2.5 Thermodynamic Engines
53
into the system from a heat reservoir at temperature T high and the heat exported from
the system into a heat reservoir at temperature T low as
((S) surr = −
Q import
T high
+
Q export
T low
.
(2.5.2)
We note that ((S) surr includes entropy produced irreversibly during the isothermal
heat transfers. If we solve Eq. (2.5.2) for Q export , we obtain
Q export =
T low
T high
Q import + T low ((S) surr .
(2.5.3)
According to the First Law of Thermodynamics, the change, U , in the internal
energy U when the working thermodynamic substance (referred to more simply as
the ‘heat engine’ or the ‘system’) is carried through a closed thermodynamic cycle,
is given in terms of the work, W ≡ −W export , done on the system, Q import , and
Q export as
((U ) cycle = 0 = −W export + Q import − Q export .
(2.5.4)
The work W export is thus given by
W export = Q import − Q export
= Q import
1 −
T low
T high
− T low ((S) surr ,
(2.5.5)
upon utilizing Eq. (2.5.3) for Q export .
Should the heat engine be reversible, so that ((S) surr = 0, then W export = W
(rev)
export
is maximal, namely,
W
(rev)
export = Q import
T high − T low
T high
= Q import
1 −
T low
T high
,
(2.5.6)
and W export may be expressed as
W export = W
(rev)
export − T low ((S) surr .
(2.5.7)
Relation (2.5.7) for W export has been termed the work-entropy relation for a heat
engine [11], and displays the role of the Second Law of Thermodynamics explicitly.
Relation (2.5.7) is obtained upon assuming that all steps in the operating cycle
of the heat engine have been carried out reversibly, so that ((S) system = 0.
Should any of the steps in the operating cycle involve irreversible changes, then
((S) surr in Eq. (2.5.7) must be replaced by ((S) engine = ((S) surr + ((S) cycle , with
((S) cycle > 0 representing the entropy created by irreversible changes occurring