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APPENDIX II
Determination of the equilibrium constant for formation of a 1:1 complex
The reaction between a metal, M, and a ligand, L, to form the ML complex can be expressed by the relationship
Kapp
Mf + Lf ⇄ MLT (II.1)
where Mf and Lf are the total concentrations of uncomplexed metal and ligand respectively, whereas MLT is the total complex concentration whose equilibrium constant is
€
Kapp= [MLT]
[Mf][Lf] (II.2)
The mass conservation equations are
CL =[Lf ]+[MLT ] (II.3)
CM =[Mf]+[MLT] (II.4)
where CL and CM the overall concentration of the ligand and metal respectively. If the Lambert & Beer law applies for a wavelength where the ligand and the complex only absorb, with a 1 cm path length cell the overall absorbance is given by the equation
€
A = εL
f[Lf] + εMLT[MLT] (II.5)
If also the metal absorbs at the used wavelength, equal amounts of metal solutions are added both to sample and reference cell at each stage of the spectrophotometric titration, in order to remove its contribution to the overall absorbance. Introduction of equation (II.3) in (II.5) yields € A = εL f(CL − [MLT]) + εMLT[MLT] (II.6) If we now define € ΔA = A −εLfCL (II.7)
Appendix II
104
€
Δε=εML
T −εLf (II.8)
equation (II.6) becomes
∆A = ∆ε[MLT ] (II.9)
Introduction of equations (II.3) and (II.4) into (II.2) yields
€
Kapp = [MLT]
(CM −[MLT])(CL−[MLT])
(II.10)
Substituting [MLT] obtained from equation (II.9) into equation (II.10) and rearranging, one obtains € CMCL ΔAbs + ΔA Δε2 = 1 KappΔε +CM + CL Δε (II.11)
which corresponds to equation (3.8) of chapter 3 and (5.22) of chapter 5. Such equation enables Kapp and ∆ε to be obtained by an iterative procedure. That is, disregarding the ∆A/∆ε2
term on first approximation, ∆ε can be calculated from the reciprocal of the slope of the straight line interpolating the data of the CMCL/∆A vs. (CM+CL). Then, introduction of this ∆ε value into equation (II.11) enables the (CMCL/∆A + ∆A/∆ε
2) term to be evaluated and new values of Kapp and ∆ε to be obtained. The procedure is repeated until convergence is reached.
