Activity coefficients introduction

Effective use of this general equation requires expHcit introduction of the compositions of the phases. This is done either through the activity coefficient, y, or the fugacity coefficient, ( ) Two procedures are in common use. By the gamma—phi approach, activity coefficients for the Hquid phase enter by equation 202 and fugacity coefficients for the vapor phase by equation 164 equation 220 then becomes equation 221 ... 

An alternative form of Eq. (4-101) follows by introduction of the activity coefficient through Eq. (4-103) ... 

For liquid-phase reaclions, Eq. (4-342) is modified by introduction of the activity coefficient, Y where x, is the liquid-phase mole... 

The thermodynamic and kinetics equations will all be written with concentrations in place of the rigorously correct activities. This usage assumes that the reactions are carried out under nearly ideal conditions. The introduction of activity coefficients for situations where this is not so is considered in Chapter 9. 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

In order to demonstrate the power of the Solver in Excel, let us return to the problem mentioned in the introduction to this chapter (p.31) What is the solubility of calcium sulphate but this time taking into account activity coefficients. As it turns out, they are far from zero, even in a saturated solution of only slightly soluble gypsum. 

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 ... 

It may be conjectured that collective behavior implies that the surfactants that make up the mixture are not too different, the presence of an intermediate being a way to reduce the discrepancy. When the activity coefficient is calculated from non-ideal models it is often taken to be proportional to the difference in solubihty parameters [42,43], which in case of a binary is the difference (3i - if the system is multicomponent, then the dil -ference is - Sm) y which is often less, because the mean value exhibits an average lower deviation. In other terms, it means that for a ternary in which the third term is close to the average of the two first terms, then the introduction of the third component reduces the nonideahty because (5i - 53) + ( 2 - < (5i - 52) -... 

Use of equation 247 for actual calculations requires explicit introduction of composition variables. As in phase-equilibrium calculations, this is normally done for gas phases through the fugacity coefficient and for liquid phases through the activity coefficient. Thus, either... 

It is worthwhile to discuss why the mass-action law on concentration basis (moles/litre) is plausible. It is beyond doubt that it is not always valid. The concentration 5.5 M of saturated aqueous sodium chloride indicates the solubility product 30 moles2/litre2. If an equal amount of such a solution is added to 12 M hydrochloric acid, the concentration of Na+ is 2.75 M and of Cl- (12 + 5.5)/2 = 8.75 M. Their product 24.06 M2 is distinctly below the solubility product, but nevertheless, more than 80 percent of the NaCl present crystallizes out. It would be to short-circuit this paradox to speak about the mass-action law on activity basis. The introduction of activity a as the product a =/Cof the activity coefficient/and the concentration is a tautological trick to keep the mass-action law valid, and it is more fruitful to try to explain why/varies more dramatically in some cases than in others. 

Nevertheless, the size of the transfer contribution ought to be considered when an exact analysis of experimental data is attempted. The problem can be formally expressed by the introduction of a transfer activity coefficient which is in the following designated by the symbol y. 

Grover, J., Chemical mixing in multicomponent solutions An introduction to the use of Margules and other thermodynamic excess functions to represent non-ideal behavior, pp. 67-97 in Thermodynamics in Geology, ed. by D. G. Fraser, D. Reidel, Dordrecht, The Netherlands, 1977. This review article provides a fine introduction to the thermodynamic theory of mixtures underlying the Margules expansion for adsorbate-species activity coefficients. 

According to a theory, based on the regular solution theory, a deviation from ideal behaviour can be described by the introduction of the activity coefficients / and f2-... 

In another attempt (Fawcett and Tikenen, 1996), the introduction of a changing dielectric constant of the solvent (although taken from experimental data) as a function of concentration has been used to estimate activity coefficients of simple 1 1 electrolyte solutions for concentrations up to 2.5 mol dm". ... 

In the highly simplified treatment of the diffusion potential that has just been presented, several drastic assumptions have been made. The one regarding the concentration variation within the transition region can be avoided. One may choose a more realistic concentration versus distance relationship either by thinking about it in more detail or by using experimental knowledge on the matter. Similarly, instead of assuming the activity coefficients to be unity, one can feed in the theoretical or experimental concentration dependence of the activity coefficients. Of course, the introduction of nonideality makes the mathematics awkward in principle,... 

