Physical Chemistry Third Edition

D Some Derivations of Formulas and Methods 1263

where the mole fractions are those in the liquid solution. We convert this equation into

a derivative equation:

x 1

P 1

(

∂P 1

∂x 1

)

T,P

+

x 2

P 2

(

∂P 2

∂x 1

)

T,P

 0 (D-12)

The total vapor pressure is the sum of the partial vapor pressures.

PvapP 1 +P 2 (D-13)

At the azeotrope, the total vapor pressure is at a maximum or a minimum with respect

tox 1 , so that

(

∂Pvap

∂x 1

)

T,P



(

∂P 1

∂x 1

)

T,P

+

(

∂P 2

∂x 1

)

T,P

 0 (D-14)

When this equation is substituted into Eq. (D-12),

x 1

P 1



x 2

P 2

(D-15)

or

x 1

x 2



P 1

P 2



y 1

y 2

(D-16)

where we denote the mole fraction in the gas phase byyand where we have used the

fact that in an ideal gas mixture the mole fraction is proportional to the partial pressure

(Dalton’s law). Each mole fraction has the same value in the solution and in the gas

phase, since they have the same ratio and must add to unity. The two curves must

coincide at the azeotrope. They are tangent to each other since they cannot cross.

D.3 Euler’s Theorem

5

A functionfthat depends onn 1 ,n 2 ,n 3 ,...nc, is said to behomogeneous of degreem

in then’s if

f(λn 1 ,λn 2 ,...,λnc)λmf(n 1 ,n 2 ,...,nc) (D-17)

for every positive real value ofλ. Euler’s theorem states that for such a function

mf

∑c

i 1

n 1

(

∂f

∂ni

)

n′

(D-18)

where the subscriptn′means that all of then′’s are held fixed except forni.

To prove the theorem, we differentiate Eq. (D-17) with respect toλ, using the chain

rule:

∑c

i 1

(

∂f

∂(λni)

)

n′

(

∂(λni)

∂(λ)

)

n′

mλm−^1 f(n 1 ,n 2 ,...,nc) (D-19)

(^5) E. A. Desloge,Statistical Physics, Holt Rinehart and Winston, New York, 1966, Appendix 10.