The Lewis Fugacity Rule

As we have seen, the fugacity coefficient (or fugacity) depends not only on T and P but also on the composition (chemical nature) of all the other species in the mixture. However, as shown in Example 7.4, we can sometimes approximate the fugacity coefficient of species i in a mixture by its pure species fugacity coefficient  

This approximation is known as the Lewis fugacity rule. It approximates all interactions in the mixture as being identical to the i-i interaction and simplifies calculations significantly because the pure species fugacity coefficient does not depend on the other species in the mixture but rather depends only on T and P (e.g., compare the complexity of Examples 7.2 and 7.4). 

Therefore, to calculate the fugacity coefficient of a mixture, we may choose from three levels of rigor. 

We can solve the full problem with compositional-dependent fugacity coefficients. To apply this approach, we need mixing rules for the equation of state parameters to account for the i-j interactions of a mixture  

As a first approximation, we can use the Lewis fugacity rule and base the fugacity coefficient on the pure species value, as discussed in Section 7.3. This approach treats all the interactions as the same i-i interactions). The advantage of this approach is that mixing rules are not needed, and it is mathematically much 

A useful and widely-used variation of Raoult s Law is obtained by expressing it in terms of fugacities instead of pressures. Thus in some homogeneous (one phase) 

As shown by Prausnitz (1969), this relation follows from Amagat s Rule, which describes the mixing situation where the volume of the solution is the same as the total volume of the individual pure components. In this case the AV of mixing is zero, and the partial molar volume Vi is equal to the molar volume of pure i, as discussed in Chapter 9. This can be demonstrated by first rewriting equation (11.9) for a pure component i as 

This relationship, the Lewis Fugacity Rule, is a kind of variation of Dalton s Law, and has been widely used to estimate fugacities in gas mixtures (Prausnitz, 1969). 

Although the Lewis Fugacity Rule is generally used for gas or supercritical solutions, it is particularly interesting to see what results when a condensed phase in equilibrium with such a phase is considered, just as we first considered Dalton s Law and then a condensed phase in equilibrium with a solution obeying Dalton s Law. 

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At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

To use Equation (13), it is first necessary to calculate the true fugacity coefficient (ft. This calculation is achieved by utilizing the Lewis fugacity rule... 

The Lewis fugacity rule is used for calculating the fugacity coefficients of the true species, and (2) the second virial co-... 

The limits of the Lewis fugacity rule are not determined by pressure but by composition the Lewis rule becomes exact at any pressure in the limit as y( - 1, and therefore it always provides a good approximation for any component i which is present in excess. However, for a component with small mole fraction in the vapor phase, the Lewis rule can sometimes lead to very large errors (P5, R3, RIO). 

An ideal liquid solution must obey Raoult s Law and the Lewis Fugacity Rule (just like an ideal gas solution). 

Fig. 11.6. A binary system (solution) that obeys the Lewis Fugacity Rule. The solution can be liquid, solid, gas, or supercritical fluid, although generally this rule is used for the latter two.

Comparison of equations (11.19) and (11.26) shows that for solutions that obey the Lewis Fugacity Rule. , ... 

Fig. 11.7. Normalized fugacities in a binary system obeying the Lewis Fugacity Rule. Figure 11.6 is converted to this diagram by dividing the fugacity of each component by the fugacity of the pure component. The mole fraction axis can now be used for the original system or for a phase equilibrated with it.

This leads to the easiest approach to understanding activities. The activity of a constituent is the ratio of the fugacity of that constituent to its fugacity in some other state, which we called a reference state. We then showed through consideration of the Lewis Fugacity Rule, which is an extension of Dalton s Law, that for ideal solutions of condensed phases, the activity of a constituent equals its mole fraction, if the reference state is the pure constituent at the same P and T. Deviations from ideal behaviour are then conveniently handled by introducing Henryan and Raoultian activity coefficients. 

Moving one step closer to reality, the Lewis Fugacity Rule (described in Chapter 11) is frequently used to approximate the behavior of real gas mixtures... 

This is the Lewis fugacity rule, and is seen to be true if Amagat s law is true. So fluids can mix ideally volumetrically, but might still be nonideal mixtures. Note that it assumes the additivity of the molar volumes of all components of... 

But it is not necessary to set / " = 1 bar, just convenient in many cases. Another option, fairly common in geochemistry though not in chemistry, is to let /° = and to define the standard state pressure as the system pressure. In this case, we compare the fugacity of i with the fugacity of pure / at the same T and P. If the Lewis fugacity rule ( 8.1.3) holds, this is the same as using the mole fraction of i, but normally this wiU be only approximately true. 

Volumetric ideal mixing (Equation 10.4) is also called Amagat s law, which we saw was connected to the Lewis fugacity rule in Chapter 8. 

Find the fugacity and fugacity coefficient of gaseous species / as a pure species and in a mixture using tables, equations of state, and general correlations. Identify the appropriate reference state. Write the Lewis fugacity rule, state the approximation on which it is based, and identify the conditions when it is likely to be valid. 

This approximation is known as the Lewis fugacity rule and will be discussed in more detail shortly. 

When is the Lewis fugacity rule a good approximation For some insight, we can look at the expression developed in Example 7.4 for a binary mixture of species a and b using the van der Waals equation of state and van der Waals mixing rules. The fugacity of species a is given by ... 

Comparison of Equation (E7.4G) with Equation (7.22) shows that the Lewis fugacity rule is a good approximation when the term in the exponential is small. This condition is valid when the following are true ... 

This analysis illustrates the use of engineering models derived from physical principles. Although the van der Waals equation has its quantitative limitations, it provides us with a valuable conceptual guideline in applying the Lewis fugacity rule. Although we should use a more accurate equation of state to quantify the behavior of a species in a mixture, this analysis tells us when we need to concern ourselves with this more dilHcult calculation and when we may omit it. 

This expression is identical to the Lewis fugacity rule [Equation (7.22)]. Recall that a species obeys the Lewis fugacity rule if all the intermolecular forces are equal. 

A gas mixture of species 1 and 2 at 300 K and 30 bar perfectly obeys the Lewis fugacity rule. What is the enthalpy of mixing Choose one of the following answers, and explain your reasoning. 

In which of the following mixtures do you expect species 1 to be better represented by the Lewis fugacity rule Explain. 

Calculate the fugacity of n-butane, / -butane, in a vapor mixture of 1 mole n-butane, 2.5 moles isobutane, 4 moles n-pentene and 1 mole n-pentane at 318.9 K and 3.79 bar. You may use the Lewis fugacity rule. 

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