Fugacity of a real gas

The effects of intermolecular forces upon the thermodynamic properties of a gas can be conveniently summarized in terms of the fugacity of the gas.  

At sufficiently low pressures all gases are perfect and the fugacity becomes equal to the pressure  

The relationships between p and other thermodynamic properties are obtained as follows. 

The fugacity may be calculated directly since (11.9) can be rewritten, using (11.53), in the form 

The integral can be evaluated either graphically, or analytically using 

---

The ratio of the fugacity/2 at the pressure P2 to the fugacity/i at the pressure Pj can be obtained by graphical or numerical integration, as indicated by the area between the two vertical lines under the isotherm for the real gas in Figure 10.6. However, as Pi approaches zero, the area becomes infinite. Hence, this direct method is not suitable for determining absolute values of the fugacity of a real gas. 

Now that we have obtained expressions for the fugacity of a real gas and its temperature and pressure coefficients, let us consider the application of the concept of fugacity to components of a mixture of real gases. 

Fugacity is a thermodynamic property related to the deviation of the p—V—T properties of the gas from those of an ideal gas. At very low pressures, the fugacity of a real gas tends to its partial pressure... 

It will be seen shortly that this postulate, which makes the fugacity of a real gas equal to its pressure at very low pressure, permits the evaluation of actual fugacities at various pressures. It may be mentioned that since gas pressures are usually expressed in atm., fugacities are recorded in the same units. 

Note that we have made use of the relation In analogy with ideal gases, we define the fugacity of a real gas as... 

If a gas is not an ideal gas, its standard state is a hypothetical state. The fugacity, /, of a real gas (a gas that is not necessarily an ideal gas) is defined by an equation with the same... 

We may consider the fugacity of a real gas in terms of the deviation from ideal behavior. We begin by integrating Equation 5.5 between two limits. 

In order to obtain the fugacity of a real gas, we must have appropriate thermodynamic property data available. We will explore three possible sources of data for pure gases ... 

One method of calculating fugacity and hence y is based on the measured deviation of the volume of a real gas from that of an ideal gas. Considet the case of a pute gas The ftee energy F and chemical potential /i changes with pressure according to the equation... 

It has already been mentioned that the state of an ideal gas at the temperature of the system and the pressure of 1 atmosphere is most frequently chosen as the standard state of gases. The idea of such an ideal gas can be explained by the imagination of a real gas which is first expanded to zero pressure and then by means of isothermal compression compressed to 1 atm. into the region of the ideal gas. As with an ideal gas pressure equals fugacity, we can substitute in equation (V-8a) p° = f° — 1, whereby the following equation is obtained ... 

Figure 7.6 Defmtion of the standard state of a real gas with the dimension of fugacity.

The function / is called the fugacity of the gas. The fugacity measures the Gibbs energy of a real gas in the same way as the pressure measures the Gibbs energy of an ideal gas. 

From the definition of the fugacity and the Gibbs-Helmholtz equation, show that the molar enthalpy, H, of a real gas is related to the molar enthalpy of the ideal gas, H°, by... 

In this subsection we introduce a ratio measure that indicates how the fugacity of a real substance deviates from that of an ideal gas. As the reference state, we choose the ideal-gas mixture at the same temperature, pressure, and composition as our real mixture. Then, on integrating the definition of fugacity (4.3.8) from the ideal-gas state to the real state, we obtain an algebraic form analogous to (4.3.12) that is, we find... 

By the introdnction of fugacity / the chemical potential p of a real gas can be expressed as... 

The fugacity fj is defined by the chemical potential of a real gas, as postulated by Lewis ... 

Fugacity is used to replace the pressure of an ideal gas by a corrected pressure / of a real gas i, to produce a universally valid relationship. At zero pressure a real gas behaves like an ideal gas, and... 

In Eq. (77) p° is the standard pressure, p° = l atm quantity p

[Pg.28]

Just as the fugacity of a pure gas is a kind of effective pressure, the fugaeity of a constituent of a gas mixture is a kind of effective partial pressure. That is, f is the partial pressure substance i would have in an ideal gas mixture that is at the same temperature as the real gas mixture and in which the chemical potential of i is the same as in the real gas mixture. 

FIGURE 4.6 A simple way of determining the fugacity coefficient of a real gas is to plot the real volume of the gas at various pressures and compare it to the expected ideal volume of the gas. The fugacity coefficient is related to the difference in the area under the curves (indicated by the shaded portion of the diagram). See equation 4.61. 

Fugacity of real gases should replace pressure in the application of laws or theoretical formulations that are defined for the ideal gas. For example, fugacity has to be used in evaluating adsorption potential at high pressures and/or low temperatures. The fugacity of a pure gas is a physical quantity that complies with the following relation ... 

---