# 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? 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. 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. 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! Aug 4 '14 at 13:41