Excess activity coefficients

Null (1970) discusses some alternate models for the excess Gibbs energy which appear to be well suited for systems whose activity coefficients show extrema. 

The molar excess enthalpy h is related to the derivatives of the activity coefficients with respect to temperature according to... 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

The difference on the left is the partial excess Gibbs energy G y the dimensionless mXio J on the right is called the activity coefficient of species i in solution, y. Thus, by definition. 

Whereas the fundamental residual property relation derives its usefulness from its direct relation to experimental PVT data and equations of state, the excess property formulation is useful because and are all experimentally accessible. Activity coefficients are found from vapor—Hquid... 

Experimentally deterrnined equiUbrium constants are usually calculated from concentrations rather than from the activities of the species involved. Thermodynamic constants, based on ion activities, require activity coefficients. Because of the inadequacy of present theory for either calculating or determining activity coefficients for the compHcated ionic stmctures involved, the relatively few known thermodynamic constants have usually been obtained by extrapolation of results to infinite dilution. The constants based on concentration have usually been deterrnined in dilute solution in the presence of excess inert ions to maintain constant ionic strength. Thus concentration constants are accurate only under conditions reasonably close to those used for their deterrnination. Beyond these conditions, concentration constants may be useful in estimating probable effects and relative behaviors, and chelation process designers need to make allowances for these differences in conditions. 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

Figure 4-2 displays plots of AH, AS, and AG as functions of composition for 6 binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-3 the activity coefficients, derived from Eq. (4-119), appear in Fig. 4-4. The properties shown here are insensitive to pressnre, and for practical pnrposes represent sohition properties at 50°C (122°F) and low pressnre (P 1 bar [14.5 psi]).

Gamma/Phi Approach For many XT E systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactoiy for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written... 

It should, however, be noted that as the concentration of the excess of precipitant increases, so too does the ionic strength of the solution. This leads to a decrease in activity coefficient values with the result that to maintain the value of Ks more of the precipitate will dissolve. In other words there is a limit to the amount of precipitant which can be safely added in excess. Also, addition of excess precipitant may sometimes result in the formation of soluble complexes causing some precipitate to dissolve. 

The activity coefficient y,fpr) is determined by differentiation of gE, the molar excess Gibbs energy at reference pressure Pr,... 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

Unless liquid phase activity coefficients have been used, it is best to use the same equation of state for excess enthalpy that was selected for the vapour-liquid equilibria. If liquid-phase activity coefficients have been specified, then a correlation appropriate for the activity coefficient method should be used. 

A particular case of electrolyte mixtures occurs if one electrolyte is present in a large excess over the others, thus determining the value of the ionic strength. In this case the ionic atmospheres of all the ions are formed almost exclusively from these excess ions. Under these conditions, the activities of all the ions present in the solution are proportional to their concentrations, the activity coefficient being a function of the concentration of the excess electrolyte alone. 

Increasing the excess of ethanol increases the conversion of acetic acid to ethyl acetate. To carry out the calculation more accurately would require activity coefficients to be calculated for the mixture (see Poling, Prausnitz and O Connell6 and Chapter 4). The activity coefficients depend on correlating coefficients between each binary pair in the mixture, the concentrations and temperature. 

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... 

Kinetic theory indicates that equation (32) should apply to this mechanism. Since the extent of protonation as well as the rate constant will vary with the acidity, the sum of protonated and unprotonated substrate concentrations, (Cs + Csh+), must be used. The observed reaction rate will be pseudo-first-order, rate constant k, since the acid medium is in vast excess compared to the substrate. The medium-independent rate constant is k(), and the activity coefficient of the transition state, /, has to be included to allow equation of concentrations and activities.145 We can use the antilogarithmic definition of h0 in equation (33) and the definition of Ksh+ in equation (34) ... 

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... 

The activity coefficient term in equation (66) contains an extra /s term, so exact linearity of the left-hand side in X would not be expected. Nevertheless, the resulting plot was almost linear, with a correlation coefficient of 0.999,246 meaning that the excess acidity treatment does in fact apply. Note that the water molecule in equation (63) is acting as a base. For the k3 step in Scheme 8, log — log Ch+ against X was found to be linear, and for the k-- step log k — log Ch+ - logaH2o was linear here the water molecule is acting as a nucleophile.246... 

For correlation of solubility, the correct thermodynamic quantities for correlation are the activity coefficient y, or the excess Gibbs free energy AG, as discussed by Pierotti et al. (1959) and Tsonopoulos and Prausnitz (1971). Examples of such correlations are given below. 

Using eq. (3.34) the excess Gibbs energy of mixing is given in terms of the mole fractions and the activity coefficients as... 

The local compostion model is developed as a symmetric model, based on pure solvent and hypothetical pure completely-dissociated liquid electrolyte. This model is then normalized by infinite dilution activity coefficients in order to obtain an unsymmetric local composition model. Finally the unsymmetric Debye-Huckel and local composition expressions are added to yield the excess Gibbs energy expression proposed in this study. 

Pitzer et al (1972, 1973, 1974, 1975, 1976) have proposed a set of equations based on the general behavior of classes of electrolytes. Pitzer (1973) writes equations for the excess Gibbs energy, AGex, the osmotic coefficient activity coefficient Y+ for single unassociated electrolytes as... 

Voltammetry and polarography are performed under diffusion control, which is ensured by keeping the solution still, and using an excess of inert electrolyte. The latter also has the effect of equalizing all activity coefficients in solution, so values of concentration, rather than activity, may be derived during measurements. 

Various functions have been used to express the deviation of observed behavior of solutions from that expected for ideal systems. Some functions, such as the activity coefficient, are most convenient for measuring deviations from ideality for a particular component of a solution. However, the most convenient measure for the solution as a whole, especially for mixtures of nonelectrolytes, is the series of excess functions (1) (3), which are defined in the foUowing way. 

Although the description of deviations from ideality in terms of the excess Gibbs function gives us one quantity instead of the two activity coefficients of the two components of a binary solution, we still need to calculate the activity coefficients first, as observed in Equation (16.57). 

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