15.1: The Chemical Potential and Fugacity of a Gas

The third law and the fugacity of a pure real gas.

In Chapter 11, we introduce the fugacity as a measure of the difference between the molar Gibbs free energy of a real gas, \(\overline{G}\left(P,T\right)\) at pressure \(P\), and that of the pure gas in its hypothetical ideal-gas standard state at the same temperature. We choose the standard Gibbs free energy of formation, \({\Delta }_fG^o\left({HIG}^o,T\right)\), to be the Gibbs free energy of the real gas in its hypothetical ideal-gas standard state. Letting the gas be \(A\), we find

\[\overline{G}\left(P,T\right)={\Delta }_fG^o\left({HIG}^o,T\right)+RT{ \ln \left[\frac{f^{\textrm{⦁}}_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ } \nonumber \]

(real gas)

where the fugacity depends on pressure according to

\[{ \ln \left[\frac{f^{\textrm{⦁}}_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ }={ \ln \left[\frac{P}{P^o}\right]\ }+\int^P_0{\left[\frac{{\overline{V}}^{\textrm{⦁}}_A}{RT}-\frac{1}{P}\right]}dP \nonumber \] (real gas)

and \({\overline{V}}^{\textrm{⦁}}_A\) is the molar volume of the pure real gas. (In Chapter 14, we introduce a solid-bullet superscript to indicate that a particular property is that of a pure substance.) Given \({\Delta }_fG^o\left({HIG}^o,T\right)\) and an equation of state for the real gas, we can calculate the fugacity and molar Gibbs free energy of the real gas at any pressure.

The fugacity of a pure ideal gas

For a pure ideal gas, we have

\[\frac{{\overline{V}}^{\textrm{⦁}}_A}{RT}-\frac{1}{P}=0 \nonumber \] (ideal gas)

The fugacity becomes equal to the ideal-gas pressure \[f^{\textrm{⦁}}_A\left(P\right)=P \nonumber \]

(ideal gas)

and the Gibbs free energy relationship becomes

\[{\overline{G}}_A\left(P,T\right)={\Delta }_fG^o\left(A,P^o,T\right)+RT{ \ln \left[\frac{P}{P^o}\right]\ } \nonumber \] (ideal gas)

For pure gases, the system pressure that appears in these equations, \(P\), is the same thing as the pressure of the gas.

The fugacity of an ideal gas in a mixture

In Chapter 13, we find that the molar Gibbs free energy of a component of an ideal gas mixture is unaffected by the presence of the other gases. For an ideal gas, \(A\), present at mole fraction \(x_A\), in a system whose pressure is \(P\), the partial pressure is \(P_A=x_AP\). Since the partial pressure is the pressure that the system would exhibit if only ideal gas \(A\) were present, the molar Gibbs free energy of an ideal gas in a mixture is

\[{\overline{G}}_A\left(x_A,P,T\right)={\Delta }_fG^o\left(A,P^o,T\right)+RT{ \ln \left[\frac{x_AP}{P^o}\right]\ } \nonumber \] (ideal gas)

The chemical potential and fugacity of real gases

In Chapter 14, we introduce the chemical potential as the partial molar Gibbs free energy. The defining relationship is \[{\mu }_A={\overline{G}}_A={\left(\frac{\partial G}{\partial n_A}\right)}_{P,T,n_{i\neq A}} \nonumber \] (any substance in any system)

When the system is a pure substance, the chemical potential is identical to the Gibbs free energy per mole of the pure substance at the same temperature and pressure. For the chemical potential of \(A\) in a system comprised of pure \(A\), we can write

\[{\mu }^{\textrm{⦁}}_A={\overline{G}}^{\textrm{⦁}}_A=\frac{G^{\textrm{⦁}}}{n_A}=\frac{dG^{\textrm{⦁}}}{dn_A} \nonumber \] (any system comprised of pure A)

From Euler’s theorem, we find that the Gibbs free energy of any system is the composition-weighted sum of the chemical potentials of the substances present:

\[G=\sum^{\omega }_{i=1}{n_i{\mu }_i} \nonumber \]

For a pure real gas, the partial molar Gibbs free energy and the molar Gibbs free energy are the same thing; we also write

\[{\mu }^{\textrm{⦁}}_A\left(P,T\right)={\Delta }_fG^o\left(A,{HIG}^o,T\right)+RT{ \ln \left[\frac{f^{\textrm{⦁}}_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ } \nonumber \] (pure real gas A)

and introducing \({\mu }^o_A\left(T\right)={\Delta }_fG^o\left(A,{HIG}^o,T\right)\), we write

\[{\mu }^{\textrm{⦁}}_A\left(P,T\right)={\mu }^o_A\left(T\right)+RT{ \ln \left[\frac{f^{\textrm{⦁}}_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ } \nonumber \] (pure real gas A)

Since \({\mu }^o_A\), \({\Delta }_fG^o\left(A\right)\), and \(f_A\left({HIG}^o\right)\) are defined to be properties of one mole of pure \(A\), it is not necessary to include either the solid-bullet superscript or the solid over-bar in these symbols.

In Section 14.11, we find that the partial molar Gibbs free energy of a component of a real-gas mixture is

\[{\mu }_A\left(P,T\right)={\mu }^o_A\left(T\right)+RT{ \ln \left[\frac{f_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ } \nonumber \] (real gas A in a mixture)

where the fugacity of \(A\), present at mole fraction \(x_A\) in a system whose pressure is \(P\), is given by

\[RT{ \ln \left[\frac{f_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ }={ \ln \left[\frac{x_AP}{P^o}\right]\ }+\int^P_0{\left[\frac{{\overline{V}}_A}{RT}-\frac{1}{P}\right]}dP \nonumber \] (real gas A in a mixture)

where \(f_A\left({HIG}^o\right)=P^o=1\ \mathrm{bar}\). The partial molar volume is a function of the system’s pressure, temperature, and composition; that is,

\[{\overline{V}}_A\left(P\right)={\overline{V}}_A\left(P,T,x_A,x_B,\dots ,x_{\omega }\right) \nonumber \]

and the fugacity depends on the same variables,

\[f_A\left(P\right)=f_A\left(P,T,x_A,x_B,\dots ,x_{\omega }\right) \nonumber \]

If the system is a mixture of ideal gases,

\[V=\left(n_A+n_B+\dots +n_{\omega }\right){RT}/{P} \nonumber \] and \[{\overline{V}}_A={\left(\partial {V}/{\partial n_A}\right)}_{PTn_{m\neq A}}={RT}/{P} \nonumber \]

The integrand becomes zero, and the fugacity relationship reduces to the ideal-gas fugacity equation introduced in Chapter 13 and repeated above.

The fugacity of a gas in any system is a measure of the difference between its chemical potential in that system and its chemical potential in its hypothetical ideal-gas standard state at the same temperature. The chemical potential of \(A\) in a particular system, \({\mu }_A\), is the change in the Gibbs free energy when the amounts of the elements that form one mole of \(A\) pass from their standard states as elements into the (very large) system as one mole of substance \(A\).