Osmotic Pressure

Osmotic pressure arises when two solutions (one pure solvent and the other containing a mixture of solvent and solutes) are separated by a membrane permeable only to the solvent. This leads to a rising of the solution inside the membrane due to an increased pressure. The pressure in turn arises due to the effort of the solvent molecules to equalize their chemical potentials on either side of the membrane. Call the solvent component one, and consider a dilute solution where the activities are all one. The chemical potential equalization is represented as

$$\mu_1^*(p) = \mu_1^*(p + \Pi) + k_B T \ln \left(1 - \sum_{i\neq1} x_i\right)$$ (1.62)

If $\delta \mu_1^* = \mu_1^*(p + \Pi) - \mu_1^*(p)$, then

$$\delta \mu_1^* = -k_B T \ln \left(1 - \sum_{i\neq1} x_i\right)$$ (1.63)

$$\approx k_B T \sum_{i\neq1} x_i$$ (1.64)

Recall Eq. 1.25 relating the derivative of the chemical potential with respect to pressure to the partial molar volume. If we identify $\delta p$ with the osmotic pressure, then

$$v\Pi = k_B T \sum_{i\neq1} x_i$$ (1.65)

where $v$ is the molar volume of the solvent. This is the van't Hoff equation for the osmotic pressure in the dilute limit (which assumes unit activities).