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The Gibbs–Duhem Equation Section 10.3
Activity coefficients of nonvolatile solutes can be found from vapor-pressure data Determination of Activities
and Activity Coefficients
using the Gibbs–Duhem equation, which we now derive. Taking the total differential
of G n m [Eq. (9.23)], we find as the change in G of the solution in any infini-
i i i
tesimal process (including processes that change the amounts of the components of
the solution)
dG d a n m a d1n m 2 a 1n dm m dn 2 a n dm a m dn i
i
i
i
i
i
i
i
i
i
i
i
i i i i i
The use of dG S dT V dP m dn [Eq. (4.73)] gives
i i i
S dT V dP a m dn a n dm a m dn i
i
i
i
i
i
i i i
a n dm S dT V dP 0 (10.17)
i
i
i
This is the Gibbs–Duhem equation. Its most common application is to a constant-
T-and-P process (dT 0 dP), where it becomes

a n dm a n dG 0 const. T, P (10.18)
i
i
i
i
i i
Equation (10.18) can be generalized to any partial molar quantity as follows. If Y

is any extensive property of a solution, then Y n Y i [Eq. (9.26)] and dY
i
i

n dY Y dn . Equation (9.25) with dT 0 dP reads dY Y dn .
i i i i i i i i i
Equating these two expressions for dY, we get

a n dY 0 or a x dY 0 const. T, P (10.19)
i
i
i
i
i i
where the form involving the mole fractions x was found by dividing by the total num-

i
ber of moles. The Gibbs–Duhem equation (10.19) shows that the Y i ’s are not all inde-

pendent. Knowing the values of r 1 of the ’s as functions of composition for a
Y
i

solution of r components, we can integrate (10.19) to find .
Y
r
For a two-component solution, Eq. (10.19) with Y V (the volume) reads

x dV A x dV B 0 or dV A (x /x ) dV B at constant T and P. Thus, dV A and dV B
B
A
B
A
must have opposite signs, as in Fig. 9.9. Similarly, dm and dm must have opposite
A
B
signs when the solution’s composition changes at constant T and P.
Activity Coefficients of Nonvolatile Solutes
For a solution of a solid in a liquid solvent, the vapor partial pressure of the solute over
the solution is usually immeasurably small and cannot be used to find the solute’s activ-
ity coefficient. Measurement of the vapor pressure as a function of solution composition
gives P , the solvent partial pressure, and hence allows calculation of the solvent activ-
A
ity coefficient g as a function of composition. We then use the integrated Gibbs–Duhem
A
equation to find the solute activity coefficient g .
B
After division by n n the Gibbs–Duhem equation (10.18) gives
B
A
x dm x dm 0 const. T, P (10.20)
A
A
B
B
From (10.6) we have m m°(T, P) RT ln g RT ln x and
A
A
A
A
dm RT d ln g 1RT>x 2 dx const. T, P
A
A
A
A
Similarly, dm RT d ln g (RT/x ) dx at constant T and P. Substitution for dm A
B
B
B
B
and dm in (10.20) gives after division by RT:
B
x d ln g dx x d ln g dx 0 const. T, P (10.21)
A
B
A
B
B
A