The experimental data and the calculated results are shown in Fig. 3-6. It will be noted that the fugacity calculations for the yfx values are in excellent agreement with the experimental results, but that RaouIt's and Dalton's laws give values of the vapor composition that are much too high. For average relative volatility these laws give 2.45; fugacity, 1.77; and experimental, 1.71.

Solution Deviations. The corrections so far considered have been limited to those associated with the fact that the vapor does not obey the perfect-gas law. A large number of mixtures, in fact most of them, do not obey the ideal solution laws even at very low pressure, and the deviations cannot be predicted by the use of gas-phase fugacity corrections. The deviations are the result of the forces between the molecules in the liquid phase, and these forces can be very Targely due

to the close packing in the dense phase. The theoretical method of attack for the liquid phase is not so simple as for the vapor phase. For the vapor-phase calculations a convenient basis was possible because at low pressure all vapor mixtures obey the perfect-gas laws. Thus the deviations could be calculated on the basis of the differences between the mixture at low pressure and at high pressure. In the case of the liquid phase no such convenient basis is possible. Thus a mixture of ethyl alcohol and water does not agree with the ideal solution rules under any practical conditions of temperature and pressure.

Basic thermodynamic relations are available for the liquid phase, but their practical application has not been so well developed as those for the vapor phase. They are helpful in formulating general concepts and are directly applicable in certain special cases. One of the most useful relations follows:

For a binary mixture dxi = — Eq. (3-20) reduces to

If the pressure is such that the vapor satisfies the perfect-gas law, then the equation can be modified as follows:

This equation is called the Duhem equation (Ref. 9). Equation (3-20) is applicable to any system of any number of components, but the Duhem equation is limited to a binary mixture under conditions such that the perfect-gas law applies to the vapor.

Theoretically these equations apply only to a process carried out at constant temperature and constant total pressure on the liquid phase. In most mixtures encountered in distillation, if one varies the composition at constant temperature, the total pressure also varies and Eqs. (3-20) to (3-22) are not strictly applicable. The equations would apply for this constant-temperature case if some method other than the vapor pressure were employed to exert pressure on the liquid which was adjusted to keep the total pressure constant; e.g., a gas insoluble in the liquid could be added to the vapor to maintain constant total pressure. Actually, these equations apply satisfactorily for most engineering purposes if they are employed at constant temperature and a variable total pressure equal to the vapor pressure. The error introduced is that due to the change in the fugacity of the liquid with the total pressure which can be calculated by Eq. (3-7). A more exact relationship for binary mixtures at constant temperature is

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