Page 87 - Modern physical chemistry

76 The First Law [or Energy

. f(X+Ax,Y+L1Y)-f(x,Y+L1Y) _ '( )_ of
hm -fx X,Y --, [4.73]
~~O Ax ~
l!.y~O

. f( x, Y + L1Y) - f( x, Y) _ ' ( ) _ of
hm -fy X,Y --, [4.74]
l!.y~O L1y oy
so that as Ax, L1y, and L1fbecome infinitesimally small, equation (4.72) becomes

[4.75]

Note that x and Y need only be functions of the independent variables. Also, an infin-
itesimal change in such a function is called an exact differential. Equation (4.75) expresses
a fundamental relationship among such differentials.
When variables X h X2,"" xn enter independently into function!, we similarly find that
df= L[~) [4.76]
ax· dxj •
J other x's
Furthermore, X h X2,'''' xn variables need only be functions of the independent variables
for the overall problem.
If we consider the internal energy E as a function of T and V, we have

dE=[OE) dT+[OE) dV. [4.77]
aT v oV T
The subscript on the partial derivative indicates the other independent variable or vari-
ables; it lists the other variable or variables held constant in the differentiation. When
energy E is considered a function of T and P, we similarly have

dE=(OE) dT+(aE) dP. [4.78]
aT p oP T

Example 4.7
Relate
(~;)p to (~;)v and (~;)T to (~~)T

for a uniform thermodynamic system.
In the system, volume V is a function of temperature T and pressure P; so

dV =(OV) dT+(OV) dP.
aT p oP T
Since we may consider the internal energy E to be a function of either T and Vor T and
P, both (4.77) and (4.78) are valid. Substituting the form for dV into (4.77) gives us