Thermodynamic Factor

The First Order Derivative of Gibbs Energy w.r.t. Molar Fraction

Let \(G\) be the molar Gibbs energy of a \(c\)-component phase and \(\mu _k\) the chemical potential of component \(k\). \begin{equation} G = G(x_1,x_2,\cdots,x_c) \end{equation} The chemical potential of the component \(k\), \(\mu _k\), is calculated by the following equation, \begin{align} \mu_k = G-\sum_{j=1}^{c} x_j \frac{\partial G}{\partial x_j} +\frac{\partial G}{\partial x_k} \quad \quad \quad \quad (k=1,2,\cdots,c) \label {A2} \end{align} If the component \(c\) is taken as the dependent component, \begin{align} x_c =1 - \sum_{j=1}^{c-1} x_j \end{align} then, the Gibbs energy is a function of \(c-1\) variables only, \begin{equation} G = G(x_1,x_2,\cdots,x_{c-1}) \end{equation} The chemical potential of the component \(k\), \(\mu _k\), is now calculated by these following equations, \begin{align} \mu_k = &G-\sum_{j=1}^{c-1} x_j \frac{\partial G}{\partial x_j} +\frac{\partial G}{\partial x_k} \quad \quad \quad \quad (k=1,2,\cdots,c-1) \label {A5} \\ \mu_c = &G-\sum_{j=1}^{c-1} x_j \frac{\partial G}{\partial x_j} \label {A6} \end{align} Please note that \(\displaystyle \frac{\partial G}{\partial x_k}\) in Eq.\eqref{A5} and Eq.\eqref{A6} is different from that in Eq.\eqref{A2}. Therefore, the first order derivative of \(G\) w.r.t molar fraction, \(x_j\), is given by \begin{align} \frac{\partial G}{\partial x_k} = \mu_k - \mu_c \quad \quad \quad \quad (k=1,2,\cdots,c-1) \label {A7} \end{align}
Thermodynamic Factor

Assume the Gibbs energy is a function of \(c-1\) independent composition variables, \begin{equation} G = G(x_1,x_2,\cdots,x_{c-1}) \end{equation} The second order derivatives of \(G\) w.r.t molar fraction can be calculated by \begin{align} \frac{\partial^2 G}{\partial x_j \partial x_k}= &\frac{\partial \mu_k}{\partial x_j} - \frac{\partial \mu_c}{\partial x_j} \\ = &\frac{\partial \mu_j}{\partial x_k} - \frac{\partial \mu_c}{\partial x_k}\quad \quad \quad \quad (j, k=1,2,\cdots,c-1) \label {A8} \end{align} where the thermodynamic factor, \(\displaystyle \frac{\partial \mu_j}{\partial x_k}\), can be evaluated numerically by the finite difference method. Thermodynamic factors are available from PanEngine.