symbol: $A_{i}\ ({\rm{where}}\ i = 1, 2, \ldots)$; unit: $\pu{mol kg^{-\textit{i}} m^{3(\textit{i}-1)}}$
synonym: virial coefficient of the chemical potential
https://doi.org/10.1351/goldbook.12239
Coefficient in the expansion of the chemical potential of the solvent, \(\mu_{\rm{s}}\), in powers of the mass concentration, \(c\), of the solute, i.e., \[\mu_{\rm{s}} - \mu_{\rm{s}}^{\circ} = -\mathit{\Pi} V_{s}^{\prime} = -RT V_{s}^{\prime} (A_{1}c + A_{2}c^{2} + A_{3}c^{3} + \ldots)\] where \(\mu_{\rm{s}}^{\circ}\) is the chemical potential of the solvent in the reference state at the temperature of the system and ambient pressure, \(\mathit{\Pi}\) is the osmotic pressure and \(V_{s}^{\prime}\) is the partial molar volume of the solvent. In solvents comprising more than one component, the definition applies to any solvent component.
Notes:
- The first osmotic virial coefficient is the reciprocal number-average molar mass, i.e., \(A_{1} = 1/M_{\rm{n}}\). The values of the second and higher virial coefficients, \(A_{2}, A_{3}, \dots\), respectively, are characteristic of polymer-solvent and polymer-polymer interactions.
- The factor \(RT\) is sometimes included in the virial coefficients, to give \[\mu_{\rm{s}} - \mu_{\rm{s}}^{\circ} = -\mathit{\Pi}V_{s}^{\prime} = -V_{s}^{\prime}(A_{1}^{\prime}c + A_{2}^{\prime}c^{2} + A_{3}^{\prime}c^{3} + \dots),\quad {\rm{where}}\ A_{i}^{\prime} = RT A_{i}\]
- To evaluate \(M_{\rm{n}}\) and \(A_{2}\), \(\mathit{\Pi}/c\) is often plotted versus \(c\), so that \[\frac{\mathit{\Pi}}{c} = \left({\frac{\mathit{\Pi}}{c}} \right)_{\circ}(1 + \mathit{\Gamma}_{2}c + \mathit{\Gamma}_{3}c^{2} + \ldots),\ {\rm{where}}\ \left(\frac{\mathit{\Pi}}{c} \right)_{\circ} = \frac{RT}{M_{\rm{n}}}\ {\rm{and,}}\ \mathit{\Gamma}_{i} = M_{\rm{n}}A_{i}\]