CHAPTER 10 . Equilibrium Analysis, the Ionic Medium Method and Activity Factors
265
Sect. 10-4- As the concentration of the ionic medium varies fairly little in a properly
designed equilibrium analytical experiment, we can assume that the activity coefficients of all reactants and products are constant. We define this constant as unity
in a given ionic medium. This means that comparison of equilibrium constants
determined in different ionic media is not trivial; it requires a method to calculate
activity coefficients of all reactants and products. This will be discussed later.
• Equilibrium analytical methods are described by Rossotti and Rossotti\ and only a
brief summary is given below, using a two-component system as an example:
The experimental data consist of sets of total concentrations of the two components
M, and L, and the concentration of the one or both of the concentrations of free M,
and L, denoted m, and l. There are two mass-balance conditions:
M;= m + [MLl + [ML2l + ... + [MLN] = m(1+ f311 + f3i+ ... + f3NI N )
(10.4)
L; = 1+ [MLl + 2 [ML2] + ... + N[MLNl = 1+ f311 + 2 f3i + ... + Nf3NI N
(10.5)
where f3n in this example denotes the different stability constants for the different
mononuclear complexes formed.
The stoichiometry and the equilibrium constants can be deduced using methods described in Rossotti and Rossotti (1961). Recently, the graphical methods have
often been replaced by least-square techniques, where a chemical model consisting of stoichiometry and equilibrium constants for different assumed complexes is
fitted to the experimental data. The least -square analysis provides a measure of the
agreement between the model and the experimental data and an estimate of the
precision of the deduced equilibrium constants. There are several reasons to be
careful when using these methods: false minima may appear in the least-squares
refinement, resulting in erroneous chemical conclusions; systematic errors in the
experimental data may be described by introducing unrealistic chemical species.
It is difficult to avoid such errors in equilibrium analytical experiments; an example
is small variations in the activity coefficients resulting from the change in composition of the test solutions during an experiment. In order to ascertain the quality
of equilibrium analytical data, it is essential to make an analysis of the amounts of
the different complexes formed throughout the experiment. Complexes that are
present in small amounts should be looked upon with suspicion, especially if they
are formed in concentration ranges where large changes in the composition of the
test solutions have been made. The least squares assigned uncertainty of the reported equilibrium constants are often underestimated. They depend on the weights
of the individual measurements and the number of data points that are used to
determine a particular equilibrium constant, and these factors are not always
analysed. In addition, the uncertainty estimate represents an estimate of the precision of the experiment, not its accuracy. A measure of the latter can only be obtained
by studying the same equilibrium system with different experimental methods.
Recent advances in NMR-spectroscopy and laser-based spectroscopic methods and
structure information based on synchrotron light sources (EXAFS) makes it possible to attain much more precise insight into equilibrium systems than offered by
the "classical" methods; some examples are given by Szabo et al. 1997, Farkas et al.
2000, and Moll et al. 1999.
265
Sect. 10-4- As the concentration of the ionic medium varies fairly little in a properly
designed equilibrium analytical experiment, we can assume that the activity coefficients of all reactants and products are constant. We define this constant as unity
in a given ionic medium. This means that comparison of equilibrium constants
determined in different ionic media is not trivial; it requires a method to calculate
activity coefficients of all reactants and products. This will be discussed later.
• Equilibrium analytical methods are described by Rossotti and Rossotti\ and only a
brief summary is given below, using a two-component system as an example:
The experimental data consist of sets of total concentrations of the two components
M, and L, and the concentration of the one or both of the concentrations of free M,
and L, denoted m, and l. There are two mass-balance conditions:
M;= m + [MLl + [ML2l + ... + [MLN] = m(1+ f311 + f3i+ ... + f3NI N )
(10.4)
L; = 1+ [MLl + 2 [ML2] + ... + N[MLNl = 1+ f311 + 2 f3i + ... + Nf3NI N
(10.5)
where f3n in this example denotes the different stability constants for the different
mononuclear complexes formed.
The stoichiometry and the equilibrium constants can be deduced using methods described in Rossotti and Rossotti (1961). Recently, the graphical methods have
often been replaced by least-square techniques, where a chemical model consisting of stoichiometry and equilibrium constants for different assumed complexes is
fitted to the experimental data. The least -square analysis provides a measure of the
agreement between the model and the experimental data and an estimate of the
precision of the deduced equilibrium constants. There are several reasons to be
careful when using these methods: false minima may appear in the least-squares
refinement, resulting in erroneous chemical conclusions; systematic errors in the
experimental data may be described by introducing unrealistic chemical species.
It is difficult to avoid such errors in equilibrium analytical experiments; an example
is small variations in the activity coefficients resulting from the change in composition of the test solutions during an experiment. In order to ascertain the quality
of equilibrium analytical data, it is essential to make an analysis of the amounts of
the different complexes formed throughout the experiment. Complexes that are
present in small amounts should be looked upon with suspicion, especially if they
are formed in concentration ranges where large changes in the composition of the
test solutions have been made. The least squares assigned uncertainty of the reported equilibrium constants are often underestimated. They depend on the weights
of the individual measurements and the number of data points that are used to
determine a particular equilibrium constant, and these factors are not always
analysed. In addition, the uncertainty estimate represents an estimate of the precision of the experiment, not its accuracy. A measure of the latter can only be obtained
by studying the same equilibrium system with different experimental methods.
Recent advances in NMR-spectroscopy and laser-based spectroscopic methods and
structure information based on synchrotron light sources (EXAFS) makes it possible to attain much more precise insight into equilibrium systems than offered by
the "classical" methods; some examples are given by Szabo et al. 1997, Farkas et al.
2000, and Moll et al. 1999.