The per cent deviations are based on [(yoalo — y©*p)/(v — *)expll00. These systems marked * are based on [(poaio — P«xp)/Pe*p] 100.
constant, B, which is not consistent with the derivation, but the use of the negative value gave satisfactory predictions. It was also found that the use of one point for the estimation of the constants of the Van Laar equation was not very satisfactory. In some cases the single-point method worked well and in other cases very poorly, and it depends upon the accuracy of the particular point in question. However, if only one point is available, such as is frequently the case when the azeotrope only is known, the Van Laar and Margules equations probably offer the best method of estimating the vapor-liquid equilibria for other conditions. When data on the azeotropic condition are not available, they can be estimated from Figs. 3 to 5 of the Appendix.
The Van Laar equation is also very useful for transferring data obtained at constant temperature to constant pressure and the reverse. It is also useful for transferring data from one temperature to another. The unsatisfactory agreement in the cases of solutions approaching immiscibility is not surprising. In these cases, the entropy of mixing cannot be equal to that for an ideal solution, and for complete immiscibility the entropy of mixing would be zero. It is therefore not surprising that solutions approaching immiscibility deviate from the Van Laar equation.
Other Applications of the Van Laar Equation. The Van Laar equation can be used to indicate qualitatively the type of phenomena encountered in liquid mixtures. Thus, it can be arranged as follows:
This equation gives the logarithm of the ratio of the vapor pressures divided by the relative volatility as a function of the Van Laar constants and the concentrations. If the solution were ideal, the logarithm term would be zero. Thus the real fact determining deviation from ideal solution is the constant B. If B is equal to zero, the relative volatility will be equal to the ratio of vapor pressures, and the y,x values will be the same as those calculated by Raoult's and Dalton's laws. If B is not equal to zero, the system is not ideal. The terms involving A and the concentrations would have about the same variation independent of the value of B.
It is interesting to consider this equation for various limits. For example, for x% equal to zero, the logarithm term equals ~B/T. For
most mixtures encountered, B is positive. (B is negative for solutions having maximum boiling azeotropes and for certain other solutions.) Thus the logarithm is negative, and the relative volatility is greater than the ratio of the vapor pressures. In these cases, it is easier to remove the component in low concentration than would be expected from the ideal solution law. At the other extreme, i.e., x% equals zero, then X\ over x2 is infinity, and the logarithm becomes B/AT, and with B positive (in all cases so far encountered A is positive) the relative volatility is less than the ideal relative volatility. These conditions are found in most common mixtures; i.e., the relative volatility at low concentration is greater than that of an ideal solution and the relative volatility at high concentration is lower. It is often expressed by saying that the components in small amount are squeezed out. Thus, at the low concentration of the volatile component, it is squeezed out and the relative volatility is high. At high concentration of the volatile component, the nonvolatile component is squeezed out, and the relative volatility is low. For the mixture in which B is negative, the reverse phenomena are true.
The Van Laar equation also would state that a mixture would agree with Raoult's law, independent of the value of B, when the ratio of the mol fractions equals 1 /y/A. For most mixtures, the value of A is somewhere between 0.5 and 2, which would indicate that the vapor-liquid equilibria at some concentration in the middle range would agree with Raoult's law. Thus, the Van Laar equation would imply that the assumption of Raoult's law for the region around a mol fraction of 0.5 would be more satisfactory than at the two ends of the curve. It should be emphasized that this relation does not state that Raoult's law is valid at this condition. It indicates that the relative volatility is equal to the ideal relative volatility at this concentration, but the temperature-total pressure relationships may be far from those indicated by Raoult's law. Thus, in the system ethyl alcohol and water, the total pressure is always higher than would be indicated by Raoult's law, but the vapor-liquid equilibrium curve crosses the Raoult's law curve. Below this intersection, ethyl alcohol is more volatile than would be indicated by Raoult's law; above this concentration, it is less volatile' than would be indicated by Raoult's law.
The Van Laar equation would indicate that the mixtures would become more ideal as the temperature increased. In other words, the ratio of B/T becomes smaller and nearer to the value for an ideal solution.
The Van Laar equation gives interesting relationships for the condi tion under which the relative volatility becomes unity, i.e., the formation of an azeotrope. For this condition, the equations can be arranged as follows:
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