Activity Coefficients at Special Compositions

Two parameter models provide sufficient flexibility with a balance of relative simplicity to provide successful VLE modeling. Determination of activity for each component permits two parameters to be fitted, and special compositions can be used.

Azeotropes

The location of an azeotrope is very important for distillation design because it represents a point at which further purification in a single distillation column is impossible. Look back at Fig. 11.1 on page 412. Looking at dilute isopropanol concentrations, note x_2-propanol = 0.01 < y_2-propanol, but near purity, x_2-propanol = 0.99 > y_2-propanol. The relative magnitudes have crossed and thus we expect y_2-propanol = x_2-propanol (i.e., there is an azeotrope) somewhere in between. If the relative sizes are the same at both ends of the composition range, then we expect that an azeotrope does not exist.¹¹ Certainly, the best way to identify an azeotrope is to plot T-x-y or P-x-y, but a quick calculation at each end of the diagram is usually sufficient.

 A simple algorithm to decide if an azeotrope exists.

Note that the relative volatility introduced in Section 10.6 on page 390 also changes significantly in an azeotropic system. For the reasons above, α_ij > 1 on one side of the azeotrope, α_ij = 1 at the azeotrope, and α_ij < 1 on the other side. Because shortcut distillation calculations fail at α_ij = 1, they must not be used if (α_ij – 1) changes sign between column ends. This means that screening systems for azeotropes such as using the algorithm above is important before blindly plugging numbers into shortcut distillation calculations.

 Relative volatility crosses 1 at an azeotrope.

 Any deviation from ideality will create an azeotrope at a Bancroft point.

We noted in Section 10.7 on page 393 that azeotropic behavior was dependent on the magnitude of deviations from ideality and the vapor pressure ratio. Look back at Fig. 11.1 on page 412 and recall that deviations from Raoult’s law create the curve in the bubble line. When the pure component vapor pressures are nearly the same then a slight curve due to non-ideality can cause an azeotrope. The same size deviation in a system with widely different vapor pressure may not have an azeotrope. A plot of logP^sat versus 1/T with both components may show a point where the two curves cross when the heats of vaporization are different. This point is called a Bancroft point. Since the vapor pressures are exactly equal at the Bancroft point, any small non-ideality generates an azeotrope. This might be avoidable if the system pressure is raised or lowered to circumvent the Bancroft point in the temperature range of a distillation column.

Many tables of known azeotropes are commonly available.¹² For systems with an azeotrope, the azeotropic pressure and composition provide a useful data point for fitting activity coefficient models because x₁ = y₁. Then ; . Then the typical single point fitting formulas are used with the azeotrope composition to find the model parameters.

 The azeotrope is a useful point to fit parameters.

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Example 11.7. Azeotrope fitting with bubble-temperature calculations

Consider the benzene(1) + ethanol(2) system which exhibits an azeotrope at 760 mmHg and 68.24°C containing 44.8 mole% ethanol. Using the two-parameter Margules model, calculate the composition of the vapor in equilibrium with an equimolar liquid solution at 760 mmHg given the following Antoine constants:

Solution

At T = 68.24°C, ; , and the azeotrope composition is known, x₁ = 0.552; x₂ = 0.448. At this composition, the activity coefficients can be calculated.

 Positive deviations from Raoult’s law, γ_i > 1.

Using Eqn. 11.38 with the composition and γ’s just tabulated, A₁₂ = 1.2947, A₂₁ = 1.8373. New activity coefficient values must be found at the composition, x₁ = x₂ = 0.5. Using Eqn. 11.37, γ₁ = 1.583; γ₂ = 1.382. The problem statement requires a bubble-temperature calculation. Using the method of Table 10.1 (a flow sheet is available in Appendix C, option (a); a MATLAB example is provided in Ex11_07.m),

 Bubble-temperature calculation.

Guess ; . For this model, the activity coefficients do not change with temperature. The K-ratio depends on the activity coefficients:

 Ex11_07.m.

Checking the sum of y_i,  is too low. Try a higher T.

After a few trials, at T = 68.262°C, ; 

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Note: The bubble temperatures at x₁ = 0.55 and 0.5 are almost the same. The T-x diagram is quite flat near an azeotrope. This has an important effect on temperature profiles in distillation columns.

Purity and Infinite Dilution

A component is said to be infinitely dilute when only a trace is present. Thus, when a binary mixture is nearly pure in component 1, it is infinitely dilute in component 2. The activity coefficients take on special values at purity and infinite dilution.

Find these limiting values in Fig. 11.5. As an example, consider the infinite dilution composition limits of Eqn. 11.37, , . Infinite dilution activity coefficients are sometimes available in the literature and can be useful for fitting if no data are available near the composition range of interest, but it should be recalled that extrapolations are less reliable than interpolations. In other words, one might experience significant errors in predictions of bubble pressures near equimolar compositions when basing parameters on infinite dilution activity coefficients. The same principle can be used with other activity coefficient models. Infinite dilution activity coefficients are especially important in applications requiring high purity. In those cases, several stages may be required in going from 99% to 99.999% purity.