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

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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.