The above formalism is useless if we cannot determine the activity coefficients.
Convention I
\begin{equation} \mu_i=\mu_{I,i}^{0}+RTln\gamma_{I,i}x_i\;\;\;\rightarrow\;\;\;\mu_i=\mu_{I ,i}^{0}+RTlna_{I,i} \end{equation} Raoult's Law, $P_i=x_iP_{i}^{\ast}$, can be generalized to real solutions simply by changing the fraction molar times the activity, $P_i=a_{I,i}P_{i}^{\ast}$ Solving for the activity, we obtain an equation that allows us to calculate it. \begin{equation} a_{I,i}=\frac{P_i}{P_{i}^{\ast}} \end{equation} The partial pressure of component i is calculated using Dalton's Law, $P_i= x_{iv}P_T$, and it is necessary to experimentally measure the total pressure, $P_T$, and the composition of the vapor over the solution, $x_{iv}$ \begin{equation} \gamma_{I,i}=\frac{ P_i}{x_{i,l}} \end{equation} This last equation allows us to calculate the activity coefficient, once the activity and composition of the liquid phase are known.
Comparing Raoult's Law for ideal solutions, $P_{i}^{id}=x_iP_i^{\ast}$, with Raoult's Law for real solutions, $P_i=\gamma_{II,i}x_iP_{i} ^{\ast}$, the following relation is obtained: \begin{equation} \gamma_{II,i}=\frac{P_i}{P_{i}^{id}} \end{equation} The activity coefficient can be greater or less than one: If $P_i>P_{i}^{id}$ then $\gamma_{II,i}>1$. In this situation, the interactions in the solution are smaller than those presented by the pure components.
Convention II
\begin{equation} \mu_i=\mu_{II,i}^{0}+RTln\gamma_{II,i}x_i \end{equation} The activity of the solute is obtained with Henry's Law $P_i=K_ia_{ II,i}$ \begin{equation} a_{II,i}=\frac{P_i}{K_i}\;\;\Rightarrow \;\; \gamma_{II,i}=\frac{P_i}{K_ix_i} \end{equation} By measuring the total pressure on the solution and its composition, $P_i$ is obtained using Dalton's Law. Henry's constant is measured in a very dilute solution.
The solvent uses Raoult's Law to calculate the activities $a_{II,A}=\frac{P_A}{P_{A}}^{\ast}$