Physical Chemistry Third Edition

4.6 Euler’s Theorem and the Gibbs–Duhem Relation 191

where we have used the relationdP 2 −dP 1 , which follows from our condition that

P 1 +P 2 Pconstant.

−

x 2

x 1

dμ 2 

x 2 RT

x 1 P 2

dP 1

From Dalton’s law of partial pressures,x 2 /P 2 x 1 /P 1 , so that

−

x 2

x 1

dμ 2 

RT

P 1

dP 1 dμ 1

The Gibbs–Duhem relation is often written as a derivative relation instead of a

differential relation. It is necessary that the partial derivatives be taken withTandP

constant since the Gibbs–Duhem relation is valid only for constantTandP. For a

two-component system,

x 1

(

∂μ 1

∂x 1

)

T,P

+x 2

(

∂μ 2

∂x 1

)

T,P

0 (constantTandP) (4.6-12)

Both derivatives must be taken with respect to the same mole fraction. For a system

with more than two components, the equation is

∑c

i 1

xi

(

∂μi

∂xk

)

T,P

0 (constantTandP) (4.6-13)

where every partial derivative must be with respect to the same variable xk.

Equation (4.6-13) is valid for any kind of changes in the mole fractions ifTandP

are constant.

The Experimental Determination of Partial Molar

Quantities

The partial molar volume is the most easily measured partial molar quantity, so we

discuss it as an example. Other partial molar quantities can be evaluated in the same way

if data are available. Assume that we measure the volume of the system as a function

of the amount of the component of interest, keeping the pressure, temperature, and

amounts of other substances fixed. If this volume can be represented by a polynomial

or some other formula, the partial molar volume can be obtained by differentiation.

If only the data points are available, the partial molar volume can be obtained by

numerical differentiation.^3

(^3) D. P. Shoemaker, C. W. Garland, and J. W. Nibler,Experiments in Physical Chemistry, 6th ed.,
McGraw-Hill, New York, 1996, pp. 757ff. See also Robert G. Mortimer,Mathematics for Physical Chemistry,
3rd ed., Academic Press, San Diego, CA, 2005, pp. 335–336.