Physical Chemistry Third Edition

182 4 The Thermodynamics of Real Systems

4.5 Multicomponent Systems

In an equilibrium open multicomponent system with one gas or liquid phase the number

of variables required to specify the equilibrium macroscopic state isc+2, where

cstands for the number of independent substances, calledcomponents. The number

of components is equal to the number of substances whose amounts can separately be

varied under the given conditions. It is also equal to the minimum number of substances

from which the system can be prepared under the given conditions. For example, if a

gaseous system at equilibrium contains NO 2 it will also contain N 2 O 4. The system can

be prepared by adding only NO 2 or by adding only N 2 O 4 and allowing the system to

equilibrate. Two substances are present, but there is only one component.

The Chemical Potential and Partial Molar Quantities

In an equilibrium one-phase simple system containingccomponents, the Gibbs energy

depends onc+2 variables, which can be chosen asT,P, and the mole fractions of all

components:

GG(T,P,n 1 ,n 2 ,...,nc) (4.5-1)

whereniis the amount of substance numberi(measured in moles). The differential

ofGis

dG

(

∂G

∂T

)

P,n

dT+

(

∂G

∂P

)

T,n

dP+

∑c

i 1

(

∂G

∂ni

)

T,P,n′

dni (4.5-2)

where the subscriptnstands for keeping the amounts of all of the components fixed

and the subscriptn′stands for keeping the amount of every component fixed except

for component numberi.

The first two partial derivatives in Eq. (4.5-2) are no different from the partial

derivatives in Eqs. (4.2-20) and (4.2-21) for a closed system. In those equations, the

amounts of all substances present were held fixed because the system was closed. In

Eq. (4.5-2), the amounts of all substances are held fixed because that is how partial

derivatives are defined. Therefore, we can write

dG−SdT+VdP+

∑c

i 1

μidni (4.5-3)

whereμiis called thechemical potentialand is defined by

μi

(

∂G

∂ni

)

T,P,n′

(definition of the

chemical potential)

(4.5-4)

Equation (4.5-3) is called theGibbs equationor thefundamental relation of chemical

thermodynamics. We could also choose to write

dG

(

∂G

∂T

)

V,n

dT+

(

∂G

∂V

)

T,n

dV+

∑c

i 1

(

∂G

∂ni

)

T,V,n′

dni (4.5-5)