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181 It follows from the additivity of affinities, AG2_() = AG2_i AG^Q, that (z+,2 - z+>1)E _i = z + , 2 ^_ 0 - z+>1 _o (3.2.24)
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This relationship (sometimes called Luther's law) for the transfer of several electrons permits us to calculate one redox potential if the others are known. Obviously, this is an analogy of the Hess law in thermodynamics. Equation (3.2.24) is not restricted to the case where the lowest oxidation state is a metal. Consider the reactions A + e<^B and B + e<= C (3.2.25)
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The sum of these reactions yields a reaction termed disproportionation: aB (3.2.26)
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where K is the disproportionation constant. An inert electrode immersed in a solution containing species A, B and C will attain a potential corresponding to both of equilibria (3.2.25): E = A,B + ^ l n ^ = Elc + ^ l n ^ (3.2.27) F aB F ac If the system initially contains only form C with concentration c0, to which the oxidant is added, then the overall degree of oxidation is (3 = (2[A] + [B])/c0. If activities are set equal to concentrations, then the potential E can be expressed in terms of j8:
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/3l [(j3l) + 4j3(2/W -j3 + 1 [03 - I) 2 + 4)3(2 - j3)*:]1/2
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This dependence is shown in Fig. 3.10. The usual dependence for n = 2 is obtained for complete disproportionation, that is K 1. With decreasing K, the E~P curve changes and at K = \ assumes the usual shape for n = 1. For small K values three inflection points appear on the curve. This situation often occurs for organic quinones. Form B is then termed the semiquinone. A typical inorganic redox system of this type involves the equilibrium between metallic copper and copper(II) ions. In the absence of a complexformer the disproportionation constant is large and a solution of Cu+ ions is very unstable. However, in the presence of ammonia, copper(I) ions are bonded in a complex, constant K decreases and the E-(3 curve exhibits three inflection points.
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-8 0.8 1.2 Overall oxidation degree,/3 1.6 2.0
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Fig. 3.10 Dependence of the potential function v = (2F/2.3RT) [E K A,B + B,C)] on the overall oxidation degree J3 for a system with disproportionation. The semiquinone formation constant K' = l/K is indicated at each curve
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3.2.5 Organic redox electrodes Quinhydrone, a solid-state associate of quinone and hydroquinone, decomposes in solution to its components. The quinhydrone electrode is an example of more complex organic redox electrodes whose potential is affected by the pH of the solution. If the quinone molecule is denoted as Ox and the hydroquinone molecule as H2Red, then the actual half-cell reaction Ox + 2e * Red2" (3.2.29) is accompanied by reactions between the hydroquinone anion Red2" and the solvent. Hydroquinone is a weak dibasic acid, with the dissociation constants
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183 For the quinhydrone electrode, the equation for the electrode potential has the form
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(3.2.31) where EOr is the formal potential. The concentration of the anion of the reduced form is substituted from the equation for the dissociation constants and from the equation for the overall concentration of hydroquinone, c Red , , = [H2Red] + [HRecT] + [Red2"]
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so that [Red2"] = cRedA:;A:2/([H+]2 + K[[H+] + K[K'2) and
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+ K[[H] + K[K
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As [Ox] = c Red for quinhydrone, the second term on the right-hand side of this equation equals zero and the first and third are combined in the constant E^h. The fourth term is simplified in different pH ranges as follows: (a) In an acid medium, [ H + ] 2 K\K'2 + K[[H% so that
(b) If K[[H+] [H+]2 + K[K'2, then
qh = * q h + ~\n K\ - ^ ^ P H (3.2.35) It Lt (c) In a rather alkaline region K[K'2 K[[H+] + [H + ] 2 , dissociation is complete and the potential is independent of pH: