Activity coefficient ratio

In normal applications of extractive distillation (i.e., pinched, closeboiling, or azeotropic systems), the relative volatilities between the light and heavy key components will be unity or close to unity. Assuming an ideal vapor phase and subcritical components, the relative volatility between the light and heavy keys of the desired separation can be written as the produc t of the ratios of the pure-component vapor pressures and activity-coefficient ratios whether the solvent is present or not ... 

An effect of ionic strength on as a consequence of effects on the activity coefficient ratio is called a primary salt effect. We will, in Section 8.3, consider this effect quantitatively. 

If the same initial solution with an activity coefficient 7 v, is used, and the final (upper) limit varied, values for the activity coefficient ratio on the left can be obtained as a function of. vl Extrapolation of this quantity to, 2 0 allow... 

Therefore, if the activity coefficient ratio, ymss/yms, does not deviate from unity and the temperature is constant, the ratio, myiSe/triMS, correlates to which is... 

Writing equation (6) in logarithmic form results in equation (7). Again by convention, the log ionization ratio, log / = log(CfiH+ /Cb), is defined, with the ionized form on top. Equations (6) and (7) are thermodynamically exact the problem with them has always been what to do about the unknown activity coefficient ratio term. The first person to tackle this problem was Hammett,21 22 who defined an acidity function, H0, as in equation (8) ... 

The second way, called the Bunnett-Olsen method,30 makes the less drastic assumption that log activity coefficient ratios such as those in equation (7) are linear functions of one another, rather than cancelling out. From the definition of H0 in equation (8) we can write equation (11), where Am refers to the primary aromatic amines used in the determination of Ho, and then any specific activity coefficient ratio, say for the weak base B, is assumed to be linear in this according to equation (12) ... 

A philosophical problem remains, however. The Bunnett-Olsen method, which assumes the linearity of activity coefficient ratios in one another, still uses H0, and H0 values are derived using the cancellation assumption The cancellation assumption is eliminated altogether in the excess acidity method (also called the Marziano-Cimino-Passerini and Cox-Yates methods, which is unfortunate since both are the same - the term excess acidity method is preferred). 

For protonation-dehydration processes, such as trityl cation formation from triphenylcarbinols, equation (24), the water activity has to be included if the formulation of the activity coefficient ratio term is to be the same as that in equation (7), which it should be if linearity in X is to be expected see equation (25). The excess acidity expression in this case becomes equation (26) this is a slightly different formulation from that used previously for these processes,36 the one given here being more rigorous. Molarity-based water activities must be used, or else the standard states for all the species in equation (26) will not be the same, see above. For consistency this means that all values of p/fR listed in the literature will have to have 1.743 added to them, since at present the custom... 

Equation (40) also predicts linearity in — H0 with unit slope if the activity coefficient ratio term cancels, and thus is indistinguishable from equation (35)... 

By analogy with equation (12), the assumption made regarding the linearity of activity coefficient ratios is equation (45) (slope parameter j), and the resulting Bunnett-Olsen equations that apply to kinetic measurements are equations (46) and (47) for unprotonated and protonated substrates, respectively.156 These apply to the A1 and A-SE2 mechanisms for the A1 and A2 mechanisms they may require correction for partial substrate protonation as in equations (25) and (26) above. For A2 reactions an additional term such as the log water activity has to be added as in equation (33). These equations have been widely tested and work quite well.155-160 The difference between the Bunnett-Olsen and the excess acidity kinetic methods (discussed below) is that the Bunnett-Olsen method features an additive combination of the slope parameters e and , whereas the excess acidity method features a multiplicative one. There seems to be no theoretical justification for the former. Also the Bunnett-Olsen method still uses H0, whereas acidity functions are not needed for the excess acidity approach see above. 

Now we make the excess acidity assumption that activity coefficient ratios such as that in equation (49) are linear in one another. The best assumption to make is that the term with the activity coefficient of the transition state is linear in the activity coefficient ratio for the same substrate S,162,163 since this is as closely similar as possible to the one in equation (49), see equation (50). We already know that the latter terms are linear in X. Thus for unprotonated substrates the relevant rate equation becomes equation (51).145,161... 

It was shown in Fig. 7 that log /, is linear in X, and if this is so then activity coefficient ratio terms like that in equation (71) will also be linear in X, see equation (50) since the substrate here is SH+,/sh+ in equation (71) corresponds to/s in equation (50). Hence the first linear region in Fig. 14. 

Table VI summarizes values of the activity coefficient ratio Ygr-/YC].- in the saturated solution for each average solid composition (as calculated from the model of Table II), the calculated provisional equilibrium distribution coefficient (Equation 12) and the provisional equilibrium aqueous solution activity ratio of Br to Cl- (Equation 13) based on the data of Table V.

