Page 91 - Modern physical chemistry

80 The First Law for Energy

~)
-
( On -V [4.99]
A,
A T,P,nB
Coefficient VA is called the partial molar volume of A, coefficient VB the partial molar
volume of B, in the solution.
Next, divide the overall equation (4.98) by dn to get

dV _ V- dnA -V; dnB [4.100]
-- A--+ B--'
dn dn dn
Then let the system be built up from nothing at constant temperature, pressure, and
concentrations. We have
dnA nA dnB = nB dV V
--=- [4.101]
dn n dn n dn n
and equation (4.100) becomes

[4.102]

whence
[4.103]
The partial molar volume of a constituent is the contribution of that constituent per mole
to the total volume. When the solution is homogeneous, this contribution depends on T,
P, and the concentrations. It is independent of the total number of moles n. Thus, it is
an intensive property.
In equation (4.102), nAln is the mole fraction of A, XA , while naln is the mole fraction
of B, X B • Thus, the equation can be rewritten as

[4.104]

But since
[4.105]
we have
V-(--)
n =VA + VB-VA X B· [4.106]
If one plots the molar volume Vln against the mole fraction X B, one generally obtains
a curve. But the tangent line drawn at a given point on the curve represents formula

V
n

FIGURE 4.5 Plot of molar volume Vln against mole
o fraction X B of B. At a particular X B • a tange~t line i~
drawn. This intersects the vertical axes at VA and VB'