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 and Partial Derivatives of Gibbs Free Energy

In Pandat, thermodynamic factors are available with the format \(\tt ThF(*,*@*)\), which is defined by \begin{equation} {\tt ThF(x_i,x_j) }= \frac{\partial \mu_i}{\partial x_j} \quad \quad \quad \quad (i,j=1,2,\cdots,c) \end{equation} where \((x_1,x_2,\cdots,x_c)\) are treated as the independent compositional variables. If the component \(1\) is assumed to be the solvent component and its molar fraction \(x_1\) is taken as the dependent variable, the independent compositional variables now are \((X_2,X_3,\cdots,X_{c})\). Here we use capital \(X\) to distinguish this set of variables from \((x_1,x_2,\cdots,x_c)\). Then we have \begin{align} \frac{\partial \mu_j}{\partial X_k}= &\frac{\partial \mu_j}{\partial x_k} - \frac{\partial \mu_1}{\partial x_k}\quad \quad \quad \quad (j, k=2,3,\cdots,c) \\ = &{\tt ThF(x_j,x_k) } - {\tt ThF(x_1,x_k) } \quad \quad \quad \quad (j, k=2,3,\cdots,c) \end{align} Now let's see how to express the second derivatives of Gibbs free energy in terms of the thermodynamic factors. If we use this compositional variable set \((X_2,X_3,\cdots,X_c)\), the first and second derivatives of \(G\) are \begin{align} \frac{\partial G}{\partial X_j}= & \sum_{i=2}^{c} \frac{\partial G}{\partial x_i} \frac{\partial x_i}{\partial x_j} + \frac{\partial G}{\partial x_1} \frac{\partial x_1}{\partial x_j} \quad \quad \quad \quad (j=2,3,\cdots,c) \\ = & \frac{\partial G}{\partial x_j} - \frac{\partial G}{\partial x_1} \quad \quad \quad \quad \quad \quad (j=2,3,\cdots,c) \\ = & \mu_j - \mu_1 \quad \quad \quad \quad \quad \quad (j=2,3,\cdots,c) \\ \frac{\partial^2 G}{\partial X_j \partial X_k} = &\sum_{i=2}^{c} \frac{\partial (\mu_j - \mu_1)}{\partial x_i} \frac{\partial x_i}{\partial x_k} + \frac{\partial (\mu_j - \mu_1)} {\partial x_1} \frac{\partial x_1}{\partial x_k} \nonumber \\ &\quad \quad\quad \quad\quad \quad\quad \quad\ (j, k=2,3,\cdots,c) \\ = & (\frac{\partial \mu_j}{\partial x_k} - \frac{\partial \mu_1}{\partial x_k}) - (\frac{\partial \mu_j}{\partial x_1} - \frac{\partial \mu_1}{\partial x_1}) \nonumber \\ &\quad \quad\quad \quad\quad \quad\quad \quad (j, k=2,3,\cdots,c) \\ = & {\tt ThF(x_j,x_k) } - {\tt ThF(x_1,x_k) } - {\tt ThF(x_j,x_1) } + {\tt ThF(x_1,x_1) } \nonumber \\ &\quad \quad\quad \quad\quad \quad\quad \quad (j, k=2,3,\cdots,c) \end{align}