2.5 Thermodynamic Engines
59
with the equality holding only for a fully reversible cycle (for which ((S) surr
= 0).
We see from the thermodynamics of the Carnot cycle for a classical ideal gas
that thermal energy supplied to the ideal gas by the hot thermal reservoir during the
isothermal expansion of Step 1 is obtained from the First Law as
Q import ≡ Q 1→2 (gas)
= −W 1→2 (gas) = Nk B T high ln
V 2
V 1
,
which is positive, as V 2 > V 1 . Similarly, the thermal energy exported to the cool
reservoir during the isothermal compression of the ideal gas is given by
Q export ≡ Q 3→4 (surr)
or
− Nk B T low ln
V 4
V 3
= Nk B T low ln
V 3
V 4
,
which is also positive, as V 3 > V 4 .
Thus, for a classical ideal gas, the efficiency, η max , for the Carnot cycle is given
explicitly as
η max = 1 −
T low
T high
ln(V 3 /V 4 )
ln(V 2 /V 1 )
.
(2.5.15)
Employment of Eq. (2.5.11) in expression for η max then gives
η max = 1 −
T low
T high
,
(2.5.16)
in accordance with the Carnot theorem.
Interlude Figure 2.2b enables a simpler derivation of the final expression for η max .
Employment of logarithmic scales for the pressure and volume axes can greatly
simplify both the geometric rendering of the Carnot cycle and the simplification of
the expression for the maximal efficiency for the Carnot cycle [12]. The condition
P V = K (K a constant) that defines an isotherm in the P V -plane is replaced by
ln P = − ln V + ln K ,
which has the form of a straight line with slope −1 and intercept ln K, while the
condition P V γ = K that defines an adiabat in the P V -plane is replaced by