Chemical potential definition for a mixture:

$${\mu }_i\equiv {\left(\frac{\partial \underset{―}{G}}{\partial n_i}\right)}_{P,T,n_{j\ne i}}$$

where ${\mu }_i$ is the chemical potential of component $i$
$\underset{―}{G}$ is the total Gibbs free energy (nG) for the mixture
$n_i$ is the number of moles of component $i$

For a pure fluid:

$$d\mu =dG=-SdT+VdP$$

where $\mu$ is the chemical potential (kJ/mol)
$G$ is the molar Gibbs free energy (kJ/mol)
$S$ is the molar entropy (kJ/mol-K)
$T$ is the temperature (K)
$V$ is the molar volume (m³/mol)
$P$ is the pressure (bar).

For mixtures:

$$d\underset{―}{G}=\underset{―}{V}dP–\underset{―}{S}dT+\sum _i{\mu }_{i}d{n}_i$$

where $\underset{―}{G}$, $\underset{―}{V}$, and $\underset{―}{S}$ are extensive variables
${\mu }_i$ is the chemical potential of component $i$
$n_i$ is the number of moles of component $i$
The summation is over all components in the mixture.

At vapor-liquid equilibrium for a binary system:

$${\mu }_1^V={\mu }_1^L{\mu }_2^V={\mu }_2^L$$

For component $i$ in a mixture:

$${\mu }_i–{\mu }_{i,pure}=RTln\left(\frac{\widehat{f_i}}{f_i}\right)$$

where $\widehat{f_i}$ is the fugacity of component $i$ in the mixture
$f_i$ is the pure component fugacity at the same temperature and pressure

For component $i$ in an ideal-gas mixture:

$${\mu }_i^{ig}=G_i^{ig}+RTln(y_i)$$

where ${\mu }_i^{ig}$ is the chemical potential of component $i$ in the mixture
$G_i^{ig}$ is the Gibbs free energy of the pure-component ideal gas
$R$ is the ideal gas constant
$T$ is the absolute temperature
$y_i$ is the mole fraction of component $i$ in the mixture.