Page 89 - Modern physical chemistry

18 The First Law for Energy

The coefficients on the right vary directly with the volume. So we rewrite equation (4.82) as
dV =aVdT- ,8VdP. [4.83J

The relative rate of change of volume with temperature at constant pressure,

1 [av) =a, [4.84J
V aT p
is called the coe.fficient of cubical expansion. For a condensed phase, this is independent
of pressure except at very high pressures. However, it does vary slowly with temperature.
The negative relative rate of change of volume with pressure at constant temperature,

-~ [ ~~)T =,8, [4.85J

is called the coe.fficient of compressibility. For a condensed phase, this is nearly inde-
pendent of temperature and pressure.
At constant volume, dV = O. Then equation (4.83) yields

a [4.86J
,8

4. 14 Exact Differentials
We have noted that dq and dw are not small changes in thermodynamic properties.
Thus, they are not exact differentials. On the other hand, dE is an exact differential. Let
us now consider how such differentials depend on the independent variables.
Suppose that u is a function of variables x and y such that
du = M(X,y) dx+N(x,y) dy. [4.87J

But formula (4.75) tells us that
au
au
du=- dx+- dy; [4.88J
ax ay
so
M=au N=au. [4.89J
ax' ay
Differentiating M with respect to y and N with respect to x yields

aM a 2u aN a 2u
-=--, -=-- [4.90J
ay ayax ax axay
Since the second derivative is independent of the order of differentiation, we have

aM aN
-=- [4.91 J
ay ax