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Molecular absorption, particularly in the UV/Vis range, has been used for a variety of different characterization studies, including determining the stoichiometry of metalâ€“ligand complexes and determining equilibrium constants.

**Characterization Applications**

Molecular absorption, particularly in the UV/Vis
range, has been used for a variety of different characterization studies, including determining the
stoichiometry of metalâ€“ligand complexes and determining equilibrium constants. Both of these
ex- amples are examined
in this section.

The stoichiometry for a metalâ€“ligand complexation reaction of the following general form

M+ *y*L < == > ML_{y}

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 Jobâ€™s
method, is the most popular.
In this method
a series of solutions is prepared such that the total moles
of metal and
ligand, *n*_{tot}, in each
solution is the
same. Thus, if (*n*_{M})* _{i} *and (

The relative amount
of ligand and metal in each solution
is expressed as the mole fraction of ligand, (*X*_{L})* _{i}*, and the mole
fraction of metal,
(

The
concentration of the metalâ€“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 metalâ€“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 metalâ€“ligand complex, ML_{y.}

If there is no wavelength where only the metalâ€“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
ML* _{y}*.

In essence, the corrected absorbance gives the change
in absorbance due to the for-
mation of the metalâ€“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 metalâ€“ligand complex be formed. To determine if this condition is true, plots
of absorbance ver- sus
*X*_{L} should
be constructed for
several different wavelengths and for several
differ- ent values of *n*_{tot}. If the maximum
absorbance does not
occur at the
same value of *X*_{L} for each set of conditions, then more than one metalâ€“ligand complex must be present. A second precaution is that the metalâ€“ligand complex
must obey Beerâ€™s
law for the range of concentrations used in constructing the plot of absorbance versus *X*_{L}. Third,
if the metalâ€“ligand complexâ€™s formation constant is relatively small, the plot of absorbance versus
*X*_{L} may show significant curvature. In this case it is often difficult to determine the stoichiometry by extrapolation. Finally,
since the stability of the metalâ€“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 metalâ€“ligand complex
absorbs. A plot of ab- sorbance as a function of the ligand-to-metal mole ratio (*n*_{L}/*n*_{M}) 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 metalâ€“ligand complexes differ and the
formation constants are sufficiently differ- ent. A typical mole-ratio plot for the
stepwise formation of ML and
ML_{2} 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 metalâ€“ligand complex
is very weak,
a plot of absorbance versus
*X*_{L} or *n*_{L}/*n*_{M} 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, *C*_{M}, is much greater
than the total
concentra- tion of ligand,
*C*_{L}. Under
these conditions we may assume
that essentially all the
ligand is complexed. The concentration of a metalâ€“ligand complex of the
general form M* _{x}*L

If absorbance is monitored at a wavelength where only M* _{x}*L

and a plot of absorbance versus *C*_{L} will be linear with a slope, *s*_{L}, 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 Beerâ€™s law
and in which only a single
complex is formed.

Another important application of molec- ular absorption is the determination of equilibrium constants. Letâ€™s consider, as a
simple example, an acidâ€“base reaction of the general
form

HIn+
H_{2}O < = = > H_{3}O+ + Inâ€“

where HIn and Inâ€“ are the conjugate weak acid and weak base forms of a visual acidâ€“base indicator. The equilibrium constant for this
reaction is

To determine the equilibrium constantâ€™s value, we prepare
a solution in which the reaction exists in a state of equilibrium and determine the equilibrium concentration of H_{3}O+, HIn, and Inâ€“. The concentration of H_{3}O+ is easily determined by measuring
the solutionâ€™s pH, whereas the concentration of HIn and Inâ€“ may be determined by measuring the solutionâ€™s absorbance.

If both HIn and Inâ€“ absorb at the selected
wavelength, then, from equation
10.6, we know that

*A *= Îµ_{HIn}*b*[HIn] + Îµ_{In}*b*[Inâ€“]** â€¦â€¦â€¦.10.20**

where Îµ_{HIn} and Îµ_{In} are the molar
absorptivities for HIn and Inâ€“. The total
concentra- tion of indicator, *C, *is given by a mass balance equation

*C *= [HIn] + [Inâ€“]** â€¦â€¦â€¦.10.21**

Solving equation 10.21 for [HIn] and substituting into equation
10.20 gives

*A *= Îµ_{HIn}*b*(*C *â€“ [Inâ€“]) + Îµ_{In}*b*[Inâ€“]

which simplifies to

*A *= Îµ_{HIn}*bC *â€“ Îµ_{HIn}*b*[Inâ€“]+
Îµ_{In}*b*[Inâ€“]

*A *=
*A*_{HIn} + *b*[Inâ€“](Îµ_{In} â€“ Îµ_{HIn})**
â€¦â€¦â€¦.10.22**

where
*A*_{HIn}, which is equal to Îµ_{HIn}*bC,** *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 Inâ€“ gives

10.23

Proceeding in the same fashion,
we can derive a similar
equation for the concentra-
tion of HIn; thus

10.24

where *A*_{In}, which is equal
to Îµ_{In}*bC, *is the absorbance when the pH is basic
enough that only Inâ€“ contributes to the absorbance. Substituting equations 10.23
and 10.24 into the equilibrium constant
expression for HIn gives

10.25

Using equation 10.25,
the value of *K*_{a} 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] >> [Inâ€“].
The absorbance of this solution
gives *A*_{HIn}. The value
of *A*_{In} is determined by adjusting the pH of the second solution
such that [Inâ€“] >> [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 *K*_{a} 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 *A*_{HIn} and *A*_{In}. Rewriting equation 10.25
in logarithmic form
and rearranging

shows that a plot of log [(*A *â€“
*A*_{HIn})/(*A*_{In} â€“ *A*)]
versus pH is linear, with
a slope of +1
and a *y*-intercept of â€“p*K*_{a}.

In developing this treatment for determining equilibrium
constants, we have considered a relatively simple system in which the absorbance of HIn and Inâ€“ were easily measured, and for which it is easy to determine
the concentration of H_{3}O+. In addition to acidâ€“base reactions, the same approach
can be applied to any reaction
of the general form

including metalâ€“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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Modern Analytical Chemistry: Spectroscopic Methods of Analysis : Characterization Applications - Ultraviolet-Visible and Infrared Spectrophotometry |

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