Partial Differential Relations
Now let us rewrite Eq. 12–3 as
(12–4)where
Taking the partial derivative of Mwith respect to yand of Nwith respect to
xyields
The order of differentiation is immaterial for properties since they are con-
tinuous point functions and have exact differentials. Therefore, the two rela-
tions above are identical:
(12–5)This is an important relation for partial derivatives, and it is used in calculus
to test whether a differential dzis exact or inexact. In thermodynamics, this
relation forms the basis for the development of the Maxwell relations dis-
cussed in the next section.
Finally, we develop two important relations for partial derivatives—the
reciprocity and the cyclic relations. The function zz(x,y) can also be
expressed as xx(y,z) if yand zare taken to be the independent variables.
Then the total differential of xbecomes, from Eq. 12–3,
(12–6)Eliminating dxby combining Eqs. 12–3 and 12–6, we have
Rearranging,
(12–7)The variables yand zare independent of each other and thus can be varied
independently. For example,ycan be held constant (dy0), and zcan be
varied over a range of values (dz0). Therefore, for this equation to be
valid at all times, the terms in the brackets must equal zero, regardless of
the values of yand z. Setting the terms in each bracket equal to zero gives
(12–8)a (12–9)0 z
0 xb
ya0 x
0 yb
za0 x
0 yb
xSa0 x
0 yb
za0 y
0 zb
xa0 z
0 xb
y 1a0 x
0 zb
ya0 z
0 xb
y 1 Sa0 x
0 zb
y110 z> 0 x (^2) y
ca
0 z
0 x
b
y
a
0 x
0 y
b
z
a
0 z
0 y
b
x
ddyc 1 a
0 x
0 z
b
y
a
0 z
0 x
b
y
d dz
dzca
0 z
0 x
b
y
a
0 x
0 y
b
z
a
0 z
0 y
b
x
ddya
0 x
0 z
b
y
a
0 z
0 x
b
y
dz
dxa
0 x
0 y
b
z
dya
0 x
0 z
b
y
dz
a
0 M
0 y
b
x
a
0 N
0 x
b
y
a
0 M
0 y
b
x
02 z
0 x 0 y
¬and¬a
0 N
0 x
b
y
02 z
0 y 0 x
Ma
0 z
0 x
b
y
¬and¬Na
0 z
0 y
b
x
dzM dxN dy
Chapter 12 | 655