# Is the activity coefficient always a constant?

The activity of a species $i$ is defined as $a_i = e^\frac{\mu_i-\mu_i^\ominus}{RT}$, with $\mu_i$ the chemical potential, and $\mu_i^\ominus$ the chemical potential under standard conditions.

Very often, for a species in a non-ideal solution, the activity is written $\gamma_i x_i$, where $x_i$ is the mole fraction and $\gamma$ is a constant known as the activity coefficient for that species. This implies that $\mu_i = \mu_i^\ominus + RT\log \gamma_i x_i$.

What I don't have a good idea of is whether the $\gamma_i x_i$ formula is exact (under some assumptions) or an approximation. Are there situations in which the activity depends on the mole fraction in some non-linear way, or upon the concentrations of other species in addition to species $i$? Is the formula $a_i = \gamma_i x_i$ an exact one under some circumstances, or are there always (in principle) nonlinear correction terms that would make it more accurate?

-

In my copy of Atkins Physical Chemistry, 6th edition (it's old!) there is a chart that shows a calculation of the activity and the activity coefficient for an example system (chloroform and acetone). It shows the results using a calculation based on Henry's law (approximating ideal-dilution) and again with one based on Raoult's law (approximating a pure substance).

In both cases, the activity coefficient is not constant over the range of concentrations.

For some evidence from a less-reputable but more easily accessible source, check out the wikipedia article on activity coefficients:

$\gamma_B$ is the activity coefficient, which may itself depend on $x_B$

($x_B$ is the mole fraction of the solute).

Now this means that the activity coefficient is not constant for a given substance in a given mixture at all concentrations, but that isn't the same as saying that the definition of activities is not exact - it is exact, because it is defined that way.

In other words,

$$\mu_i = \mu_i^\ominus + RT\log \gamma_i x_i$$

is an exact relationship, even though $\gamma_i$ is not constant.

-
Ok, that sounds convincing. I'll ask the following as a separate question if you don't know the answer, but is it possible to have a situation where $\gamma_i$ depends not only on $x_i$ but also on $x_j$, the concentration of a different substance? – Nathaniel Aug 1 '14 at 23:05
I believe that is possible, but I haven't found a reference for it yet. I think I remember looking at that case in a stat mech class several years ago. – thomij Aug 1 '14 at 23:34
Actually - this has to be true, at least in a binary mixture. If $x_j$ changes, $x_i$ changes as well. The only question is whether the change is large enough to matter. For a multi-component mixture, I believe it is also true, simply because the non-ideality is a result of intermolecular interactions, and all molecules in a multi-component solution "see" each other. – thomij Aug 1 '14 at 23:39
I meant in a non-binary mixture of course. It'd be nice to have a specific example, but this is more than enough help for now. Thank you! – Nathaniel Aug 4 '14 at 13:41