Page 90 - Modern physical chemistry

4.15 Partial Molar Properties 79

Next suppose that functions M and N are given satisfying condition( 4.91). Construct
the relation
M=av [4.92]
ax'
which differentiates to
aM _ a v _ a [av) [4.93]
2
ay - ayax - ax ay .
Use (4.91) to replace the derivative of M with that of N:
~~=a~(~~} [4.94]

Integrate equation (4.94) with y constant:

[4.95]

But A may vary with y. Write it in the form d¢J(y)/dy to get

N = av + diP . [4.96]
ay dy
Finally, construct du as in (4.87) and reduce:

du = M dx + N dy = av dx + av dy + diP dy = dV + diP. [4.97]
ax ay dy
Since dVand d¢J are exact, differential du is exact.
We see that equation (4.91) is a necessary and sufficient condition that du in formula
(4.87) be exact.

4. 15 Partial Molar Properties

Any extensive thermodynamic property of a homogeneous material varies directly,
linearly, with the amount of material considered. When the material is a solution, the
property also varies linearly with the amount of each constituent.
As a simple example, consider the volume V of a homogeneous mixture of substances
A and B. Let n A be the number of moles of A, n B the number of moles of B, and n the
total number of moles in the solution.
At a given temperature T and pressure P, volume V is a function of nA and nB . Then
application of formula (4.75) yields

av)
[av)
--
dV = [ anA dnA + an B dnB = VA dnA + VB dnB, [4.98]
T,P,nB T,P,nA
where