12239 IUPAC Gold Book - osmotic virial coefficient 10.1351/goldbook.12239 12239 current synonym: virial coefficient of the chemical potential 1 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. 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 \[\frac{\mathit{\Pi}}{c} = \left({\frac{\mathit{\Pi}}{c}} \right)_{\circ}(1 + \mathit{\Gamma}_{2}c + \mathit{\Gamma}_{3}c^{2} + \ldots),\ {\rm{where}}\ \left(\frac{\mathit{\Pi}}{c} \right)_{\circ} = \frac{RT}{M_{\rm{n}}}\ {\rm{and,}}\ \mathit{\Gamma}_{i} = M_{\rm{n}}A_{i}\] number-average molar mass https://goldbook.iupac.org//terms/view/12216 PAC, 2015, 87, 71. 'Definitions of terms relating to individual macromolecules, macromolecular assemblies, polymer solutions, and amorphous bulk polymers (IUPAC Recommendations 2014)' on page 93 (https://doi.org/10.1515/pac-2013-0201) https://goldbook.iupac.org/terms/view/12239/html https://goldbook.iupac.org/terms/view/12239/json https://goldbook.iupac.org/terms/view/12239/plain Citation: 'osmotic virial coefficient' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.12239 The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) for individual terms. If you are interested in licensing the Gold Book for commercial use, please contact the IUPAC Executive Director at executivedirector@iupac.org . The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using. 2026-06-13T16:05:01+00:00