Calculation of activity coefficient

We consider Pitzer s model for the calculation of activity coefficients in aqueous electrolyte solutions (Pitzer, 1991). It is the most widely used thermodynamic model for electrolyte solutions. 

Eqn. 2.66 is also of interest for the calculation of activity coefficients in other solution media, where as a consequence of considerably different densities (g0) and dielectric constants (a) another value for A is calculated, and also the extended equation log f = - A z+z P/l + I has been applied23. 

Calculation of activity coefficients from mutual solubility data... 

Fredenslund, A., Gmehling, J., Michelsen, M. L., Rasmussen, P. and Prausnitz, J. M. (1977a) Ind. Eng. Chem. Proc. Des. and Dev. 16, 450. Computerized design of multicomponent distillation columns using the UNIFAC group contribution method for calculation of activity coefficients. 

Helgeson, H. C., D. H. Kirkham and G. C. Flowers, 1981, Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high temperatures and pressures, IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600 °C and 5 kB. American Journal of Science 281, 1249-1516. 

Roughly half of the data on the activities of electrolytes in aqueous solutions and most of the data for nonelectrolytes, have been obtained by isopiestic technique. It has two main disadvantages. A great deal of skill and time is needed to obtain reliable data in this way. It is impractical to measure vapor pressures of solutions much below one molal by the isopiestic technique because of the length of time required to reach equilibrium. This is generally sufficient to permit the calculation of activity coefficients of nonelectrolytes, but the calculation for electrolytes requires data at lower concentrations, which must be obtained by other means. 

The methods most generally used for the calculation of activity coefficients at intermediate pressures are the Wilson (1964) and UNIQUAC (Abrams and Prausnitz, 1975) equations. Wilson s equation was used by Sato et al. (1985) to predict the composition of fhe condensate gas stripped from a packed bed fermenter at 30°C, whilst Beck and Bauer (1989) used the UNIQUAC equation, with temperature-dependent parameters given by Kolbe and Gmehling (1985), for their model of an anaerobic gas-solid fluidized bed fermenter at 36°C. In this case it was necessary to go beyond the temperature range of fhe source data down to 16°C in order to predict the composition of the fluidizing gas leaving the condenser. 

Solvent effects on esterification equilibria have also been described using UNIFAC calculations of activity coefficients. This method was claimed to give... 

The Calculation of Activity Coefficients in the Binary System. The experimental quantity A was calculated from the relation ... 

The Calculation of Activity Coefficients in the Ternary System. The polynomial with 10 constants was used to describe the experimental quantity A(m 1,7712) in the whole concentration range... 

From the above equations it is evident that in any calculation of activity coefficients ionic strength is a key parameter and as such it is useful if it can be calculated from experimental data. Ionic strength can be calculated from electrical conductivity measurements using the Babcock equation which is given by Sposito (1989) as... 

Appendix B Calculation of Activity Coefficients for Multicomponent Mixtures Containing Hard Spheres... 

In case of low charged species, and approximately below 3 mol kg 1 the Specific Ion interaction Theory (SIT) [29] can be applied for the calculations of activity coefficients. Data available on interaction coefficients are scarce. But, paradoxically for actinide ions such data are relatively well known. However, in certain cases, they can be estimated from the model developed by L. Ciaviatta [33,34],... 

Analytical representation of the excess Gibbs energy of a system impll knowledge of the standard-state fugacities ft and of the frv. -xt relationshi Since an equation expressing /, as a function of x, cannot recognize a solubili limit, it implies an extrapolation of the /i-vs.-X[ curve from the solubility I to X) = 1, at which point /, = This provides a fictitious or hypothetical va for the fugadty of pure species 1 that serves to establish a Lewis/ Randall 1 for this species, as shown by Fig. 12.21. ft is also the basis for calculation of activity coefficient of species 1 ... 

These equations allow calculation of activity coefficients based on Henry s law from activity coefficients based on the Lewis/Randall rule. In the limit as Xi - 0,... 

This simple equation is adequate to our present purpose, allowing easy calculation of activity coefficients from experimental low-pressure VLE data. For comparison, when a system obeys Raoult s law, y(P = P at, and yf = 1. 

Hydrogeochemical models are dependent on the quality of the chemical analyses, the boundary conditions presumed by the program, theoretical concepts (e.g. calculation of activity coefficients) and the thermodynamic data. Therefore it is vital to check the results critically. For that, a basic knowledge about chemical and thermodynamic processes is required and will be outlined briefly in the following chapters on hydrogeochemical equilibrium (chapter 1.1), kinetics (chapter 1.2), and transport (chapter 1.3). Chapter 2 gives an overview on standard... 

Thus, the Debye-Hiickel ionic-cloud model for ion-ion interactions has permitted a theoretical calculation of activity coefficients resulting in Eq. (3.60). 

Thirdly, another corollary of the first limitation, is the inconsistency and inadequacy of activity coefficient equations. Some models use the extended Delbye-Huckel equation (EDH), others the extended Debye-Huckel with an additional linear term (B-dot, 78, 79) and others the Davies equation (some with the constant 0.2 and some with 0.3, M). The activity coefficients given in Table VIII for seawater show fair agreement because seawater ionic strength is not far from the range of applicability of the equations. However, the accumulation of errors from the consideration of several ions and complexes could lead to serious discrepancies. Another related problem is the calculation of activity coefficients for neutral complexes. Very little reliable information is available on the activity of neutral ion pairs and since these often comprise the dominant species in aqueous systems their activity coefficients can be an important source of uncertainty. 

Calculation of Activity Coefficients for Ionic Species with Different Charge Numbers... 

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