60
2 Macroscopic Thermodynamics
ln P = −γ ln V + ln K
,
which similarly defines a straight line, but with slope −γ and intercept ln K . A
Carnot cycle is thus represented simply by a parallelogram in the P V -plane (but
with logarithmic scales), one pair of sides having slopes −1, the other pair of sides
having slopes −γ , as seen in Fig. 2.2b, with intersection points identified by the
integers 1, 2, 3, and 4, as in the more traditional representation of the Carnot cycle
illustrated in Fig. 2.2a.
We see directly from this figure that the geometric property of the equality of
the lengths of the opposite sides of a parallelogram translates directly into the
equality of their projections onto the volume axis: for example, the projection of
the upper isotherm onto the ln V -axis is ln V 2 − ln V 1 , while the projection of the
lower isotherm onto the ln V - axis is ln V 3 − ln V 4 . The equality of these projections
gives directly
ln V 2 − ln V 1 = ln V 3 − ln V 4
or
ln
V 2
V 1
= ln
V 3
V 4
.
Of course, there is nothing special about our selection of the parallelogram sides
representing the two isotherms. Precisely the same result is obtained from the
equality of the projections onto the ln V -axis of the parallelogram sides representing
the two adiabats, namely, ln V 4 − ln V 1 = ln V 3 − ln V 2 .
The Carnot Cycle for a Quantum Ideal Gas
A specific Carnot cycle for a photon gas (as the primary exemplar of a quantum ideal
gas) is illustrated in Fig. 2.3. An isothermal change corresponding to the transition
from (V 1 = 0, P 1 ) to (V 2 , P 2 ) in this Carnot diagram has already been discussed in
Example 2.4.
We may utilize Eq. (1.2.12) for the internal energy of the photon gas to obtain
the changes U for each of the four segments of the Carnot cycle, namely,
U 12 = u(T expn )V 2 , ,U 23 = −
1
3
V 2 +V
V 2
u(T ) dV ,
U 34 = −(V 2 + V )u(T comp ) , ,U 41 = 0 .
Now, as U(T , V ) is a function of state, the net change in the internal energy for a
closed-loop cycle must be zero: hence, by summing the above four contributions to
U for the full Carnot cycle, we obtain
2 Macroscopic Thermodynamics
ln P = −γ ln V + ln K
,
which similarly defines a straight line, but with slope −γ and intercept ln K . A
Carnot cycle is thus represented simply by a parallelogram in the P V -plane (but
with logarithmic scales), one pair of sides having slopes −1, the other pair of sides
having slopes −γ , as seen in Fig. 2.2b, with intersection points identified by the
integers 1, 2, 3, and 4, as in the more traditional representation of the Carnot cycle
illustrated in Fig. 2.2a.
We see directly from this figure that the geometric property of the equality of
the lengths of the opposite sides of a parallelogram translates directly into the
equality of their projections onto the volume axis: for example, the projection of
the upper isotherm onto the ln V -axis is ln V 2 − ln V 1 , while the projection of the
lower isotherm onto the ln V - axis is ln V 3 − ln V 4 . The equality of these projections
gives directly
ln V 2 − ln V 1 = ln V 3 − ln V 4
or
ln
V 2
V 1
= ln
V 3
V 4
.
Of course, there is nothing special about our selection of the parallelogram sides
representing the two isotherms. Precisely the same result is obtained from the
equality of the projections onto the ln V -axis of the parallelogram sides representing
the two adiabats, namely, ln V 4 − ln V 1 = ln V 3 − ln V 2 .
The Carnot Cycle for a Quantum Ideal Gas
A specific Carnot cycle for a photon gas (as the primary exemplar of a quantum ideal
gas) is illustrated in Fig. 2.3. An isothermal change corresponding to the transition
from (V 1 = 0, P 1 ) to (V 2 , P 2 ) in this Carnot diagram has already been discussed in
Example 2.4.
We may utilize Eq. (1.2.12) for the internal energy of the photon gas to obtain
the changes U for each of the four segments of the Carnot cycle, namely,
U 12 = u(T expn )V 2 , ,U 23 = −
1
3
V 2 +V
V 2
u(T ) dV ,
U 34 = −(V 2 + V )u(T comp ) , ,U 41 = 0 .
Now, as U(T , V ) is a function of state, the net change in the internal energy for a
closed-loop cycle must be zero: hence, by summing the above four contributions to
U for the full Carnot cycle, we obtain