3.3.2 Quantification of Entropy in Thermodynamics
Carnot in 1824 defined entropy for a hypothetical reversible engine, where he shows
the mechanical equivalence of heat energy. Entropy is a variable defined by
ds ¼ À
dQ
T
for a reversible process
ð
Þ
ð 3:31Þ
where Q is the heat energy in Joule or Calorie and T is temperature in Kelvin. In
1851, William Thomson (Baron Kelvin) and Clausius published Carnot’s work
posthumously, where it is shown that entropy is a state variable. “If the integral of
a quantity around any closed path is zero, that quantity is called a state variable, that
it has a value that is characteristic only of the state of the system regardless of how
that state was arrived at” Halliday and Resnick (1966):
I
ds ¼ 0
reversible process
ð
Þ
ð 3:32Þ
We should point out that neither heat energy Q nor temperature T is a perfect
differential of any function. However, ds ¼
dQ
T is a perfect differential of a function.
The fact that entropy is a state function can easily be proven for an ideal gas. Entropy
will return to its initial value whenever the temperature returns to its initial value, for
any reversible process. Of course, reversible process does not exist in real life. It is
just an imaginary idealization created for expedience in formulation.
3.3.2.1 Gibbs-Duhem Relation
The fundamental equation for total energy U is a function of entropy, volume, and
extensive parameters given by
U ¼ U S, V, N 1 , . . . N r
ð
Þ
ð 3:33Þ
Taking the first differential of the total energy, we can write
dU ¼
∂U
∂S
dS þ
∂U
∂V
dV þ
∂U
∂N 1
dN 1 þ ⋯ þ
∂U
∂N r
dN r
ð3:34Þ
Partial derivatives for individual terms are defined as follows:
3.3 Second Law of Thermodynamics
85