Ideal binary mixture

Favorable Vapoi Liquid Equilibria. The suitabiHty of distiUation as a separation method is strongly dependent on favorable vapor—Hquid equiHbria. The absolute value of the key relative volatiHties direcdy determines the ease and economics of a distillation. The energy requirements and the number of plates required for any given separation increase rapidly as the relative volatiHty becomes lower and approaches unity. For example given an ideal binary mixture having a 50 mol % feed and a distillate and bottoms requirement of 99.8% purity each, the minimum reflux and minimum number of theoretical plates for assumed relative volatiHties of 1.1,1.5, and 4, are... 

The calculation for a point on the flash curve that is intermediate between the bubble point and the dew point is referred to as an isothermal-flash calculation because To is specified. Except for an ideal binary mixture, procedures for calculating an isothermal flash are iterative. A popular method is the following due to Rachford and Rice [I. Pet. Technol, 4(10), sec. 1, p. 19, and sec. 2, p. 3 (October 1952)]. The component mole balance (FZi = Vy, + LXi), phase-distribution relation (K = yJXi), and total mole balance (F = V + L) can be combined to give... 

FIG. 13-107 Binary distiUatio n column dynamic distillation of ideal binary mixture. 

Figure 8-16. Approximate solution for N and ly/D in distillation of ideal binary mixtures. Used by permission, Faasen, J.W., Industrial Eng. Chemistry, V. 36 (1944), p. 248., The American Chemical Society, all rights reserved.

FIGURE 8.35 The vapor pressures of the two components of an ideal binary mixture obey Raoult s law. The total vapor pressure is the sum of the two partial vapor pressures (Dalton s law). The insets below the graph represent the mole fraction of A. 

Consider an ideal binary mixture of the volatile liquids A and B. We could think of A as benzene, C6H6, and B as toluene (methylbenzene, C6H< CH ), for example, because these two compounds have similar molecular structures and so form nearly ideal solutions. Because the mixture can be treated as ideal, each component has a vapor pressure given by Raoult s law ... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

For a non-ideal binary mixture the partial pressure may be expressed as ... 

It is important to note that during this discussion of degrees of freedom, we have said nothing about the number or types of chemical components involved. If we are separating an ideal binary mixture, we... 

Binary copolymerization resembles distillation of a bicomponent liquid mixture, with a reactivity ratio corresponding to the ratio of vapor pressures of the pure components in the latter case. The vapor-liquid composition curves of ideal binary mixtures have no inflection points and neither do the polymer-composition curves for random copolymerizations, in which/ r2 — 1 (Fig. 7-1). For this reason, such comonomer systems are sometimes called ideal. 

By applying the Clapeyron equation and Trouton s rule to an ideal binary mixture, Rose derived the relation... 

The present paper is devoted to the local composition of liquid mixtures calculated in the framework of the Kirkwood—Buff theory of solutions. A new method is suggested to calculate the excess (or deficit) number of various molecules around a selected (central) molecule in binary and multicomponent liquid mixtures in terms of measurable macroscopic thermodynamic quantities, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. This method accounts for an inaccessible volume due to the presence of a central molecule and is applied to binary and ternary mixtures. For the ideal binary mixture it is shown that because of the difference in the volumes of the pure components there is an excess (or deficit) number of different molecules around a central molecule. The excess (or deficit) becomes zero when the components of the ideal binary mixture have the same volume. The new method is also applied to methanol + water and 2-propanol -I- water mixtures. In the case of the 2-propanol + water mixture, the new method, in contrast to the other ones, indicates that clusters dominated by 2-propanol disappear at high alcohol mole fractions, in agreement with experimental observations. Finally, it is shown that the application of the new procedure to the ternary mixture water/protein/cosolvent at infinite dilution of the protein led to almost the same results as the methods involving a reference state. 

Ideal Binary Mixture. Using eq 13 and relations (A-4 and A-5) from the Appendix, one can write the following expressions for the excesses A/iy values of ideal binary mixtures around a central molecule 1... 

These equations show that the Ariij values for an ideal binary mixture become zero only when the molar volumes of the pure components are the same, otherwise the excesses and deficits have nonzero values and can be calculated with eqs 14 and 15. 

Figure 1. The excess (or deficit) number of molecules i (i = 1, 2) around a central molecule 1 for a binary ideal mixture with vj = 30 cmVmol and = 60 cmVmol. Line 1 is Ann, and line 2 is An2i-Shown are (a) Anij values of an ideal binary mixture calculated with the new eq 13, and (b) An values of an ideal binary mixture calculated with eq 1 (the KBIs were provided by eqs A-4 and A-5 in which was taken as zero).

