Physical Chemistry , 1st ed.

For systems that have more than one chemical component, we will have to give
the chemical potential a label (typically a number or a chemical formula) to
specify which component. The chemical potential for a single component i
assumes that only the amount of the ith component,ni, varies, and the amounts
of all other components nj,ji, remain constant. Equation 4.46 is therefore
written

i

n

G

i



T,p,nj(ji)

(4.47)

If we want to consider the infinitesimal change in Gnow, we must broaden it
by considering possible changes in amount of substance, too. The general ex-
pression for dGnow becomes

dG

G

T



p,n’s

dT (^) 

G

p



T,n’s

dp (^)
0
i 

n

G

i



T,p,nj(ji)

dni

or

dGS dT V dp (^)
0
i
idni (4.48)
where the summation has as many terms as there are different substances in
the system. Equation 4.48 is sometimes referred to as the fundamental equa-
tion of chemical thermodynamics,since it embodies all state variables of condi-
tions and amounts.
The chemical potential iis the first example of a partial molar quantity.It
expresses the change in a state variable, the Gibbs free energy, versus molar
amount. For pure substances, the chemical potential is simply equal to the
change in the Gibbs free energy of the system as the amount of material
changes. For systems of more than one component, the chemical potential
does not equal the change in free energy of the pure material because each
component interacts with the other, which affects the total energy of the sys-
tem. If all components were ideal, this wouldn’t happen, and partial molar
quantities would be the same for any component in any system.*
Because of the relationships between the various energies defined by ther-
modynamics, chemical potential can also be defined in terms of the other en-
ergies, but with different state variables held constant:
i
n

U

i



S,V,nj(ji)

(4.51)

4.8 The Chemical Potential and Other Partial Molar Quantities 109

*Partial molar quantities can be defined for any state variable. For example, the partial
molar change in entropy Siis defined as

Si nS

i



nj(ji)

(4.49)

and for whatever other conditions remain constant. Similarly, a partial molar volume Viis
defined as

Vi

n

V

i



T,p,nj(ji)

(4.50)

The partial molar volume is an especially useful concept for condensed phases. It is also the
reason why the mixing of 1 L of water and 1 L of alcohol yields a solution whose volume
is not 2 L (it’s a little less than 2 L): from the strict thermodynamic sense, volumes are not
directly additive, but partial molar volumesare. [Note that partial molar quantities (except
for ) have the same symbolism as molar quantities, that is, the line over the variable. Thus,
care should be exercised when using these two quantities.]