Published on *PNG 520: Phase Behavior of Natural Gas and Condensate Fluids* (https://www.e-education.psu.edu/png520)

We have seen that, for a closed system, the Gibbs energy is related to pressure and temperature as follows:

$$dG=VdP-SdT$$

For a constant temperature process,

$$dG=VdP\text{@constantT}$$

For an ideal gas,

$$dG=\frac{RT}{P}dP$$

$$dG=RTd\text{InP@constantT}$$

This expression by itself is strictly applicable to ideal gases. However, Lewis, in 1905, suggested extending the applicability of this expression to all substances by defining a new thermodynamic property called fugacity, f, such that:

$$dG=RTd\text{Inf@constantT}$$

This definition implies that for ideal gases, ‘f’ must be equal to ‘P’. For mixtures, this expression is written as:

$$d\overline{{G}_{i}}=RTd{\text{Inf}}_{i}\text{@constantT}$$

where $\overline{{G}_{i}}{\text{andf}}_{i}$ are the partial molar Gibbs energy and fugacity of the i-th component, respectively. Fugacity can be readily related to chemical potential because of the one-to-one relationship of Gibbs energy to chemical potential, which we have discussed previously. Therefore, the definition of fugacity in terms of chemical potential becomes:

For a pure substance,

$$d\text{In}f=\frac{d\mu}{RT}@\text{constantT}$$

$$\underset{P\to 0}{\mathrm{lim}}\text{}f=P\text{(idealgaslimit)}$$

For a component in a mixture,

$$d\text{In}{f}_{i}=\text{}\frac{d{\mu}_{i}}{RT}@\text{constantT}$$

$$\underset{P\to 0}{\mathrm{lim}}\text{}{f}_{i}={y}_{i}P\text{=partialpressure(idealgaslimit)}$$

The fugacity coefficient (${\varphi}_{i}$ ) is defined as the ratio of fugacity to its value at the ideal state. Hence, for pure substances:

$$\varphi =\frac{f}{P}$$

and for a component in a mixture,

$${\varphi}_{i}=\frac{{f}_{i}}{{y}_{i}P}$$

The fugacity coefficient takes a value of unity when the substance behaves like an ideal gas. Therefore, the fugacity coefficient is also regarded as a *measure of non-ideality*; the closer the value of the fugacity coefficient is to unity, the closer we are to the ideal state.

Fugacity turns out to be an auxiliary function to chemical potential. Even though the concept of thermodynamic equilibrium which we discussed in the previous section is given in terms of chemical potentials, above definitions allow us to restate the same principle in terms of fugacity. To do this, previous expressions can be integrated for the change of state from liquid to vapor at saturation conditions to obtain:

$$\underset{l}{\overset{v}{\int}}d\text{In}{f}_{i}=\frac{1}{RT}{\displaystyle \underset{l}{\overset{v}{\int}}d{\mu}_{i}}$$

$$\text{In}{f}_{i}^{(v)}-\text{In}{f}_{i}^{(v)}=\frac{1}{RT}\left({\mu}_{i}^{(v)}-{\mu}_{i}^{(l)}\right)$$

For equilibrium, ${\mu}_{i}^{(l)}={\mu}_{i}^{(v)}$ ,hence,

$$In\left(\frac{{f}_{i}^{(v)}}{{f}_{i}^{(l)}}\right)=0$$

Therefore:

$${f}_{i}^{(l)}={f}_{i}^{(v);\text{}i=1,\text{}2,\text{}...\text{}{n}_{c}}$$

For equilibrium, fugacities must be the same as well! This is, for a system to be in equilibrium, both the fugacity and the chemical potential of each component in each of the phases must be equal. Conditions (16.14) and (16.25) are equivalent. Once one of them is satisfied, the other is satisfied immediately. Using ${\mu}_{i}^{(l)}={\mu}_{i}^{(v)}\text{or}{f}_{i}^{(l)}={f}_{i}^{(v)}$ to describe equilibrium is a matter of choice, but generally the fugacity approach is preferred.