It is not certain that the theoretical arguments, which led to the introduction of the term C t, are completely satisfactory, but it seems to be established that the experimental data require a term of this type. The aggregation of solvent molecules in the vicinity of an ion is the factor responsible for the so-called salting-out effect, namely, the decrease in solubility of neutral substances frequently observed in the presence of salts the constant C is consequently called the salting-out constant. The activity coefficient of a non-electrolyte, as measured by its solubility in the presence of electrolytes, is often given by an expression of the form log / = CV this is the result to which equation (62) would reduce for the activity of a non-electrolyte, i.e., when z+ and z arc zero, in a salt solution of ionic strength... 

Equation (7-4) indicates that the solubility product includes an activity-coefficient term, a term which has been assumed to be unity up to this time. The introduction to this chapter pointed out that errors arising from neglect of the effects of the activity coefficient are usually small when compared with several uncertainties or side reactions. The activity coefficient in Equation (7-4) depends on the kind and concentration of all electrolytes in solution, not merely those involved directly with the precipitate. The correction to solubility calculations that must be made to account for the activity-coefficient effect is known as the diverse ion effect. The appropriate background is discussed in Chapter 2, and Problems 2-1,2-2, and 2-3 are examples of the calculations. For 1 1 electrolytes in solution, activity coefficients can usually be assumed to be unity when concentrations are much less than 0.1 M. Common ion and diverse ion effects can be significant at the same time, for example, when a large excess of common ion is added in a precipitation. The diverse ion effect is one of the reasons that the haphazard addition of a large excess of precipitant should be avoided. 

An alternative fonn of Eq. (11.86) follows introduction of the activity coefficient tinougliEq. (11.88) ... 

With tlie introduction of activity coefficients, this becomes ... 

The experimental method by which S0rensen proposed to measure pH did not, however, actually provide an unequivocal value for the hydrogen ion concentration in solutions of unknown composition. Introduction of the concept of activity (a) and the activity coefficient (y, concentration scales or y, molality scale) led to a modified definition (10) for which a modified symbol pan was first suggested ... 

The introduction of the activity and activity coefficients enables a comparison to he made quite simply between the properties of a given system and those of the ideal reference system. 

If we compare this with (7.52) we see that the standard chemical potential is the same as before, but the chemical potential of mixing is altered the activity ai—Xiyi replaces the mole fraction Xi, This may be generalized to other thermodynamic quantities. The standard properties of a non-ideal system are the same as those of the corresponding ideal reference system. It is only the quantities dependent upon composition that are altered by the introduction of activity coefficients. This is illustrated by table 7.2 which is to be compared with table 7.1. 

Depending on the nature of the class, the instructor may wish to spend more time with the basics, such as the mass balance concept, chemical equilibria, and simple transport scenarios more advanced material, such as transient well dynamics, superposition, temperature dependencies, activity coefficients, redox energetics, and Monod kinetics, can be skipped. Similarly, by omitting Chapter 4, an instructor can use the text for a water-only course. In the case of a more advanced class, the instructor is encouraged to expand on the material suggested additions include more rigorous derivation of the transport equations, discussions of chemical reaction mechanisms, introduction of quantitative models for atmospheric chemical transformations, use of computer software for more complex groundwater transport simulations, and inclusion of case studies and additional exercises. References are provided... 

FIG. 4-6 Activity coefficients at 50 C for six binary liquid systems (a) chloroform(l)/n-heptanef2) (b) acetone l)/methanol(2) (c) acetone(l)/chloroform(2) (d) ethanol(l)/n-heptane(2) (e) ethanol(l)/chloro-form(2) (/) ethanol l)/water(2). [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 445, McGraw-Hill, New York (2005).]... 

It may be noted that if the gases taking part in the reaction were ideal, the activity coefficient factor in equation (32.9) would be unity further, the partial pressure pi of each gas would be equal to NtP, by equation (5.8). In this event, the result would be identical with the expression in (32.13), so that the quantity defined by Kp would represent the true equilibrium constant. When the departure of the reacting gases from ideal behavior is not large, e.g., when the total pressures are of the order of 1 atm. or less, a very satisfactory approximation to the equilibrium constant Kf can be obtained from equation (32.13). At high pressures, however, the values of Kp deviate considerably from constancy, as will be seen below, but a great improvement is possible by the introduction of the activity coefficient factor, as in equation (32.14). 

For dilute electrolyte solutions, Lewis and Randall observed that the mean activity coefficient of a strong electrolyte does not depend on the kind of ion, but only on the concentration and charge numbers of all ions present in solution. So, the individual properties of the ions are not decisive for interionic interactions in dilute electrolyte solutions. These observations paved the way for the introduction of the concept of ionic strength / ... 

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