Lewis Randall

Lewis-Randall rule Lewis relation Lewmet 55 Lexan... 

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

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. 

The excess Gibbs energy is of particular interest. Equation 160 may be written for the special case of species / in an ideal solution, with fSidt replaced by xj in accord with the Lewis-Randall rule ... 

Limiting Laws. Simple laws that tend to describe a narrow range of behavior of real fluids and substances, and which contain few, if any, adjustable parameters are called limiting laws. Models of this type include the ideal gas law equation of state and the Lewis-Randall fugacity rule (10). 

Equation 22 is a special application of the general Lewis-Randall ideal solution model (3,10) that is typically used for near-ambient pressures and... 

Thermodynamics of Vapoi—Liquid Equilibrium. Assuming ideal vapor and choosing the Lewis-Randall standard state, a chemical... 

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 ... 

This expression is known as the Lewis/Randall rule. 

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 ... 

The straight dashed line in Fig. 12.18 that represents the Lewis/Randall is the only model of ideal-solution behavior so far considered. Alternative me also express the direct proportionality between /, and x, represented by (11.61), but with different proportionality constants. We may express this... 

Figure 12.18 Composition dependence of /, showing relation to Henry s law and the Lewis/ Randall rule.

When x,- = 1, ff is equal to the fugadty of pure species i in some state at the mixture T and P. Such states are called standard states, and they may be either real or imaginary. When/ = f, Eqs. (11.61) and (12.45) are identical thus the standard state associated with the Lewis/ Randall rule is the real state of species i at the T and P of the mixture. 

This equation is the exact expression of the Lewis/ Randall rule as it appli real solutions. It shows that Eq. (11.61) is valid in the limit as xt 1 and this equation is approximately correct for values of x( near unity. 

Equations (12.46) and (12,47) imply two models of solution ideality. first is based on the Lewis/ Randall rule, for which the standard-state fuga... 

For ideality in the sense of the Lewis/ Randall rule, this equation is identical with Eq. (11.59). For ideality in the sense of Henry s law, it becomes... 

The Gibbs/ Duhem equation provides a relation between the Lewis/Randall rule and Henry s law. Substituting dGt from Eq. (11.28) for dAft in Eq. (11.8) gives, for a binary solution at constant T and P,... 

This is the Lewis/ Randall rule for species 2, and the derivation shows holds whenever Henry s law is valid for species 1. Similarly, fi - f Xx when fl k2X2 ... 

Consider a binary liquid solution of species 1 and 2, wherein species 1 dissolves up to some solubility limit at a specified T and P. Data for the solution can therefore exist only up to this limit, and a plot like Fig. 12.18 is necessarily truncated, as indicated by Fig. 12.20. Clearly, the Lewis/ Randall line for species 2, representing the relation... 

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. 

Analytical representation of the excess Gibbs energy of a system impll knowledge of the standard-state fugacities ft and of the frv. -xt relationshi Since an equation expressing /, as a function of x, cannot recognize a solubili limit, it implies an extrapolation of the /i-vs.-X[ curve from the solubility I to X) = 1, at which point /, = This provides a fictitious or hypothetical va for the fugadty of pure species 1 that serves to establish a Lewis/ Randall 1 for this species, as shown by Fig. 12.21. ft is also the basis for calculation of activity coefficient of species 1 ... 

These equations allow calculation of activity coefficients based on Henry s law from activity coefficients based on the Lewis/Randall rule. In the limit as Xi - 0,... 

When Henry s law is taken as the model of ideality for the solute (species 1) and the Lewis/ Randall rule provides the model of ideality for the solvent... 

This equation relates the excess Gibbs energy based on the asymmetric treatme of solution ideality to the excess Gibbs energy based entirely on the Lewis Randall rule. 

In Sec. 10.4 we wrote down equations for an ideal solution by analogy to for an ideal gas. We wish here to formalize development of the equations f ideal solution. We define an ideal solution as a fluid which obeys Eq. (11 the Lewis/ Randall rule,... 

A mixture of ideal gases is a special case ofan ideal solution for which the Lewis/Randall rule [Eq. (11.61)] simplifies to f a = y,P. Equation (11.58) then reduces to... 

For species known to be present in high concentration, the equation at = x, is usually nearly correct, because the Lewis/ Randall rule always becomes valid for a species as its concentration approaches xf = 1, as discussed in Sec. 12.7. 

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