Van Laar form

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

AH adopted in Ref.9) is generally incorrect. We will see below that for the dilute solution of rods, the main contribution to AH appears to stem from the configurations in which two rods are almost parallel and have a great area of contact. However, the role of these configurations is seriously underestimated if AH is written in the Van-Laar form. 

This is just the well-known van Laar expression for the heat of mixing in any two-component system. Thus, within the limits of the approximations used, the polymeric character of the solute does not alter the form of the heat of mixing expression. To generalize somewhat, let the solvent molecule contain Xi segments instead of only one. Eq. (19) is then replaced by... 

The Wilson equation is superior to the familiar Van-Laar and Margules equations (see Volume 2, Chapter 11) for systems that are severely non-ideal but, like other three suffix equations, it cannot be used to represent systems that form two phases in the concentration range of interest. 

The activity coefficients were assumed to satisfy van Laar equations of the form... 

Figure 3.9D shows the form of the curve of the excess Gibbs free energy of mixing obtained with Van Laar parameters variable with T. the mixture is subregular— i.e., asymmetric over the binary compositional field. 

J. J. van Laar has shown how the form of the vap. press, curves of a liquid mixture can furnish an indication, not a precise computation, of the degree of dissociation of any compound which maybe formed, on the assumption that the different kind of molecules in the liquid—12, Br2, and IBr—possess partial press, each of which is equal to the product of the vap. press, of a given component in the unmixed state and its fractional molecular concentration in the liquid. It is assumed that in the liquid, there is a balanced reaction 2IBr I2-)-Br2, to which the law of mass action applies, where K is the equilibrium constant, and Clt C2, and C respectively denote the concentration of the free iodine, free bromine, and iodine bromide. From this, P. C. E. M. Terwogt infers that at 50 2°, K for the liquid is 7j and that for iodine monobromide about 20 per cent, of the liquid and about 80 per cent, of the vapour is dissociated. That the vapour of iodine monobromide is not quite dissociated into its elements is evident from its absorption spectrum, which shows some fine red orange and yellow lines in addition to those which characterize iodine and bromine. In thin layers, the colour of the vapour is copper red. 0. Ruff29 could uot prove the formation of a compound by the measurements of the light absorption of soln. of iodine and bromine in carbon tetrachloride. 

Na (g). We have calculated the heat of sublimation of sodium to form the monatomic gas from the vapor pressure-temperature data, taking due account of the appreciable amount of Na2 molecules contained in the actual vapor at equilibrium. The vapor pressure data used are those of Edmonson and Egerton,1-2 Rodebush and Walters,1 Rodebush,2 Rodebush and de Vries,1 Rodebush and Henry,1 Haber and Zisch,1 Ladenberg and Minkowski,1 and Gibhart.1 See also Kroner,1 Hackspill,1 van Laar,9 and Simon and Zeidler.1 Our value for the heat of sublimation, Na (c) = Na (g), is —25.9 at 18°. Sherman1 calculated —25.8. 

Related Calculations. These calculations show how to use vapor-liquid equilibrium data to obtain parameters for activity-coefficient correlations such as those of Van Laar and Wilson. (Use of liquid-liquid equilibrium data for the same purpose is shown in Example 1.20.) If the system forms an azeotrope, the parameters can be obtained from a single measurement of the azeotropic pressure and the composition of the constant boiling mixture. If the activity coefficients at infinite dilution are available, the two parameters for the Van Laar equation are given directly, and the two in the case of the Wilson equation can be solved for as shown in the example. 

In principle, the parameters can be evaluated from minimal experimental data. If vapor-liquid equilibrium data at a series of compositions are available, the parameters in a given excess-free-energy model can be found by numerical regression techniques. The goodness of fit in each case depends on the suitability of the form of the equation. If a plot of GE/X X2RT versus X is nearly linear, use the Margules equation (see Section 3). If a plot of Xi X2RT/GE is linear, then use the Van Laar equation. If neither plot approaches linearity, apply the Wilson equation or some other model with more than two parameters. 

So that an azeotrope with acetone does not form, the alcohol used must have a high enough boiling point. This requirement is reliably established only if vapor-liquid equilibrium data for at least two, preferably three, of the members of the series with acetone are known. The Pierotti-Deal-Derr method (4) (discussed later) or the Tassios-Van Winkle method (5) can be used in this case. In the latter method a log-log plot of y°i vs. P°i should yield a straight line. Figure 1 presents results for n-alco-hols and benzene from the isobaric (760 mm Hg) data of Wehe and Coates (6). Reliable infinite dilution activity coefficients are established for the other n-alcohols from data for at least two, and preferably three, of them. These y° values are used with equations like those of Van Laar or Wilson (7) to generate activity coefficients at intermediate compositions and to check for an existing azeotrope or a difficult separation (x-y curve close to the 45° line). 

