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A perpetual motion machine of the second kind is a machine which
spontaneously converts thermal energy into mechanical work. When the
thermal energy is equivalent to the work done, this does not violate the
law of conservation of energy. However it does violate the more subtle

second law of thermodynamics . The signature of a perpetual motion
machine of the second kind is that there is only one heat reservoir
involved, which is being spontaneously cooled without involving a

transfer of heat to a cooler reservoir. This conversion of heat into useful
work, without any side effect, is impossible, according to the second law
of thermodynamics.
Recall that a perpetual motion machine of the first kind produces

work without the input of energy. It, thus, violates the first law of
thermodynamics: the law of conservation of energy

2.12. Entropy
In this and the following sections we show that the second law of
thermodynamics is related to a thermodynamic variable called entropy, S,

and that we can express the second law quantitatively in terms of this
variable.
We start by considering a Carnot cycle. For such a cycle we can write

Q  Q T  T Q Q
1 2  1 2  1  2 (2.12.1)
Q T T T
1 1 1 2
We now discard the absolute value notation, recognizing in the process
that whether the Carnot cycle is carried out clockwise, as an engine, or

counterclockwise, as a refrigerator, Q and Q always have opposite signs.
2
1
We can therefore write
Q Q
1  2  0 (2.12.2)
T 1 T 2

This equation states that the sum of the algebraic quantities Q/T is zero
for a Carnot cycle.
As a next step, we want to generalize Eq. 2.12.2 to any reversible cycle,

not just a Carnot cycle.,. To. do this, we approximate any reversible cycle
as an assembly of Carnot cycles. Figure 2.12.1shows an arbitrary
reversible cycle superimposed on a family of isotherms. We can
approximate the actual cycle by connecting the isotherms by suitably

chosen short segments of adiabatic lines, thus forming an assembly of thin

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