For liquid solutions, the osmotic coefficient is often used to calculate the salt activity coefficient from the solvent activity, or vice versa. For example, freezing point depression measurements, or measurements of deviations from ideality for other colligative properties, allows calculation of the salt activity coefficient through the osmotic coefficient.

In a single solute solution, the (molality based) osmotic coefficient and the solute activity coefficient ${\displaystyle \gamma }$  are related to the excess Gibbs free energy ${\displaystyle G^{E}}$  by the relations:

${\displaystyle RTm(1-\phi )=G^{E}-m{\frac {dG^{E}}{dm}}}$ 

${\displaystyle RT\ln \gamma ={\frac {dG^{E}}{dm}}}$ 

and there is thus a differential relationship between them (temperature and pressure held constant):

${\displaystyle d((\phi -1)m)=md(\ln \gamma )}$ 

Liquid electrolyte solutions edit

For a single salt solute with molal activity (${\displaystyle \gamma _{\pm }m}$ ), the osmotic coefficient can be written as ${\displaystyle \phi ={\frac {-\ln(a_{A})}{\nu mM_{A}}}}$ where ${\displaystyle \nu }$  is the stochiometric number of salt and ${\displaystyle a_{A}}$  the activity of the solvent. ${\displaystyle \phi }$  can be calculated from the salt activity coefficient via:^[2]

${\displaystyle \phi =1+{\frac {1}{m}}\int _{0}^{m}md\left(\ln(\gamma _{\pm })\right)}$ 

Moreover, the activity coefficient of the salt ${\displaystyle \gamma _{\pm }}$  can be calculated from:^[3]

${\displaystyle \ln(\gamma _{\pm })=\phi -1+\int _{0}^{m}{\frac {\phi -1}{m}}dm}$ 

According to Debye–Hückel theory, which is accurate only at low concentrations, ${\textstyle (\phi -1)\sum _{i}m_{i}}$  is asymptotic to ${\textstyle -{\frac {2}{3}}AI^{3/2}}$ , where I is ionic strength and A is the Debye–Hückel constant (equal to about 1.17 for water at 25 °C).

This means that, at least at low concentrations, the vapor pressure of the solvent will be greater than that predicted by Raoult's law. For instance, for solutions of magnesium chloride, the vapor pressure is slightly greater than that predicted by Raoult's law up to a concentration of 0.7 mol/kg, after which the vapor pressure is lower than Raoult's law predicts. For aqueous solutions, the osmotic coefficients can be calculated theoretically by Pitzer equations^[4] or TCPC model.^{[5][6] [7][8]}

• Bromley equation
• Pitzer equation
• Davies equation
• van 't Hoff factor
• Law of dilution
• Thermodynamic activity
• Ion transport number