# 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

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

(real gas)

where the fugacity depends on pressure according to

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

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

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

(ideal gas)

and the Gibbs free energy relationship becomes

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

${\overline{G}}_A\left(x_A,P,T\right)={\Delta }_fG^o\left(A,P^o,T\right)+RT{ \ln \left[\frac{x_AP}{P^o}\right]\ }$ (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 ${\mu }_A={\overline{G}}_A={\left(\frac{\partial G}{\partial n_A}\right)}_{P,T,n_{i\neq A}}$ (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

${\mu }^{\textrm{⦁}}_A={\overline{G}}^{\textrm{⦁}}_A=\frac{G^{\textrm{⦁}}}{n_A}=\frac{dG^{\textrm{⦁}}}{dn_A}$ (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:

$G=\sum^{\omega }_{i=1}{n_i{\mu }_i}$

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

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

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

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

${\mu }_A\left(P,T\right)={\mu }^o_A\left(T\right)+RT{ \ln \left[\frac{f_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ }$ (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

$RT{ \ln \left[\frac{f_A\left(P\right)}{f_A\left({HIG}^o\right)}\right]\ }={ \ln \left[\frac{x_AP}{P^o}\right]\ }+\int^P_0{\left[\frac{{\overline{V}}_A}{RT}-\frac{1}{P}\right]}dP$ (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,

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

and the fugacity depends on the same variables,

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

If the system is a mixture of ideal gases,

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

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 $$A$$.

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