Page 39 - Physical Chemistry

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Chapter 1 minimum point (Fig. 1.12). Hence to locate an extremum, we look for the points
Thermodynamics where dy/dx 0.
2
2
The function dy/dx is the first derivative of y. The second derivative d y/dx is
2
2
defined as the derivative of the first derivative: d y/dx d(dy/dx)/dx.
Partial Derivatives
In thermodynamics we usually deal with functions of two or more variables. Let z be a
function of x and y; z f(x, y). We define the partial derivative of z with respect to x as
0z f1x ¢x, y2 f1x, y2
Figure 1.12 a 0x b lim ¢x (1.29)
¢xS0
y
Horizontal tangent at maximum This definition is analogous to the definition (1.25) of the ordinary derivative, in that
and minimum points.
if y were a constant instead of a variable, the partial derivative (
z/
x) would become
y
just the ordinary derivative dz/dx. The variable being held constant in a partial deriv-
ative is often omitted and (
z/
x) written simply as
z/
x. In thermodynamics there
y
are many possible variables, and to avoid confusion it is essential to show which vari-
ables are being held constant in a partial derivative. The partial derivative of z with re-
spect to y at constant x is defined similarly to (1.29):
0z f1x, y ¢y2 f1x, y2
a b lim
0y x ¢yS0 ¢y
There may be more than two independent variables. For example, let z g(w, x, y).
The partial derivative of z with respect to x at constant w and y is
0z g1w, x ¢x, y2 g1w, x, y2
a b lim
0x w,y ¢xS0 ¢x
How are partial derivatives found? To find (
z/
x) we take the ordinary derivative
y
yx
2 3
of z with respect to x while regarding y as a constant. For example, if z x y e ,
2 2
yx
yx
3
then (
z/
x) 2xy ye ; also, (
z/
y) 3x y xe .
y x
Let z f(x, y). Suppose x changes by an infinitesimal amount dx while y remains
constant. What is the infinitesimal change dz in z brought about by the infinitesimal
change in x? If z were a function of x only, then [Eq. (1.26)] we would have dz
(dz/dx) dx. Because z depends on y also, the infinitesimal change in z at constant y is
given by the analogous equation dz (
z/
x) dx. Similarly, if y were to undergo an
y
infinitesimal change dy while x were held constant, we would have dz (
z/
y) dy.
x
If now both x and y undergo infinitesimal changes, the infinitesimal change in z is the
sum of the infinitesimal changes due to dx and dy:
0z 0z
dz a b dx a b dy (1.30)*
0x y 0y x
In this equation, dz is called the total differential of z(x, y). Equation (1.30) is often
used in thermodynamics. An analogous equation holds for the total differential of a
function of more than two variables. For example, if z z(r, s, t), then
0z 0z 0z
dz a b dr a b ds a b dt
0r s,t 0s r,t 0t r,s
Three useful partial-derivative identities can be derived from (1.30). For an infin-
itesimal process in which y does not change, the infinitesimal change dy is 0, and
(1.30) becomes
0z
dz a b dx y (1.31)
y
0x y