Molecular absorption, particularly in the UV/Vis range, has been used for a variety of different characterization studies, including determining the stoichiometry of metal–ligand complexes and determining equilibrium constants. Both of these ex- amples are examined in this section.
The stoichiometry for a metal–ligand complexation reaction of the following general form
M+ yL < == > MLy
can be determined by one of three methods: the method of continuous variations, the mole-ratio method, and the slope-ratio method.
Of the three methods, the method of continuous variations, also called Job’s method, is the most popular. In this method a series of solutions is prepared such that the total moles of metal and ligand, ntot, in each solution is the same. Thus, if (nM)i and (nL)i are, respectively, the moles of metal and ligand in the i-th solution, then
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,
The concentration of the metal–ligand complex is determined by the limiting reagent, with the greatest concentration occurring when the metal and ligand are mixed stoichiometrically. If the reaction is monitored at a wavelength where only the metal–ligand complex absorbs, a plot of absorbance versus the mole fraction of ligand will show 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 occurs when a stoichiometric mixing of metal and ligand is reached. The mole fraction of ligand at this intersection is used to determine the value of y for the metal–ligand complex, MLy.
If there is no wavelength where only the metal–ligand complex absorbs, then the measured absorbances must be corrected for the absorbance that would be exhib- ited if the metal and ligand did not react to form MLy.
In essence, the corrected absorbance gives the change in absorbance due to the for- mation of the metal–ligand complex. An example of the application of the method of continuous variations is shown in Example 10.7.
Several precautions are necessary when using the method of continuous varia- tions. First, the method of continuous variations requires that a single metal–ligand complex be formed. To determine if this condition is true, plots of absorbance ver- sus XL should be constructed for several different wavelengths and for several differ- ent values of ntot. If the maximum absorbance does not occur at the same value of XL for each set of conditions, then more than one metal–ligand complex must be present. A second precaution is that the metal–ligand complex must obey Beer’s law for the range of concentrations used in constructing the plot of absorbance versus XL. Third, if the metal–ligand complex’s formation constant is relatively small, the plot of absorbance versus XL may show significant curvature. In this case it is often difficult to determine the stoichiometry by extrapolation. Finally, since the stability of the metal–ligand complex may be influenced by solution conditions, the compo- sition of the solutions must be carefully controlled. When the ligand is a weak base, for example, the solutions must be buffered to the same pH.
In the mole-ratio method the moles of one reactant, usually the metal, are held constant, while the moles of the other reactant are varied. The absorbance is moni- tored at a wavelength at which the metal–ligand complex absorbs. A plot of ab- sorbance as a function of the ligand-to-metal mole ratio (nL/nM) has two linear branches that intersect at a mole ratio corresponding to the formula of the complex. Figure 10.34a shows a mole-ratio plot for the formation of a 1:1 complex in which the absorbance is monitored at a wavelength at which only the complex absorbs. Figure 10.34b shows a mole-ratio plot for a 1:2 complex in which the metal, the ligand, and the complex absorb at the selected wavelength. Unlike the method of continuous variations, the mole-ratio method can be used for complexation reac- tions that occur in a stepwise fashion, provided that the molar absorptivities of the metal–ligand complexes differ and the formation constants are sufficiently differ- ent. A typical mole-ratio plot for the stepwise formation of ML and ML2 is shown in Figure 10.34c.
Both the method of continuous variations and the mole-ratio method rely on an extrapolation of absorbance data collected under conditions in which a linear re- lationship exists between absorbance and the relative amounts of metal and ligand. When a metal–ligand complex is very weak, a plot of absorbance versus XL or nL/nM may be curved, making it impossible to determine the stoichiometry by extrapola- tion. In this case the slope ratio may be used.
In the slope-ratio method two sets of solutions are prepared. The first set con- sists of a constant amount of metal and a variable amount of ligand, chosen such that the total concentration of metal, CM, is much greater than the total concentra- tion of ligand, CL. Under these conditions we may assume that essentially all the ligand is complexed. The concentration of a metal–ligand complex of the general form MxLy is
If absorbance is monitored at a wavelength where only MxLy absorbs, then
and a plot of absorbance versus CL will be linear with a slope, sL, of
A second set of solutions is prepared with a fixed concentration of ligand that is much greater than the variable concentration of metal; thus
The mole ratio of ligand-to-metal is determined from the ratio of the two slopes.
An important assumption in the slope-ratio method is that the complexation reac- tion continues to completion in the presence of a sufficiently large excess of metal or ligand. The slope-ratio method also is limited to systems that obey Beer’s law and in which only a single complex is formed.
Another important application of molec- ular absorption is the determination of equilibrium constants. Let’s consider, as a simple example, an acid–base reaction of the general form
HIn+ H2O < = = > H3O+ + In–
where HIn and In– are the conjugate weak acid and weak base forms of a visual acid–base indicator. The equilibrium constant for this reaction is
To determine the equilibrium constant’s value, we prepare a solution in which the reaction exists in a state of equilibrium and determine the equilibrium concentration of H3O+, HIn, and In–. The concentration of H3O+ is easily determined by measuring the solution’s pH, whereas the concentration of HIn and In– may be determined by measuring the solution’s absorbance.
If both HIn and In– absorb at the selected wavelength, then, from equation 10.6, we know that
A = εHInb[HIn] + εInb[In–] ……….10.20
where εHIn and εIn are the molar absorptivities for HIn and In–. The total concentra- tion of indicator, C, is given by a mass balance equation
C = [HIn] + [In–] ……….10.21
Solving equation 10.21 for [HIn] and substituting into equation 10.20 gives
A = εHInb(C – [In–]) + εInb[In–]
which simplifies to
A = εHInbC – εHInb[In–]+ εInb[In–]
A = AHIn + b[In–](εIn – εHIn) ……….10.22
where AHIn, which is equal to εHInbC, is the absorbance when the pH is acidic enough that essentially all the indicator is present as HIn. Solving equation 10.22 for the concentration of In– gives
Proceeding in the same fashion, we can derive a similar equation for the concentra- tion of HIn; thus
where AIn, which is equal to εInbC, is the absorbance when the pH is basic enough that only In– contributes to the absorbance. Substituting equations 10.23 and 10.24 into the equilibrium constant expression for HIn gives
Using equation 10.25, the value of Ka can be determined in one of two ways. The simplest approach is to prepare three solutions, each of which contains the same amount, C, of indicator. The pH of one solution is made acidic enough that [HIn] >> [In–]. The absorbance of this solution gives AHIn. The value of AIn is determined by adjusting the pH of the second solution such that [In–] >> [HIn]. Finally, the pH of the third solution is adjusted to an intermediate value, and the pH and ab- sorbance, A, are recorded. The value of Ka can then be calculated by making appro- priate substitutions into equation 10.25.
A second approach is to prepare a series of solutions, each of which contains the same amount of indicator. Two solutions are used to determine values for AHIn and AIn. Rewriting equation 10.25 in logarithmic form and rearranging
shows that a plot of log [(A – AHIn)/(AIn – A)] versus pH is linear, with a slope of +1 and a y-intercept of –pKa.
In developing this treatment for determining equilibrium constants, we have considered a relatively simple system in which the absorbance of HIn and In– were easily measured, and for which it is easy to determine the concentration of H3O+. In addition to acid–base reactions, the same approach can be applied to any reaction of the general form
including metal–ligand complexation and redox reactions, provided that the con- centration of the product, Z, and one of the reactants can be determined spec- trophotometrically and the concentration of the other reactant can be determined by another method. With appropriate modifications, more-complicated systems, in which one or more of these parameters cannot be measured, also can be treated.
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