Activity and Activity Coefficients

We have now seen and used several expressions for the chemical potential of a substance or component in a mixture. For a one-component ideal gas we had

(1)
where μo is the chemical potential when p = po and po is usually one atmosphere.

For a mixture of ideal gases it can be shown that for each component, i, the chemical potential is given by,

(2)
For a nonideal gas we used the fugacity, f, instead of the pressure and the chemical potential for one component is,
(3)
For a mixture of nonideal gases it can be shown that,
(4)
only now the fugacity of component i is a function of the pressures of all the gases in the mixture,
(5)
For ideal solutions we found that,
(6)
These expressions for chemical potential all have the form of a reference or standard state chemical potential plus RT times the logarithm of something related to pressure or concentration. This form turns out to be very important, so important that G. N. Lewis used it to give the most general case of chemical potential as
(7)
The quantity, ai, is called the "activity" of component i and Equation 7 should be regarded as the definition of activity. Notice that the activity has no units.

All of the special cases we have been considering so far can be reconciled to this definition of activity. Thus, for an ideal gas mixture,

for a nonideal gas mixture,
for an ideal solution,
and so on. In a nonideal solution we would have to just write Equation 7 again,
(7)
In Equation 7 all the nonidealities of the solution are absorbed into the activity. We will see a more convenient way to write this below under the heading "activity coefficient."

We can find the activity of a component of a nonideal solution from measurements of the vapor pressure of that component in the vapor in equilibrium with the solution. We know that the chemical potential of a component must be the same in the vapor as in the liquid. that is, from Equations 2 and 7 we obtain,

(8a, b)
but for pure component i, we must have,
(9)
(Note: When dealing with liquid solutions it is customary to write the chemical potential of the pure liquid as μ*il instead of the usual μoil, which means that the standard state for a liquid is the pure liquid itself.    pi* is the vapor pressure of the pure liquid.)

Subtracting Equation 9 from Equation 8b yields,
(10a, b)
so that
(11)
If the solution were ideal pi would be given by Raoult's law and the activity would be just the mole fraction.

Activity Coefficient

Sometimes it is convenient to write the activity as the product of an ideal part times a nonideality correction part. For example, in a real solution we might write,

(12)
or, in a real gas we would write,
(13)
so that
(14)
In the latter case we already know that
(15)
so that
(16)
In the case of the nonideal gas γ → 1 as p → 0. In the case of a nonideal solution γi → 1 as Xi → 1. From Equations 11 and 12 we see that,
(17)
from which we conclude that for a nonideal solution,
(18)

In general, the activity coefficient is a unitless parameter that contains all of the nonideality of a system.

WRS

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