Mixtures of strong electrolytes in .NET

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1.3.5 Mixtures of strong electrolytes
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The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter a> it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central ion.
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If the validity of Eq. (1.3.31) is assumed for the mean activity coefficient of a given electrolyte even in a mixture of electrolytes, and quantity a is calculated for the same measured electrolyte in various mixtures, then different values are, in fact, obtained which differ for a single total solution molality depending on the relative representation and individual properties of the ionic components. A number of authors have suggested various mixing rules, according to which the quantity a could be calculated for a measured electrolyte in a mixture, starting from the known individual parameters of the single electrolytes and the known composition of the solution. However, none of the proposed mixing relationships has found broad application. Thus, the question about the dependence of the mean activity coefficients of the individual electrolytes on the relative contents of the various electrolytic components was solved in a different way. The approach introduced by E. A. Guggenheim and employed by H. S. Harned, G. Akerlof, and other authors, especially for a mixture of two electrolytes, is based on the Br0nsted assumption of specific ion interactions: in a dilute solution of two electrolytes with constant overall concentration, the interaction between ions with charges of the same sign is non-specific for the type of ion, while interaction between ions with opposite charges is specific. Guggenheim used this assumption to employ Eq. (1.3.38) for the activity coefficient of the electrolyte, where the product aB was set equal to unity and the specific interaction between oppositely charged ions was accounted for in the term CI. Consider a mixture of two uni-univalent electrolytes AlBl and AnBn with overall molality m and individual representations yY = mjm and yu = mu/m, where mx and mu are molalities of individual electrolytes. According to Guggenheim, In y, = -A* + [2y,6 u + (ftIU + 6 UI )(1 In Yn = -A* + [2(1 - y ) 6 + (6 +
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where the fo's are specific interaction constants (blu is the parameter of the interaction of Ax with Bu, bu is the parameter of the interaction of Ax with Bl9 etc.) and>4* = lnlO>lV7/(l + V7). Two limiting activity coefficients can be obtained from Eq. (1.3.44) for each electrolyte: lim In y, = In y = -A* 4- (bu x + bx n)m o lim In y, = In y\ = -A* + 2bllm lim In yu = In y n = -A* + (bu { 4- bY n)m lim In Yn =
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Yu = ~^* + 2 ^n n m
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Thus, lny = lny n . Coefficients or, and au are introduced, describing the combination of the interaction constants: m ! o (1-3.46) In 10 = 2bUM - V , - 6,.n = '" y " " '" y " m Combination of Eqs (1.3.44) to (1.3.46), introduction of molalities mY and mn and conversion to decadic logarithms yield the equations
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log yY = log y + alml = log y\ - <xxmYl ^
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log Yu = log y ! + aumn = log y}T - auml These relationships are termed the Harned rules and have been verified experimentally up to high overall molality values (e.g. for a mixture of HC1 and KC1 up to 2mol-kg~ 1 ). If this linear relationship between the logarithm of the activity coefficient of one electrolyte and the molality of the second electrolyte in a mixture with constant overall molality is not fulfilled, then a further term is added, including the square of the appropriate molality:
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