Physical Chemistry Third Edition

4.6 Euler’s Theorem and the Gibbs–Duhem Relation 189

kwith independent variablesn 1 ,n 2 ,...,nc,Euler’s theoremstates that

kf

∑c

i 1

ni

(

∂f

∂ni

)

n′

(Euler’s theorem) (4.6-2)

where the subscriptn′stands for holding all of then’s constant except forni. A proof

of this theorem is found in Appendix D.

LetYstand for any extensive quantity. SinceYis homogeneous of degree 1 in the

n’s ifTandPare constant, Euler’s theorem becomes

Y

∑c

i 1

ni

(

∂Y

∂ni

)

T,P,n′



∑c

i 1

niYi (4.6-3)

whereYiis the partial molar quantity for substanceiand wherecis the number of

components. Remember thatTandPare held constant in the differentiations of the

partial molar quantities. Two important examples of Eq. (4.6-3) are

G

∑c

i 1

niμi (4.6-4)

and

V

∑c

i 1

niVi (4.6-5)

Equation (4.6-3) is a remarkable relation that gives the value of an extensive quantity

as a weighted sum of partial derivatives. An unbiased newcomer to thermodynamics

would likely not believe this equation without its mathematical proof.

Euler’s theorem can also be written in terms of themean molar quantityYm, defined

byYmY/n, wherenis the total amount of all components:

Ym

1

n

∑c

i 1

niYi

∑c

i 1

xiYi (4.6-6)

wherexiis themole fractionof substance numberi, equal toni/n.

EXAMPLE4.21

In a solution of acetone (component 1) and chloroform (component 2)x 1  0 .531 and

V 1  74 .2cm^3 mol−^1 .IfVm77.0 cm^3 mol−^1 at this composition, findV 2.

Solution

From Euler’s theorem

V 2 

Vm−x 1 V 1

x 2



77 .0cm^3 mol−^1 −( 0. 531 )

(

74 .2cm^3 mol−^1

)

0. 469

 80 .2cm^3 mol−^1