58
2 Macroscopic Thermodynamics
If we note from Fig. 2.2 that (V 1 , P 1 ) and (V 4 , P 4 ) lie on the same adiabat, and
that according to Eq. (2.3.12b) for the entropy of an ideal classical (monatomic) gas,
V T
3
2 is constant for an adiabatic process, then it follows that V 4 and V 1 are related
by
V 4 = V 1
T high
T low
3
2
.
Moreover, as (V 2 , P 2 ) and (V 3 , P 3 ) are also points on a common adiabat, V 3 and V 2
are similarly related by
V 3 = V 2
T high
T low
3
2
.
From these two conditions, we see that
V 4
V 3
=
V 1
V 2
,
(2.5.11)
so that W (cycle) for an ideal monatomic gas becomes
W (cycle) = Nk B (T low − T high ) ln
V 2
V 1
≡ −W export ,
(2.5.12)
with W export representing the exportable work from the cycle.
The efficiency η, as defined via the first half of Eq. (2.5.8a) can be written as
η =
Q import − Q export
Q import
= 1 −
Q export
Q import
.
(2.5.13)
This expression satisfies the requirements of the First Law of Thermodynamics,
but does not yet incorporate restrictions introduced by the Second Law that led
to the second half of Eq. (2.5.8a). Specifically, the entropy transferred from the
working substance (system) to the cold reservoir during the third step of the Carnot
cycle must be greater than (or, for a reversible change, at least equal to) the entropy
transferred from the hot reservoir to the working substance in the first step. Thus,
we may write
Q export
T low
≥
Q import
T high
, or
Q export
Q import
≥
T low
T high
,
so that
η ≤ 1 −
T low
T high
,
(2.5.14)