Fugacity - Definition in Terms of Chemical Potential

Definition in Terms of Chemical Potential

The fugacity of a real gas is formally defined by an equation analogous to the relation between the chemical potential and the pressure of an ideal gas.

In general, the chemical potential (μ) is defined as the partial molar Gibbs free energy. However for any pure substance, it is equal to the molar Gibbs free energy, and its variation with temperature (T) and pressure (P) is given by . At constant temperature, this expression can be integrated as a function of . We must also set a reference state. For an ideal gas the reference state depends only on pressure, and we set = 1 bar so that

Now, for an ideal gas

\mu - \mu ^\circ = \int_{P^\circ }^P {\frac{{RT}} {P}dP} = RT\ln \frac{P} {{P^\circ }}

Reordering, we get

\mu = \mu ^\circ + RT\ln \frac{P} {{P^\circ }}

This gives the chemical potential for an ideal gas in an isothermal process, with a reference state is = 1 bar.

For a real gas, we cannot calculate because we do not have a simple expression for a real gas’ molar volume. Even if using an approximate expression such as the van der Waals equation, the Redlich-Kwong or any other equation of state, it would depend on the substance being studied and would be therefore of very limited utility.

Additionally, chemical potential is not mathematically well behaved. It approaches negative infinity as pressure approaches zero and this creates problems in doing real calculations.

We would like the expression for a real gas’ chemical potential to be similar to the one for an ideal gas. We therefore define a magnitude, called fugacity, so that the chemical potential for a real gas becomes

\mu = \mu ^\circ + RT\ln \frac{f} {{f^\circ }}

with a given reference state to be discussed later. This is the usual form of the definition, but it may be solved for f to obtain the equivalent explicit form

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