Modern Engineering Thermodynamics

7.7 Clausius’s Definition of Entropy 219
T
An infinite
number of
infinitesimal
heat engine
cycles
dT
An arbitrary
thermodynamic
cycle for a
closed system
FIGURE 7.11
An infinite number of infinitesimal heat engine cycles approximating an arbitrary closed system thermodynamic cycle.
A
operating over an infinitely small temperature difference, T H ≈ T L = T and ðQ H Þ rev − jQ L j rev ≈ dQ rev .Then,in
the limit, the previous equation becomes
" lim ∑ # I
Q dQ
=
= 0 (7.21)
n!∞
n cycles
T
T
The temperature T in this equation is the absolute temperature at the point where the heat transfer dQ occurs.
Clausius then noted the remarkable result that, since, by definition,
I
ðAny thermodynamic property differentialÞ = 0
cycle
the argument of the integral in Eq. (7.21) must define a thermodynamic property. That is,
dQ
= Differential of some thermodynamic property
T
rev
But, which property? The term dQ by itself is a path function and thus cannot be a thermodynamic property
rev
differential. However, when dQ is divided by T, a property differential results. Clausius realized that he had
rev
discovered a new thermodynamic property and he chose to name it entropy 7 and represent the total entropy of a
system by the symbol S, where
dS =
dQ
(7.22)
T
or
or
S 2 − S 1 =
s 2 − s 1 = S 2 − S 1
m
Z 2
1
cycle
= 1 m
rev
dQ
T
Z 2
1
rev
rev
dQ
T
rev
(7.23)
(7.24)
Be careful to note that Eqs. (7.22)–(7.24), which define entropy, are for a closed system of fixed mass m only. The
effect of mass flow on system entropy is taken up in a separate section of this chapter.
The use of a relative temperature scale in a grouping of units can sometimes be confusing. For example, when a
temperature unit appears in the denominator of a units grouping, it can be written either as °F or R (or °C orK
in SI) because only the degree size there is important. Therefore, Eq. (7.24) indicates that the units of specific
entropy can be written correctly in either of the following forms:
s ½in Btu/ ðlbm.°FÞŠ≡ s ½in Btu/ ðlbm.RÞŠ
7 Here is a translation of how Clausius, in 1865, described why he chose the word entropy for the name of his new property. “We
might call S the transformational content of the body, just as we termed the quantity U the heat and work content of the body. But since
I believe it is better to borrow terms for important quantities from the ancient languages so that they may be adopted unchanged in all
modern languages, I propose to call the quantity S the entropy of the body, from the Greek word ητροπή, meaning a transformation.”