Nonvolatile solute activity coefficients can be determined from vapor pressure data using the Gibbs-Duhem equation. We start from the equation that gives us G for a solution as the sum of the product of the moles of each component times its chemical potential.
\begin{equation} G=\sum_{i}n_i\mu_i \end{equation} Differentiating:
\begin{equation} dG=\sum_{i}n_id\mu_i-\sum_{i}\mu_idn_i \end{equation}
Writing the Gibbs equation for dG
\begin{equation} dG=-SdT+VdP+\sum_{i}\mu_idn_i \end{equation}
Equating both equations
\begin{equation} \sum_{i}n_id\mu_i-\cancel{ \sum_{i}\mu_idn_i}=-SdT+VdP+\cancel{\sum_{i}\mu_idn_i} \end{equation}
If T and P are constants dT=dP=0
\begin{equation} \sum_{i} =n_id\mu_i=0 \end{equation}
This last equation is known as the Gibbs-Duhem equation and can be generalized to any partial molar magnitude.