where d in pf is the fractional change in effective pressure of component

1 due to the change in total pressure on the liquid. It is calculated by Eq. (3-7).

where t = total pressure on liquid phase

V\ = partial molal volume of component 1 R = gas law constant T = absolute temperature The value of d In is calculated in the same manner with v2 instead of V\. In most cases these corrections at constant temperature are small in comparison to the change in fugacity due to the change in composition; for this reason Eq. (3-22) is frequently accepted as applying to mixtures under their own vapor pressure. It can be in serious error at high pressure, and Eq. (3-23) would be more exact.

For constant pressure, variable temperature conditions, Eq. (3-22) generally is unsatisfactory owing to the rapid change in fugacity or vapor pressure of a liquid with changes in temperature. Equation (3-22) can be modified to compensate for this effect as follows:

Xi where d In P[ is the fractional change in the partial pressure of component 1 at a composition $lf with the change in temperature. It is evaluated by d In P' AH'

dT RT2

where AH' is the heat of vaporizing 1 mol of the component from the solution into a vacuum.

Equations (3-23) and (3-25) can be combined and expressed in the more exact fugacity form,

Xi where d In f° is the fractional change of the fugacity of the component in the liquid phase at the given composition and is calculated by Eq. (3-26) for constant pressure changes and by Eq. (3-24) for constant temperature changes.

The Duhem equation as such cannot be integrated, but if a relationship between the pressure of one of the components and the mol frac tion is available, it is possible to calculate the relationship between the partial pressure of the other component and the mol fraction. This is not of any real engineering utility for predicting vapor-liquid equilibria since in general when the partial pressure of one component is known that of the other component is also known. However, the Duhem equation is useful in checking experimental data and also in guiding the development of correlations. For example, Eqs. (3-22) to (3-27) can be used to evaluate the accuracy of vapor-liquid equilibrium data. Consider the case of a binary mixture at constant temperature for which the liquid-vapor data are available. By Eq. (3-23),

If one experimental value of pi/p* is taken as a base, then the value of this ratio can be determined at other compositions by the integration of the right-hand side of the equation. This integration must usually be performed graphically, and a convenient method is plotting x2/(l — x%) vs. In (pn/p*) and determining the area under the curve. The experimental data for p2 as a function of x2 and one value of pi allow the value of pi to be predicted at other compositions. If corrections for total pressure on pf and p$ are to be included, they can be evaluated from the experimental total pressure data or by summing the calculated value of pi and in the latter case, the integration becomes trial and error.

Constant pressure conditions are more important in distillation calculations than constant temperature, and for this condition Eq. (3-25) can be modified to

The evaluation of the P' terms involves the heat of vaporizing 1 mol of the component from the mixture. At moderate pressure this is equal to the latent heat of vaporization at the same temperature plus the heat effects of mixing the liquids at this temperature and bringing them to the total pressure. As an approximation the latent heat of vaporization of the pure liquids can be employed in Eq. (3-26), but it will not give satisfactory results if (1) the heat of mixing is large or (2)

the total pressure is so high that the pressure-enthalpy corrections for the vapor are large.

One of the cases in which the Duhem equation is of real utility in determining vapor-liquid equilibria is where the analysis of the composition of the two phases offers serious difficulties. In such a case, if it is possible to prepare known mixtures of the two components and to determine their equilibrium total pressure at a given temperature, these data can be used with the Duhem equation to calculate the composition of the equilibrium vapor.

Activity Coefficient. Because there are no convenient conditions for the liquid mixture upon which to base calculations, it is customary to use the pure liquids before they are mixed as the basis, and then calculate the deviations that result from the mixing operation. The deviations in the liquid phase are summed up in what is termed an activity coefficient. -Thus Raoult's law and the idealized fugacity law are modified by the insertion of a factor on the right-hand side.

The value of the activity coefficient, 7, is the factor that will make these equations correct for the case in question. All deviations other than those associated with the gas law are lumped into the one value, and the problem is thus made into one of predicting the activity coefficient. These factors will be different for each component, but they are interrelated by Eq. (3-20) and, for a binary mixture, Eqs. (3-21), (3-22), (3-30), and (3-31) become

The methods that have been used for predicting the activity coefficient are either empirical or semitheoretical, the theoretical part being the use of thermodynamic equations to direct the development of empirical rules. A number of rules have been proposed (Refs. 3, 5,13, 15, 28), but the two most commonly used methods of estimating the activity coefficients for solutions of the type employed in distillation are the Margules and Van Laar equations.

Example Illustrating the Use of Duhem Equation. Data on the vapor-liquid equilibria of benzene-n-propanol and ethanol-water are given in the accompanying tables. Using the Duhem equation, check the consistency of the data.

Mol fraction in liquid |
Partial pressure, mm. Hg | ||

Benzene |
n-Propanol |
Benzene |
n-Propanol |

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