Standard state Lewis-Randall

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

We emphasize that (5.2.3)-(5.2.8) only apply when we use the Lewis-Randall standard state (5.1.5) for the ideal solution. 

When pure components are mixed at constant T and P, an energy balance shows that measures the heat effect. In the Lewis-Randall standard state, ideal solutions have no heat effect on mixing, = h = 0 but for real mixtures, the heat effect may be exothermic q = h < 0) or endothermic q = h > 0). 

Lastly, we emphasize that the definition of the excess properties (5.2.1) is completely general in that it can be used to measure deviations from any kind of ideal solution. In this section we have illustrated that definition using ideal solutions based on the Lewis-Randall standard state (5.1.5). This is a typical application however, other kinds of ideal solutions, based on other standard states, can be defined and prove useful in special situations. In those cases, the generic definition of the excess property (5.2.1) still applies. 

In 4.5.5 we computed residual properties for gaseous mixtures of methane and sulfur hexafluoride mixtures at 60°C and 20 bar. In 5.3.1 and 5.3.2 we computed excess properties for this same mixture. We can also compute residual properties for the ideal solution (Lewis-Randall standard state). Comparisons of these three kinds of difference measures are shown in Table 5.1 for equimolar mixtures. We see that the equimolar mixture of methane and sulfur hexafluoride exhibits positive deviations... 

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. 

We emphasize that in writing (5.4.10) we have specified the temperature and pressure of the standard state, but we still have not made a unique choice for the standard state because we have not yet specified its phase. One common choice is the Lewis-Randall standard state, defined in (5.1.5), in which each standard-state fugacity is taken to be that for the pure component in the same phase and at the same temperature T and pressure P as the mixture. 

Figure 5.7 Activity coefficients for each component in three binary liquid mixtures, all at 60°C. Top acetone-chloroform. Middle acetone-methanol. Bottom methanol-chloroform. Note the scale change from one ordinate to the next. These y,- are based on the Lewis-Randall standard state and were comjnited using the Margules model, with parameters from Table E.2. Note in the top panel that y, < 1, while in the middle and bottom panels y, > 1. After a similar figure in Prausnitz et al. [2] and based on original data in Severns et al. [9].

To obtain an expression for the rhs in terms of activity coefficients, we choose the Lewis-Randall standard state, and then we use (5.2.7) to eliminate g in favor of g, ... 

Consider a binary mixture containing components 1 and 2, and let us choose the Lewis-Randall standard state (5.1.5) to define an ideal solution. Our objective is to obtain a functional model to represent the composition dependence of the excess Gibbs energy. If we look back at Figures 5.2-5.5, we see that, for binary liquid mixtures at fixed T, is nearly parabolic in Xp even when and are not parabolic. This suggests a first approximation to g (xi). 

The Margules expressions for activity coefficients are based on the Lewis-Randall standard state (5.1.5), and therefore they must obey the pure-component limit (5.4.12). In addition, as with Porter s equations, the parameters A and A2 are simply related to the activity coefficients at infinite dilution. In particular, when we apply the dilute-solution limit (5.4.13) to (5.6.12) and (5.6.13), we obtain... 

Consider a binary mixture that has = 0 and h /RT = 0.6 ci c2 with the ideal solution relative to the Lewis-Randall standard state. Find the expression for the composition dependence of g , the change in Gibbs energy on mixing. 

That is, at low pressures we ignore the pressure dependence of all activity coefficients and all standard-state fugacities. In the 3 phase, values for the activity coefficients depend on the choice made for the standard-state fugacity for example, if the Lewis-Randall standard state is chosen for all components (5.1.5), then the y,- would be obtained from a model for the excess Gibbs energy. Common choices for the standard state are discussed in 10.2. In the a phase, values for the fugacity coefficients are obtained from a volumetric equation of state now, either pressure-explicit or volume-explicit models may be chosen. Fortunately at low pressures, either the ideal-gas law or a virial equation may be sufficiently accurate. 

Otherwise we usually have y < 1, as suggested by Figure 10.5 in contrast, in the Lewis-Randall standard state we usually have y, > 1, also suggested by Figure 10.5. This means that deviations from Lewis-Randall ideal-solution behaviors differ qualitatively from deviations from Henry s law ideal-solution behaviors. 

This establishes a simple relation between two standard-state fugadties that in the Lewis-Randall standard state (/J,urei) solute-free dilute-solution stan-... 

Lewis-Randall standard state. If we have a value for the pure 1 fugacity at 25°C and 1 bar, then we can use the activity coefficient in the Lewis-Randall standard state to evaluate fi. A value for / ure 1 itiight be obtained from a correlation, an estimate, or a reduction of experimental data. For this situation we find /pure 1 = 1-4 bar. Then... 

This is the same value found in (10.2.63) using the Lewis-Randall standard state. 

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