Modern Engineering Thermodynamics

7.8 Numerical Values for Entropy 221
WHAT ARE PERPETUAL MOTION MACHINES?
Devices that supposedly operate using processes that violate either the first or second laws of thermodynamics or that are
required to be reversible represent various forms of perpetual motion machines. When the operation of a device depends on
the violation of the first law of thermodynamics it is called a perpetual motion machine of the first kind (e.g., a heat engine
that produces power but does not absorb heat from the environment). When the operation of a device depends on the violation
of the second law, it is called a perpetual motion machine of the second kind (e.g., an adiabatic air compressor in which
the air exits at a lower temperature than it entered), and when it requires a reversible process to operate it is called a perpetual
motion machine of the third kind (e.g., a wheel on a shaft that, once started, continues to rotate indefinitely).
No perpetual motion machines operate as proposed and have for centuries been the source of frauds brought on the
unsuspecting public by unscrupulous or naive inventors.
and dividing by the appropriate absolute temperature and rearranging gives
dS =
dQ
= dQ
+ dW rev − dW act
T
rev
T
act
T
(7.25)
For a work-producing heat engine, dW act ≤ dW and both are positive work quantities, since they represent
rev
energy leaving the system; Eq. (7.25) can be rearranged to produce
dS >
dQ
(7.26)
T
Equation (7.26) is known as the Clausius inequality. ItisClausius’s mathematical form of the second law of
thermodynamics for a closed system. Dropping the subscript on the bracketed term and thus allowing it
to represent either a reversible or actual process produces the following somewhat more general mathematical
second law expression:
dS ≥
dQ
(7.27)
T
and
where the equality sign is used for a reversible heat transport of energy.
I
cycle
act
dQ
≤ 0 (7.28)
T
7.8 NUMERICAL VALUES FOR ENTROPY
In Chapter 3, we discussed five methods for finding numerical values for properties: thermodynamic equations
of state, thermodynamic tables, thermodynamic charts, direct experimental measurements, and the formulae of
statistical thermodynamics. The same five methods can be used to find numerical values for the specific entropy.
In this section, we focus on the use of thermodynamic equations of state, tables, and charts.
Energy and entropy are thermodynamic properties and therefore mathematical point functions. Consequently,
the energy and entropy changes of a system depend only on the beginning and ending states of a process and
not on the actual thermodynamic path taken by the process between these states. Therefore, for a closed system,
we can write the differential energy and entropy balances as
and
ðdEÞ rev = ðdEÞ act = dE = dQ rev − dW rev = dQ act − dW act
ðdSÞ rev
= ðdSÞ act
= dS =
dQ T
Combining the “reversible” path parts of these two equations, we get
rev
dQ rev = TdS = dE + dW rev
(7.29)