The conventional method based on eq 1 provides umeason-able results, such as nonzero excesses (or deficits) for single components, all negative excesses for an ideal binary mixture A—B when aU three KBIs are negative, and all negative excesses in some concentration ranges for some real binary mixtures. 

The new method eliminates the above inconsistencies It provides a zero excess for pure components, and excesses (or deficits) which satisfy the volume conservation condition (for both ideal and real mixtures). The derived eq 13 allows one to calculate the excess (or deficit) for an ideal binary mixture (Figure 1) and shows that they become zero only when the molar volumes of the components are equal. 

Fig. 1 Application of eqn (2) to an ideal binary mixture A-B with equal volumes of the components. Excess (or deficit) number of molecules i i = A, B) around a central molecule A. 1 is An a a and 2 is A ba (KBIs were provided by eqns (A 1-4) and (A 1-5) in Appendix...

A modified local composition (LC) expression is suggested, which accounts for the recent finding that the LC in an ideal binary mixture should be equal to the bulk composition only when the molar volumes of the two pure components are equal. However, the expressions available in the literature for the LCs in binary mixtures do not satisfy this requirement. Some LCs are examined including the popular LC-based NRTL model, to show how the above inconsistency can be eliminated. Further, the emphasis is on the modified NRTL model. The newly derived activity coefficient expressions have three adjustable parameters as the NRTL equations do, but contain, in addition, the ratio of the molar volumes of the pure components, a quantity that is usually available. The correlation capability of the modified activity coefficients was compared to the traditional NRTL equations for 42 vapor—liquid equilibrium data sets from two different kinds of binary mixtures (i) highly nonideal alcohol/water mixtures (33 sets), and (ii) mixtures formed of weakly interacting components, such as benzene, hexafiuorobenzene, toluene, and cyclohexane (9 sets). The new equations provided better performances in correlating the vapor pressure than the NRTL for 36 data sets, less well for 4 data sets, and equal performances for 2 data sets. Similar modifications can be applied to any phase equilibrium model based on the LC concept. 

In this Article, a modified LC expression is suggested. This modification is a result of the observation that the traditional expressions for the LCs " are inconsistent with the expressions for the excesses around molecules in ideal binary mixtures (see the next section). The new LCs will be used to obtain expressions for the activity coefficients of binary mixtures using the NRTL equations for illustration. The traditional and corrected NRTL equations wUl be used to correlate the vapor—liquid equilibria (VLE) for alcohol + water binary mixtures and binary mixtures containing benzene, hexafiuorobenzene, toluene, and cyclohexane. It is shown that the modified LCs provide a moderate improvement of the NRTL results. 

The contradiction between the expressions obtained by the authors for the excesses and the LC expressions is resolved by suggesting a simple modification of the latter expressions. The modification is based on the observation that the LCs for an ideal binary mixture should be equal to the bulk composition only when the components of the mixture have equal molar volumes. The new expressions for the LCs are used to improve the NRTL expressions for the activity coefficients. 

The Solubility of Poorly Soluble Solids in a Binary Solvent by Combining Equation 7 with Various Expressions for the Activity Coefficients of the Constituents of the Binary Solvent. 1) The mixed solvent is an ideal binary mixture... 

However, as noted by Matteoli and Lepori [30] and Matteoli [31], the above expression leads for an ideal binary mixture to non-zero values, even though they are expected to vanish. For the above reasons, Eq. (15) was replaced by [27,30,31] ... 

For ideal binary mixtures, relative volatility, a, is the ratio of vapor pressures ... 

Graphical depiction of an ideal binary mixture results in the equilibrium phase diagram shown in Fig. 2.32. 

A non-ideal binary mixture may exist as a single liquid phase at certain compositions, temperatures, and pressures, or as two liquid phases at other conditions. Also, depending on the conditions, a vapor phase may or may not exist at equilibrium with the liquid. When two immiscible liquid phases coexist at equilibrium, their compositions are different, but the component fugacities are equal in both phases. 

Although by starting with Eq. 11.2-2 one can proceed directly to the calculation of the liquid-liquid phase equilibrium state, this equation does not provide in.sight into the reason that phase separation and critical solution temperature behavior occur. To obtain this insight it is necessary to study the Gibbs energy versus composition diagram for various mixtures. For an ideal binary mixture, we have (Table 9.3-1)... 

FIGURE 10.2 Equilibrium partial pressures of the components of an ideal binary mixture as a function of the mole fraction of A, xA. 

For an ideal binary mixture, the derivative of y with respect to Xg is found from Eq. (71) to be... 

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