Equilibrium criteria for solvent where in general the left side (L) can be a solvent/solute mixture (or pure solvent) and the right side (R) is a solvent/solute mixture.

\[\hat{f}^L _{solvent} = \hat{f}^R _{solvent}\]

In a mixture

\[\hat{f} _{solvent} = x_{solvent} \gamma _{solvent} P\]

where \(x_{solvent}\) = mole fraction of the solvent

\(\gamma _{solvent}\) = activity coefficient of the solvent, and 

\(P\) = pressure of the mixture.

Poynting correction for fugacity dependence on pressure for a liquid

\[f_i(P_2) = f_i(P_1)exp \left( \frac{V_i(P_2 – P_1)}{RT} \right) \]

where \(f_i(P_2)\) = fugacity of pure component \(i\) at pressure \(P_2\),

\(f_i(P_1)\) = fugacity of pure component \(i\) at pressure \(P_1\)

\(V_i\) = molar volume of the pure liquid \(i\)

\(R\) = gas constant

\(T\) =temperature

Osmotic pressure equation

\[\Pi V^{liquid} _{solvent} = x_{solute} RT\]

where \(\Pi\) = osmotic pressure,

\(x_{solute}\) = mole fraction of the solute (if the solute dissociates into ions, the mole fraction of the ions is used in this equation). For example, if the mole fraction of NaCl is 0.02, the mole fraction of ions is 0.04 if NaCl completely dissociates, 

\(V^{liquid} _{solvent}\) = molar volume of the liquid solvent

Reverse osmosis

\[N_{solvent} = \frac{P_M}{L} (\Delta P – \Pi)\]

where \(N_{solvent}\) = flux of solvent through a semi-permeable membrane,

\(P_M\) = membrane permeability

\(L\) = membrane thickness

\(\Delta P\) = pressure drop across the membrane

\(\Pi\) = osmotic pressure of the mixture