# Method of Continuous Variations

The method of continuous variations, also called Job’s method, is used to determine the stoichiometry of a metal-ligand complex. In this method we prepare a series of solutions such that the total moles of metal and ligand, ntotal, in each solution is the same. If (nM)i and (nL)i are, respectively, the moles of metal and ligand in solution i, then

ntotal = (nM)i + (nL)i

The relative amount of ligand and metal in each solution is expressed as the mole fraction of ligand, (XL)i, and the mole fraction of metal, (XM)i,

(XL)i = (nL)i/ntotal

(XM)i = 1 – (nL)i/ntotal = (nM)i/ntotal

The concentration of the metal–ligand complex in any solution is determined by the limiting reagent, with the greatest concentration occurring when the metal and the ligand are mixed stoichiometrically. If we monitor the complexation reaction at a wavelength where the metal–ligand complex absorbs only, a graph of absorbance versus the mole fraction of ligand will have two linear branches—one when the ligand is the limiting reagent and a second when the metal is the limiting reagent. The intersection of these two branches represents a stoichiometric mixing of the metal and the ligand. We can use the mole fraction of ligand at the intersection to determine the value of y for the metal–ligand complex MLy.

y = (nL/nM) = (XL/XM) = (XL/1 –XM)

The illustration below shows a continuous variations plot for the metal–ligand complex between Fe2+ and o-phenanthroline. As shown here, the metal and ligand form the 1:3 complex Fe(o-phenanthroline)32+. 