Physical Chemistry Third Edition

260 6 The Thermodynamics of Solutions

The Activity and Activity Coefficient of a Nonideal Gas

The standard state for a nonideal gas is defined to be the hypothetical ideal gas

at the standard-state pressureP◦. From Eq. (5.4-6) we see that Eq. (6.3-6) applies

to a nonideal gas if we define the activity of a nonideal gas to be the ratio of the

fugacity toP◦:

aifi/P◦ (nonideal gas) (6.3-15)

We define theactivity coefficientγiof a nonideal gas as

γi

ai(real)

ai(ideal)

(6.3-16)

so that the activity coefficient of a gas equals the ratio of the fugacity to the pressure:

γi

fi/P◦

Pi/P◦



fi

Pi

(6.3-17)

The activity coefficient of an ideal gas equals unity. The chemical potential of a nonideal

gas can be written

μiμ◦i+RTln(ai)μ◦i+RTln

(

fi

P◦

)

μ◦i+RTln

(

γiPi

P◦

)

(6.3-18)

The activity coefficient of a gas is also known as thefugacity coefficientand is some-

times denoted byφiinstead of byγi. If the value of the activity coefficient of a real

gas is greater than unity, the gas has a greater activity and a greater chemical potential

than if it were ideal at the same temperature and pressure. If the value of the activity

coefficient is less than unity, the gas has a lower activity and a lower chemical potential

than if it were ideal.

Activities and Activity Coefficients in Solutions

We now obtain formulas for solutions that are neither ideal nor dilute. The activities are

specified in different ways, depending on whether one of the components is designated

as the solvent and depending on the composition variables used. There are two different

schemes that use mole fractions as composition variables, called convention I and

convention II.

Convention I

In this treatment all of the components are treated in the same way, with no substance

designated as the solvent. The standard state for each component is the pure substance

at pressureP◦and at the temperature of the solution, just as with an ideal solution.

As before, we obtain working formulas for volatile substances by assuming that the

solution is at equilibrium with an ideal gas mixture. The chemical potential of substance

iin the solution is equal to its chemical potential in the vapor phase. If we write the

chemical potential in the solution in terms of the activity as in Eq. (6.3-6),

μiμ◦i(I)+RTlnai(I)μ(g)i μ◦i(g)+RTln

(

Pi

P◦

)

(6.3-19)