Fugacity

Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The real gas pressure and fugacity are related through the dimensionless fugacity coefficient $${\displaystyle \phi =\frac{f}{P}.}$$

For an ideal gas, fugacity and pressure are equal, and so φ = 1. Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to RT ln φ.

The fugacity is closely related to the thermodynamic activity. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the standard state and normally chosen as 1 atmosphere or 1 bar.

Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The thermodynamic condition for chemical equilibrium is that the total chemical potential of reactants is equal to that of products. If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar reaction quotient form (or law of mass action) except that the pressures are replaced by fugacities.

For a condensed phase (liquid or solid) in equilibrium with its vapor phase, the chemical potential is equal to that of the vapor, and therefore the fugacity is equal to the fugacity of the vapor. This fugacity is approximately equal to the vapor pressure when the vapor pressure is not too high.

Pure substance

Fugacity is closely related to the chemical potential μ. In a pure substance, μ is equal to the Gibbs energy G_m for a mole of the substance, and $${\displaystyle d\mu =d{G}_{\mathrm{m}}=-S_{\mathrm{m}}dT+V_{\mathrm{m}}dP,}$$ where T and P are the temperature and pressure, V_m is the volume per mole and S_m is the entropy per mole.

Gas

For an ideal gas the equation of state can be written as $${\displaystyle V_{\mathrm{m}}^{\text{ideal}}=\frac{RT}{P},}$$ where R is the ideal gas constant. The differential change of the chemical potential between two states of slightly different pressures but equal temperature (i.e., dT = 0) is given by $${\displaystyle d\mu =V_{\mathrm{m}}dP=RT\frac{dP}{P}=RTd\ln P,}$$where ln p is the natural logarithm of p.

For real gases the equation of state will depart from the simpler one, and the result above derived for an ideal gas will only be a good approximation provided that (a) the typical size of the molecule is negligible compared to the average distance between the individual molecules, and (b) the short range behavior of the inter-molecular potential can be neglected, i.e., when the molecules can be considered to rebound elastically off each other during molecular collisions. In other words, real gases behave like ideal gases at low pressures and high temperatures. At moderately high pressures, attractive interactions between molecules reduce the pressure compared to the ideal gas law; and at very high pressures, the sizes of the molecules are no longer negligible and repulsive forces between molecules increases the pressure. At low temperatures, molecules are more likely to stick together instead of rebounding elastically.

The ideal gas law can still be used to describe the behavior of a real gas if the pressure is replaced by a fugacity f, defined so that $${\displaystyle d\mu =RTd\ln f}$$ and $${\displaystyle \underset{P\to 0}{lim}\frac{f}{P}=1.}$$ That is, at low pressures f is the same as the pressure, so it has the same units as pressure. The ratio $${\displaystyle \phi =\frac{f}{P}}$$ is called the fugacity coefficient.

If a reference state is denoted by a zero superscript, then integrating the equation for the chemical potential gives $${\displaystyle \mu -{\mu }^0=RT\ln \frac{f}{P^0},}$$ Note this can also be expressed with ${\displaystyle a=f/P^0}$, a dimensionless quantity, called the activity.

Numerical example: Nitrogen gas (N₂) at 0 °C and a pressure of P = 100 atmospheres (atm) has a fugacity of f = 97.03 atm. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at 97.03 atm. The fugacity coefficient is ⁠97.03 atm/100 atm⁠ = 0.9703.

The contribution of nonideality to the molar Gibbs energy of a real gas is equal to RT ln φ. For nitrogen at 100 atm, G_m = G_m,id + RT ln 0.9703, which is less than the ideal value G_m,id because of intermolecular attractive forces. Finally, the activity is just 97.03 without units.

Condensed phase

The fugacity of a condensed phase (liquid or solid) is defined the same way as for a gas: $${\displaystyle d{\mu }_{\mathrm{c}}=RTd\ln f_{\mathrm{c}}}$$ and $${\displaystyle \underset{P\to 0}{lim}\frac{f_{\mathrm{c}}}{P}=1.}$$ It is difficult to measure fugacity in a condensed phase directly; but if the condensed phase is saturated (in equilibrium with the vapor phase), the chemical potentials of the two phases are equal (μ_c = μ_g). Combined with the above definition, this implies that $${\displaystyle f_{\mathrm{c}}=f_{\mathrm{g}}.}$$

When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. At constant temperature, the change in fugacity as the pressure goes from the saturation press P_sat to P is $${\displaystyle \ln \frac{f}{f_{\mathrm{s}\mathrm{a}\mathrm{t}}}=\frac{V_{\mathrm{m}}}{RT}{\int }_{P_{\mathrm{s}\mathrm{a}\mathrm{t}}}^{P}dp=\frac{V\left(P-P_{\mathrm{s}\mathrm{a}\mathrm{t}}\right)}{RT}.}$$ This fraction is known as the Poynting factor. Using f_sat = φ_sat p_sat, where φ_sat is the fugacity coefficient, $${\displaystyle f={\phi }_{\mathrm{s}\mathrm{a}\mathrm{t}}{P}_{\mathrm{s}\mathrm{a}\mathrm{t}}\exp \left(\frac{V\left(P-P_{\mathrm{s}\mathrm{a}\mathrm{t}}\right)}{RT}\right).}$$ This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1.

Unless pressures are very high, the Poynting factor is usually small and the exponential term is near 1. Frequently, the fugacity of the pure liquid is used as a reference state when defining and using mixture activity coefficients.