Modern Engineering Thermodynamics

11.4 Maxwell Equations 367
11.4 MAXWELL EQUATIONS
Two sets of equations are named after the Scottish physicist James Clerk Maxwell (1831−1879): the electromagnetic
field equations and the thermodynamic property equations. The thermodynamic Maxwell equations allow
additional numerical information to be obtained about the nonmeasurable properties u, h, and s from accurately
measured p, v, and T data.
Consider an arbitrarily continuous function of the form
Then, we can write its total differential as
where we set
dz =
∂z
∂x y
dx +
z = zx, ð yÞ
∂z
dy = Mdx + Ndy (11.11)
∂y
x
M =
∂z
∂x y
and
N =
∂z
∂y
x
If we now differentiate M with respect to y while holding x constant and differentiate N with respect to x while
holding y constant, we get
∂M
= ∂2 z
∂y
x
∂y ∂x
and
∂N
= ∂2 z
∂x y ∂x ∂y
Since we require z(x, y) to be a continuous function, it follows that
or that
∂ 2 z
∂y ∂x =
∂M
∂y
x
∂2 z
∂x ∂y
≡ ∂N
∂x y
(11.12)
Recall that the thermodynamic state of any pure substance is fixed by any pair of independent intensive thermodynamic
properties of that substance. That is, any property of a pure substance can be written as a function of
any other two independent properties of that substance. Consequently, if x and y are such independent properties,
then z also is a property, provided that Eq. (11.12) is satisfied.
EXAMPLE 11.4
Suppose we make a series of measurements in the laboratory and think we discovered a new thermodynamic property, call
it z. Our experimental data provide an empirical equation of the form: dz = pdv + v 2 dp. Isz a new property?
Solution
The unknown is whether or not z is a new thermodynamic property. Equation (11.11) here has the form
dz = Mdx + Ndy = pdv+ v 2 dp
so M = p, N = v 2 , x = v, and y = p. The cross differentials in Eq. (11.12) are
∂M
= ∂p
= 1
∂y
x
∂p
v
(Continued )