Appendix IV
157 APPENDIX IV
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
Mf + Lf MLT
Kapp
(IV.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 ] [ ] [ ] [ f f T app L M ML K × = (IV.2)
The mass conservation equations are ] [ ] [ f T L L ML C = + (IV.3) ] [ ] [ f T M M ML C = + (IV.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 ] [ ] [ f ML T Lf L ML Abs T ε ε + = (IV.5)
Appendix IV 158 ] [ ]) [ ( L T MLT T Lf C ML ML Abs=ε − +ε (IV.6) If we now define L LfC Abs Abs= −ε ∆ (IV.7) Lf MLT ε ε ε = − ∆ (IV.8)
equation (IV.6) becomes ]
[MLT
Abs=∆ε
∆ (IV.9)
Introduction of equations (IV.3) and (IV.4) into (IV.2) yields
]) [ ( ]) [ ( ] [ T L T M T app ML C ML C ML K − × − = (IV.10)
Substituting [MLT] obtained from equation (IV.9) into equation (IV.10) and
rearranging one obtains
ε ε ε ∆ + + ∆ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ ∆ + ∆ ) ( 1 2 L M app L M C C K Abs Abs C C (IV.11)
which corresponds to equation (5.2) of chapter 5.
Such equation enables Kapp and ∆ε to be obtained by an iterative procedure. That
is, disregarding the ∆Abs/∆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/∆Abs vs. (CM+CL). Then, introduction of this ∆ε value into equation (IV.11)
enables the (CMCL/∆Abs + ∆Abs/∆ε2) term to be evaluated and new values of Kapp