Using the experimental solution compositions (Table IV) and the calculated aqueous solution activity coefficient ratio Ygr-/Y( - (Table VI), Figure 4 shows the slopes of log K... 

Because HDO or H2 0 molecules cannot be expected to have the same solubility in the brine of the main solvent species H2 0 (hereafter simply denoted H2O), their differential behavior must be accounted for by introduction of the appropriate activity coefficient ratio (or isotope salt effect cf Horita et ah, 1993a,b) F ... 

If it is assumed, in a first approximation, that the activity coefficient ratios /ha/ /a- and /ArCHo//ArCH(OH)o do not differ very much, the difference between H and J for a given solution would be expressed by H- — J = —log flH2o-However a calculation of the difference by the latter formula, using the only available information (33) about the activity of water in water-DMSO mixtures, leads to values much smaller than those found in practice. This would indicate that the ratio /ha//a-> for the H indicator acids, is considerably larger than the ratio of /ArCHo//ArCH(OH)o- for the benzaldehydes used as J indicators. Probably this difference in activity coefficient ratios is caused by a larger extent of charge delocalization in the A anion compared to the geminal diol anion, which causes an extra stabilization of A- (manifested by a decrease in f ) by dispersion interaction with DMSO. 

The substituent constant of the Hammett equation has been related successfully to the logarithm of the activity coefficient ratio at infinite dilution for a series of meta and para isomers of phenol. Hammett stated that a free energy relationship should exist between the equilibrium or rate behavior of a benzene derivative and a series of corresponding meta and para monosubstitut-ed benzene derivatives. The Hammett equation may be written... 

Once established, the H scale is used to find pKa values for weak acids. A number of measurements have been made by various groups.53-57 The results obtained at first appeared to disagree with Streitwieser s, but revision of values for some compounds on the basis of further measurements brought the results of the two methods into fairly good agreement. At the same time, however, it became clear that the problems discussed in the previous sections relating to the different behavior of substances of different structural type also apply to the // scale work.58 The activity coefficient ratio evidently is not the same for carbon acids as for the nitrogen acids used to establish the scale.59 Thus the pAa values found by these methods, while probably internally consistent for similar compounds, are not on a firm basis with respect to their absolute relationship to the water scale. 

The derivation is analogous to that given in Section 3.3 for H0.) The method will be successful only if the activity coefficient ratio-is the same for all acids investigated as it is for the acids used as indicators. Since this requirement is evidently not met, it may not be possible to establish a unique solution pA scale for weak acids by using //. ... 

Acceptors, 274 Activity coefficient ratio (T s) determination of experimental value, 157 values for Al-Ga-Sb, 157, 158/ values for Ga-In-Sb, 157, 159/... 

The interfacial pd selectivity coefficient, the factor multiplying a is determined by the ratio K /K, by the activity coefficient ratio, and by the mobility ratio, when the internal diffusion potential contribution is added. Clearly interferences should correlate with the ratio K /1C, which can be determined from salt extraction coefficients K KX/K K for a series of positive drugs, using common anion salts. This result is well documented in the literature (7,8). A curious correlation for N-based drugs studied by us and by Freiser ( ) is a trend in selectivity... 

Figure 1.4a and 6 shows the effect of composition on the activity coefficient ratio. According to Eq. (1.12), this ratio represents the main effect of composition on relative volatility. When a system shows positive deviations, relative volatility decreases as the concentration of the MVC increases, The reverse applies for negative-deviation systems. 

Activity coefficient ratio ( /ocstont f /chloroform ) Activity coefficient ratio (fn-propanol /watsr ... 

Figure 1.4 Activity coefficient ratios (a in the positive-deviation system, n-propanol-water (6) in the negative-deviation system, acetone-chloroform.

It is well known that outside the pH range almost all predeterminations for unexcited molecules have been based on the Hammett indicator method (Hammett and Deyrup, 1932). Even in concentrated acid (or alkaline) solutions, where uncertainties in the value of b//bh become very serious, it is easy to measure the ratio of protonated to unprotonated indicator concentrations spectrophotometrically when the absorption peaks are sufficiently resolved. The difficulty arises in trying to extrapolate beyond the measurable [BH+]/[B] range to zero electrolyte concentration. Hammett and Deyrup assumed that the activity coefficient ratios of the type /b//bh+ were very similar for different indicators in the same acid solution. For the equilibrium (51) between two indicators A and B, a comparison of the concentration ratios [BH+]/[B] and [AH+]/[A] over an acidity range in which both could be measured would lead to a direct estimate of the difference in p-K-values using (52) and (53). Beginning with 4-nitroaniline, which is about half-... 

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