Modern Engineering Thermodynamics

362 CHAPTER 11: More Thermodynamic Relations
We begin this chapter by rounding out our list of useful thermodynamic properties by defining two new properties,
the Helmholtz and Gibbs functions. We then move on to develop a series of general mathematical results, called
the Maxwell equations, that relate a number of thermodynamic properties. We end this chapter by using the principle
of corresponding states to develop a set of generalized thermodynamic property charts that are valid for many
real gases.
11.2 TWO NEW PROPERTIES: HELMHOLTZ AND GIBBS FUNCTIONS
If we consider a stationary closed system containing a pure substance subjected only to a moving boundary
mechanical work mode, then the combined energy and entropy balance is given by Eq. (7.30) as
du = Tds− pdv (11.1)
Since any two independent properties fix the thermodynamic state of a pure substance subjected to only one
work mode (see Chapter 4), we can take the two independent properties here to be s and v, or
u = us, ð vÞ
The total differential of this composite function, then, has the form
du =
∂u
ds +
∂u
dv (11.2)
∂s v ∂v s
Comparing Eqs. (11.1) and (11.2) we see that
T =
∂u
∂s v
and
p = − ∂u
∂v s
For this system, we can also write, from Eq. (7.31),
and, in this case, we take the two independent properties to be s and p, so that
whose total differential is
On comparing Eqs. (11.3) and (11.4), we see that
and
dh = Tds+ vdp (11.3)
h = hs, ð pÞ
dh =
∂h
ds +
∂s p
T =
v =
∂h
∂s p
∂h
∂p
∂h
∂p
s
dp (11.4)
s
EXAMPLE 11.1
To illustrate the relation between the constant volume and constant pressure specific heats and entropy, begin with Eqs.
(11.1) and (11.3) and show that the constant volume and constant pressure specific heats are related to specific entropy by:
c v = T
∂s
∂T v
and
c p = T
∂s
∂T P