expn V 2 ) − P comp (V 4 − V 3 )
+ 3(P expn V 1 − P comp V 4 )
or
W (cycle) = 4[P expn (V 1 − V 2 ) + P comp (V 3 − V 4 )] .
For the Carnot cycle illustrated in Fig. 2.3, V 1 = V 4 = 0 and, as P expn =
4σ T 4
expn /(3c), while P comp = 4σ T 4
comp /(3c), W (cycle) becomes
W (cycle) =
16σ
3c
[T
4
comp V 3 − T
4
expn V 2 ] .
The efficiency η max for taking a photon gas (specifically, blackbody radiation)
through the Carnot cycle is hence given by
η max =
W (cycle)
Q 1→2 (surr)