Chapter 12, Problem 12.21P

Interpretation Introduction

Interpretation:

The derivation of spreadsheet equation using fractions of free metal ion and metal complex and should be explained.

Concept introduction:

Spreadsheet equation for auxiliary complexing agent:

In the EDTA titration, the titration curve is drawn by a single equation, it is known as spreadsheet equation for auxiliary complexing agent and it applies whole titration because the reaction is carried out by at constant pH.

The spreadsheet equation for auxiliary complexing agent is,

$\varphi \text{=}\frac{{\text{C}}_{\text{EDTA}}{\text{V}}_{\text{EDTA}}}{{\text{C}}_{\text{M}}{\text{V}}_{\text{M}}}$

Where,

${\text{C}}_{\text{M}}$ is total concentration of metal ion

${\text{C}}_{\text{EDTA}}$ is concentration of metal EDTA

Fraction of free metal ion:

In the metal ligand equilibria the number of moles of metal and ligand are equilibrium with number of mole of complex formed.

$\begin{array}{l}\text{M+}\text{L}\underset{}{\rightleftharpoons }\text{ML}{\text{β}}_1\text{=}\frac{\text{[ML]}}{\text{[M][L]}}\\ \text{M+}2\text{L}\underset{}{\rightleftharpoons }{\text{ML}}_2{\text{β}}_2\text{=}\frac{{\text{[ML}}_2\text{]}}{{\text{[M][L]}}^2}\\ \text{M+}\text{nL}\underset{}{\rightleftharpoons }{\text{ML}}_{\text{n}}{\text{β}}_{\text{n}}\text{=}\frac{{\text{[ML}}_{\text{n}}\text{]}}{{\text{[M][L]}}^{\text{n}}}\end{array}$

Where,

${\text{β}}_{\text{n}}$ is cumulative formation constant

Fraction of free metal ion ${\text{α}}_{\text{M}}$ is,

${\text{α}}_{\text{M}}\text{=}\frac{\text{[M]}}{{\text{[M}}_{\text{tot}}\text{]}}$

$\begin{array}{l}{\text{α}}_{\text{M}}\text{=}\frac{\text{[M]}}{{\text{[M]{1+β1[L]+β}}_{\text{2}}{\text{[L]}}^{\text{2}}\text{}}}\\ =\frac{1}{{\text{1+β1[L]+β}}_{\text{2}}{\text{[L]}}^{\text{2}}}\end{array}$

Where,

$\text{M}\text{is free metal ion}$