The results obtained in the solution of a sample problem are summarized here to illustrate the application of the method. An extractive distillation problem from Oliver (6) was used in which methylcyclo-hexane is separated from toluene by adding phenol. The column contains 11 stages (including the reboiler and condenser) and has a feed of 0.4 moles/unit time of methylcyclohexane and 0.6 moles/unit time of toluene to the fourth stage from the reboiler and 4.848 moles/unit time of phenol to the fourth stage from the condenser. We used the same physical property correlations as Oliver. The activity coefficients were obtained from a multicomponent form of the Van Laar Equation (7). 

Conglomerates that are equimolecular mixtures of two crystalline enantiomers are easily separated by cristallization. There are two phases in the solid state and only one phase in the liquid state (miscibility). The equation of Schroder-Van Laar in its simplified form correlates the composition of mixtures to the end of fusion 7 ... 

This is the equation for the crystallization curve of a solution provided that the solution is ideal, that no mixed crystals are formed, and that the difference between the heat capacities in the liquid and solid states is small enough to justify neglecting the second term in (22.4). This important equation is due to Schroder and van Laar.j ... 

The dependence of the curvature of the freezing point curve at the origin upon the entropy of fusion of the solvent was first pointed out by van Laar. The most common case corresponds to (22.17) and fig. 22.1 (c/. chap. XIV, 5, table 14.6). On the other hand, for spherical solvent molecules (c/. table 14.5) for which the entropy of fusion is abnormally small, the case (22.18) is realized. The form of the freezing point curve at the origin is thus a useful criterion, in the absence of calorimetric data, for the identification of those compounds which have a low entropy of fusion, f... 

Another form of the general empirical equations for the variation of the fugacity with composition, which is said to be more convenient for certain purposes, was proposed by J. J. van Laar (1910) thus,... 

At this stage the program attempts to optimize the two model parameters of the van Laar model, and the intermediate results are continuously displayed on the screen in the form of an error bar, When the optimization is complete, a message displaying summary of results appears on the screen for inspection. Press RETURN to continue, The results given below appear on the screen.)... 

The multi-component van Laar equation takes the following form (Carlson and Colburn, 1942 Wohl, 1946 Prausnitz et al., 1967) ... 

Hydrofluoric acid and water form an azeotrope at 120°C and atmospheric pressure, with a hydrofluoric acid mole fraction of 0.35. Calculate the activity coefficients of hydrofluoric acid and water at the azeotrope, and file van Laar constants. Assume ideal gas behavior in the vapor phase. The vapor pressures of hydrofluoric acid and water are given as follows ... 

Benzene (1) and water (2) form a heterogeneous azeotrope at 450 K and 2000 kPa. The benzene-rich liquid phase has 95.7% mole benzene, the water-rich phase has 97.7% mole water, and the vapor phase has 48% mole benzene. Use these data to determine the van Laar parameters for the benzene-rich and the water-rich phases. An approximation of ideal gas behavior in the vapor phase may be assumed. The vapor pressures at 450 K are 936 kPa for water and 978 kPa for benzene. 

In an ethyl acetate (l)-ethyl alcohol (2) separation process, it is required to determine if this mixture forms an azeotrope. If it does, it is required to determine the azeotropic temperature and composition as a function of the azeotropic pressure between 70 and 120 kPa at 10 kPa intervals. Solve for the composition and temperature by doing a single computation operation at each pressure value. Additionally, for any one of the pressure points determine if the azeotrope is minimum- or maximum-boiling. Use the van Laar equation for the liquid activity coefficients and assume ideal gas behavior for the vapor phase. The following data are given ... 

R.S.T. has the same form as the van Laar Equation, so that proof follows from (iii) above. 

The form of the Wohl Equation which leads to the van Laar Equation is... 

Perhaps the most important term in Eq. (5.2-3) is the liquid-phase activity coefficient, and mathods for its prediction have been developed in many forms and by many workers. For binery systems die Van Laar [Eq. (1.4-18)]. Wilson [Eq. (1.4-23)]. NRTL (Eq. (1.4-27)], and UNIQUAC [Eq. (1.4-3 )] relationships are useful for predicting liquid-phase nonidealities, but they require some experimental data. When no data are available, and an approximate nonideality correction will suffice, the UNiFAC approach Eq-(1.4-31)], which utilizes functional group contributions, may be used. For special cases Involving regular solutions (no excess entropy of mixing), the Scatchard-Hiidebmod mathod provides liquid-phase activity coefficients based on easily obtained pane-component properties. 

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