
The Second and Third Laws 61
second cycle is (1 −
η
2
)Q
H
=(1−
η
1
)(1 −
η
2
)Q
H
. This means the efficiency of the
composite cycle is
η
12
=1−(1 −
η
1
)(1 −
η
2
) and this is not
η
1
+
η
2
. In fact, the
composition rule for the series case is (1−
η
12
)=(1−
η
1
)(1 −
η
2
).Theefficiencies
in a serially connected case is not the sum because the heat input for the second is
not the same as the first, but is reduced.
More explicitly, if Q
H
(T) is the heat absorbed at the higher temperature T, the
work done in an infinitesimal cycle is dW =
Q
H
(T)
T
dT. So for the temperatures
T
H
,T
1
,T
2
,...,T
L
characterizing many cycles connected in series resulting in a com-
posite cycle operating between T
H
and ...