Fugacity Lewis-Randall rule

This equation, known as the Lewis-Randall rule, applies to each species in an ideal solution at all conditions of temperature, pressure, and composition. It shows that the fugacity of each species in an ideal solution is proportional to its mole fraction the proportionality constant is the fugacity of pure species / in the same physical state as the solution and at the same T and P. Ideal solution behavior is often approximated by solutions comprised of molecules similar in size and of the same chemical nature. 

Equation 2-40 is known as the Lewis Randall rule, where f is the fugacity of pure component i at the same temperature and total pressure as the mixture. 

Assume at 1 bar that the vapor is an ideal gas. The vapor-phase fugacities are then equal to the partial presures. Assume the Lewis/Randall rule applies to concentrated species 2 and that Henry s law applies to dilute species 1. Then ... 

Figure 12.17 Fugacities /, and f2 for the system methyl ethyl ketone(l)/toluene(2) at 50°C. dashed lines represent the Lewis/Randall rule.

The straight dashed lines represent Eq. (11.61), the Lewis/Randall rule, w expresses the composition dependence of the component fugacities in an i solution ... 

Figure 12.18 is drawn for a species that shows positive deviations from ideality in the sense of the Lewis/ Randall rule. Negative deviations from ideality are also common, and in this case the /j-vs.-x, curve lies below the Lewis/Randall line. In Fig. 12.19 we show the composition dependence of the fugacity of acetone in two different binary solutions at 50°C. When the second component is methanol, acetone shows positive deviations from ideality. On the other hand, when the second component is chloroform, acetone shows negative deviations from ideality. The fugacity of pure acetone f—<— is of course the same regardless of the second component. However, Henry s constants, represented by the slopes of the two dotted lines, are very different for the two cases. 

Related Calculations. If the gas is not ideal, the fugacity coefficients , will not be unity, so the activities cannot be represented by the mole fractions. If the pressure is sufficient for a nonideal solution to exist in the gas phase, , will be a function of y, the solution to the problem. In this case, the y, value obtained for the solution with Lewis-Randall rule for... 

The value of the parameter 2 13 in a gas mixture can he calculated from PVT data using any traditional EOS. Eor the mixtures that obey the Lewis-Randall rule [16] (the fugacity of a species in a gaseous mixture is the product of its mole fraction and the fugacity of the pure gaseous component at the same temperature and pressure), the fugacity coefficients of the components of the mixture are independent of composition. In such cases, the KB equation [13] for the binary mixtures 1-3 ... 

The Lewis—Randall rule for the fugacity of a species in a gas mixture will be adopted hence, the fugacity of a component in a mixture is obtained by multipl5dng its fugacity as a pure gas with its mole fraction. In addition, for the sake of simplicity, the solubilities of both gases will be assumed small, and the concentration of the solvent in the gas phase will be neglected. Therefore, for the fugacities of the two species of the gas mixture, one can write... 

Determine by the graphical method the fugacity of nitrogen in the mixture at the various total pressures, and compare the results with those obtained for the pure gases hence, test the Lewis-Randall rule for the fugacity of a gas in a mixture [cf. Merz and Whittaker, J, Am. Chem, Soc., 50, 1522 (1928)]. 

Use the Lewis-Randall rule and Fig. 18 to calculate the fugacities of carbon monoxide, oxygen and carbon dioxide in a mixture containing 23, 34 and 43 mole %, respectively, of these gases at 400 C and a total pressure of 250 atm. 

This result, which relates the fugacity of a species in a gaseous mixture only to its mole fraction and.the fugacity of the pure gaseous component at the same ternperature and pressure, is known as the Lewis-Randall rule. 

Approximate Species Fugacity Calculation Using the Lewis-Randall Rule... 

Compute the fugacities of ethane and /i-butane in an equimolar mixture at 323.15 K at 1, 10, and 15 bar total pressure assuming that the Lewis-Randall rule is correct. 

Compute the fugacity of both carbon dioxide and methane in an equimolar mixture at 500 K and 500 bar using (a) the Lewis-Randall rule and (b) the Peng-Robinson equation of state. 

To estimate the fugacity of a species in a gaseous mixture using the Lewis-Randall rule. 

In low- to moderate-density vapors, mixture nonidealities are not very large, and therefore equations of state of the type discussed in this text can generally be used for the prediction of vapor-phase fugacities of all species. [However, mixtures containing species that associate (i.e., form dimers, trimers, etc.) in the vapor phase, such as acetic acid, are generally described using the virial equation of state with experimentally determined virial coefficients.] The Lewis-Randall rule should be used only for approximate calculations it is best to use an equation of state to calculate the vapor-phase fugacity of vapor mixtures. 

The concept of ideal solution signifies no interactions between molecules. The only information regards pure component properties and mixture composition. Following the Lewis Randall rule, the component fugacity in an ideal solution is obtained by multiplying the pure species fugacity at given T and P by its molar fraction. Thus, for the vapour phase we may write ... 

For an ideal solution we may apply the Lewis-Randall rule and write the fugacity as fi =. It follows that ... 

Another reference-state for the solute / may be its pure liquid fugacity, fj. This state is a virtual one, because in practice x, 1. If the actual liquid mixture has as reference an ideal solution obeying the Lewis-Randall rule, we may define the reference-state f- as the limit of component fugacity at jc, 1 ... 

Ideal solution is often chosen as reference in analysis. In this case we may write for the fugacity of a component the relations f/ (HL) = XjHj and fj LR) = X/fk vhere HL stands for Henry law and L-R for Lewis-Randall assumption. The problem is now how to express the phase equilibrium A possible approach would be to use Henry law for solute, and Lewis-Randall rule for solvent. For this reason such definition of Ai-values is considered asymmetric. 

Because of the unit value of this activity coefficient in the ideal mixture (see (4.437), (4.444)) this may be expressed by fugacities through Lewis or the Lewis-Randall rule equivalently as... 

Many choices for the standard state are possible, and in fact, we need not even choose the same standard state for all spedes in a mixture. But to have an example for use throughout this chapter, we introduce the most common choice the Lewis-Randall rule [3], in which the standard state for each component is taken to be the pure substance in the same phase and at the same T and P as the mixture. With this choice, each standard-state fugacity is given by... 

In the land of pure-component standard states, the Lewis-Randall rule (5.1.5) is but a district. The two differ in their standard-state pressures and phases. The Lewis-Randall standard-state pressure and phase are always those of the mixture, but in a generic pure-component standard state, the standard-state pressure and phase need not be the same as those of the mixture. In general, the choice for standard-state is dictated by the availability of a value for the pure-component fugacity either from a reduction of experimental data, or from a correlation, or from an estimate. We caution that other authors may make other distinctions, and some may make no distinction between the Lewis-Randall rule and the pure-component standard state. 

Using the Lewis-Randall rule (5.4.11) for the standard state fugacity in (5.4.5), the resulting expression for the activity coefficient jj approaches unity as the mixture is made more nearly pure in component i ... 

Fugacity Formula 2. If we have, from experiment or correlation, the value of a standard-state fugacity f° at the mixture temperature and pressure, so we can use the Lewis-Randall rule (5.1.5), then we recast FFF 1 into an alternative form. First multiply and divide (6.4.1) by the known standard-state fugacity f°,... 

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