# 16.5: Expressing the Activity Coefficient as A Deviation from Henry's Law

Even if Henry’s law is valid only for solute concentrations very close to zero, we can use it to express the activity of the real system as a function of solute concentration. Let us suppose that we have data on the mole fraction of $$A$$, $$x_A$$, in a gas whose pressure is $$P$$ and which is at equilibrium with a solution in which its mole fraction is $$y_A$$. In the preceding section, we find that we can choose the solute’s standard state so that its activity in any state is $$\tilde{a}_A=x_AP/{\kappa }_A$$. Introducing the activity coefficient, defined by $$\tilde{a}_A=y_A{\gamma }_A$$, we have $$x_AP/{\kappa }_A=y_A{\gamma }_A$$. The activity coefficient is

${\gamma }_A=\frac{x_AP}{y_A{\kappa }_A}$ (Henry’s law activity coefficient)

and the chemical potential is

${\mu }_A={\widetilde{\mu }}^o_A\left(Hyp\ \ell ,{\kappa }_A\right)+RT{ \ln y_A{\gamma }_A\ }$

Just as when we define the activity coefficient using the deviation from Raoult’s law, this development provides a way to recast the available information in a way that makes the solute mole fraction, $$y_A$$, the independent variable in the chemical-potential equation. In Section 16.4, we note that Raoult’s law is the special case of Henry’s law in which $$P^{\textrm{⦁}}_A={\kappa }_A$$. If we make this substitution into the Henry’s-law based activity coefficient, we recover the Raoult’s-law based activity coefficient.