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3.1 THE MATHEMATICAL PROPERTIES OF STATE FUNCTIONS 47

EXAMPLE PROBLEM 3.1

a. Calculate
0f 0f
0a b 0a b
2
2
0f 0f 0 f 0 f £ 0x y≥ £ 0y x ≥
a b , a b , a b , a b , , and
0x y 0y x 0x 2 y 0y 2 x 0y x 0x y
x
for the function f(x, y) = ye + xy + xln y .
b. Determine if f(x, y) is a state function of the variables x and y.
c. If f(x, y) is a state function of the variables x and y, what is the total differential df?

Solution
0f 0f x
a. a b = ye + y + ln y, a b = e + x +
x
x
0x y 0y x y
2
2
0 f 0 f x
a b = ye , a b =-
x
0x 2 y 0y 2 x y 2
0f 0f
0a b 0 a b
£ 0x y ≥ 1 £ 0y ≥ 1
x
= e + 1 + , x = e + 1 +
x
0y x y 0x y y
b. Because we have shown that
0f 0f
0a b 0 a b
£ 0x y ≥ £ 0y ≥
= x
0y x 0x y
f(x, y) is a state function of the variables x and y. Generalizing this result, any
well-behaved function that can be expressed in analytical form is a state function.
c. The total differential is given by

0f 0f
df = a b dx + a b dy
0x y 0y x
x
x
x
= (ye + y + ln y)dx + ae + x + bdy
y
Two other important results from differential calculus will be used frequently.
Consider a function z = f(x, y) that can be rearranged to x = g(y, z) or y = h(x, z) .
For example, if P = nRT>V , then V = nRT>P and T = PV>nR . In this case

0x 1
a b = (3.6)
0y z a 0y b
0x z
The cyclic rule will also be used:

0x 0y 0z
a b a b a b =-1 (3.7)
0y z 0z x 0x y
It is called the cyclic rule because x, y, and z in the three terms follow the orders x, y, z;
y, z, x; and z, x, y. Equations (3.6) and (3.7) can be used to reformulate Equation (3.3)
shown next below:
0P 0P
dP = a b dT + a b dV
0T V 0V T