15331 IUPAC Gold Book - Gibbs–Konovalov equations 10.1351/goldbook.15331 15331 current synonyms: Gibbs–Konovalow equations, van der Waals’ equations 1 Pair of equations for a binary mixture of components \(\ce{A}\) and \(\ce{B}\) that relate the variables \(T\), \(p\), in one phase of variable composition, \(\upalpha\), to the variables \(T\), \(p\), in a coexisting equilibrium phase of variable composition, \(\upbeta\): \[\begin{array}{c} -\frac{\left(1 - x_{\ce{B}}^{\beta} \right)\Delta_{\alpha}^{\beta} H_{\ce{A}} + x_{\ce{B}}^{\beta} \Delta_{\alpha}^{\beta} H_{\ce{B}}}{T}{\rm{d}}T + \left[\left(1 - x_{\ce{B}}^{\beta} \right)\Delta_{\alpha}^{\beta}V_{\ce{A}} + x_{\ce{B}}^{\beta}\Delta_{\alpha}^{\beta}V_{\ce{B}} \right]{\rm{d}}p +\left(x_{\ce{B}}^{\alpha} - x_{\ce{B}}^{\beta} \right)\left(\frac{\partial^{2} G_{\rm{m}}^{\alpha}}{\partial x_{\ce{B}}^{\alpha 2}} \right)_{T,p} {\rm{d}}x_{\ce{B}}^{\alpha} = 0 \\ -\frac{\left(1 - x_{\ce{B}}^{\alpha} \right)\Delta_{\alpha}^{\beta} H_{\ce{A}} + x_{\ce{B}}^{\alpha} \Delta_{\alpha}^{\beta} H_{\rm{B}}}{T}{\rm{d}}T + \left[\left(1 - x_{\ce{B}}^{\alpha} \right)\Delta_{\alpha}^{\beta}V_{\ce{A}} + x_{\ce{B}}^{\alpha}\Delta_{\alpha}^{\beta}V_{\ce{B}} \right]{\rm{d}}p +\left(x_{\ce{B}}^{\alpha} - x_{\ce{B}}^{\beta} \right)\left(\frac{\partial^{2} G_{\rm{m}}^{\alpha}}{\partial x_{\ce{B}}^{\beta 2}} \right)_{T,p} {\rm{d}}x_{\ce{B}}^{\beta} = 0 \end{array}\] where \(\Delta_{\upalpha}^{\upbeta} H_{\ce{A}} = H_{\ce{A}}^{\upbeta} - H_{\ce{A}}^{\upalpha}\), \(\Delta_{\upalpha}^{\upbeta} V_{\ce{A}} = V_{\ce{A}}^{\upbeta} - V_{\ce{A}}^{\upalpha}\) are the enthalpy and volume of transfer of component \(\ce{A}\) from phase \(\upalpha\) to phase \(\upbeta\), and similarly for component \(\ce{B}\). \(\left(\frac{\partial^{2} G_{\rm{m}}}{\partial x_{\ce{B}}^{2}} \right)_{T,p} \gt 0\) (condition for diffusional stability). This quantity may also be expressed in terms of the derivatives of the chemical potentials, using \[\left(\frac{\partial^{2} G_{\rm{m}}}{\partial x_{\ce{B}}^{2}} \right)_{T,p} = -\frac{1}{x_{\ce{B}}} \left(\frac{\partial\mu_{\ce{A}}}{\partial x_{\ce{B}}} \right)_{T,p} = \frac{1}{x_{\ce{A}}} \left(\frac{\partial\mu_{\ce{B}}}{\partial x_{\ce{B}}} \right)_{T,p}\] These equations show that an extremum occurs for each phase equation when the compositions of the two phases are equal, and that the slope of the \(T\)-composition or \(p\)-composition curve is zero for each phase equation at the extremum. Sometimes the German transliteration Konovalow is found. components https://goldbook.iupac.org/terms/view/16268 mixture https://goldbook.iupac.org/terms/view/16270 PAC, 2008, 80, 233. 'Glossary of terms related to solubility (IUPAC Recommendations 2008)' on page 248 (https://doi.org/10.1351/pac200880020233) https://goldbook.iupac.org/terms/view/15331/html https://goldbook.iupac.org/terms/view/15331/json https://goldbook.iupac.org/terms/view/15331/plain Citation: 'Gibbs–Konovalov equations' 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.15331 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. 2026-05-09T20:41:09+00:00