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 \(\ce{B}\).