Basic Thermodynamic Principles

Fugacity and Fugacity Coefficient

Fugacity can be thought of as the corrected partial pressure ($p_i=u_i\cdot p$ where $u_i$ is either $z_i$, $x_i$ or $y_i$) of a component in a mixture where the fugacity coefficient is the correction term to the partial pressure for non-ideal conditions. Based on this description, you would expect the fugacity coefficient ($\phi_i$) to tend to 1 as the fluid acts ideal, as is the case.

$\begin{equation}\label{eq:fugacity} f_i = \phi_i \cdot p_i \end{equation}$

where

$\begin{equation}\label{eq:fug_coeff} \ln{(\phi_i)} = \int_\infty^V(\frac{\partial p}{\partial n_i} - \frac{RT}{V})dV - \ln{(Z)} \end{equation}$

Chemical Potential

$\begin{equation}\label{eq:chem_pot} \mu_i = RT \cdot \ln{(f_i)} + \lambda_i(T) \end{equation}$

Gibbs Free Energy

$\begin{equation}\label{eq:phase_gibbs} g = \sum_{i=1}^{N_c} u_i \ln{(\phi_i)} \end{equation}$

where $u_i$ can be either the liquid molar composition ($x_i$) or the vapor molar composition ($y_i$).

$\begin{equation}\label{eq:mix_gibbs} g_{mix} = g_VF_V + g_L(1-F_V) \end{equation}$

Thermodynamic Equilibrium

Equal Chemical Potential Criteria

$\begin{equation}\label{eq:eql_chem_pot} \mu_i^L = \mu_i^V \end{equation}$

Equal Fugacity Criteria

$\begin{equation} RT \cdot \ln{(f_i^L)} + \lambda_i(T) = RT \cdot \ln{(f_i^V)} + \lambda_i(T) \end{equation}$

$\begin{equation} \ln{(f_i^L)} = \ln{(f_i^V)} \end{equation}$

$\begin{equation}\label{eq:eql_fug} f_i^L = f_i^V \end{equation}$