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The retention model for anionic chromatography can be split into two distinct models, one for describing eluents with a single anion, and the other for describing eluents with complexing agents present. Given an eluent anion or an analyte anion, two phases are observed, the stationary phase (denoted by S) and the mobile phase (denoted by M). As such, there is equilibrium between the two phases for both the eluent anions and the analyte anions that can be described by Equation \ref{1}.
\[ y*[A^{x-}_{M}]\ +\ x*[E^{y-}_{S}]\ \Leftrightarrow \ y*[A^{x-}_{S}]\ +\ x*[E^{y-}_{M}] \label{1} \]
This yields an equilibrium constant as given in Equation \ref{2} .
\[ K_{A,E} = \frac{ [A^{x-}_{S}]^{y} [E^{y-}_{M}]^{x} \gamma ^{y} _{A^{x-}_{S} } \gamma ^{x} _{E^{y-}_{S}} }{ [A^{x-}_{M}] ^{y} [E^{y-}_{S}]^{x} \gamma ^{y} _{A^{x-}_{M}} \gamma ^{x} _{E^{y-}_{S}}} \label{2} \]
Given the activity of the two ions cannot be found in the stationary or mobile phases, the activity coefficients are set to 1. Two new quantities are then introduced. The first is the distribution coefficient, DA, which is the ratio of analyte concentrations in the stationary phase to the mobile phase, Equation \ref{3} . The second is the retention factor, k1A, which is the distribution coefficient times the ratio of volume between the two phases, Equation \ref{4} .
\[ D_{A} \ =\ \frac{[A_{S}]}{[A_{M}]} \label{3} \]
\[k_{A}^{1} \ = \ D_{A} * \frac{V_{S}}{V_{M}} \label{4} \]
Substituting the two quantities from Equation \ref{3} and Equation \ref{4} into Equation \ref{2} , the equilibrium constant can be written as Equation \ref{5}
\[ K_{A,E} \ = (k_{A}^{1} \frac{V_{M}}{V_{S}})^{y} * (\frac{[E_{M}^{y-} ]}{[E^{y-}_{S}]})^{x} \label{5} \]
Given there is usually a large difference in concentrations between the eluent and the analyte (with magnitudes of 10 greater eluent), equation 4 can be re-written under the assumption that all the solid phase packing material&#;s functional groups are taken up by Ey-. As such, the stationary Ey- can be substituted with the exchange capacity divided by the charge of Ey-. This yields Equation \ref{6}
\[ K_{A,E} \ = (k_{A}^{1} \frac{V_{M}}{V_{S}})^{y} * (\frac{Q}{\gamma })^{-x} [E_{M}^{y-}] \label{6} \]
Solving for the retention factor Equation \ref{7} is developed.
\[ z*[A^{x-}_{M}] \ +\ x*[B^{z-}_{S}] \Leftrightarrow z* [A^{x-}_{S}] \ +\ x*[B^{z-}_{M}] \label{7} \]
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Equation \ref{8} shows the relationship between retention factor and parameters like eluent concentration and the exchange capacity, which allows parameters of the ion chromatography to be manipulated and the retention factors to be determined. Equation \ref{9} only works for a single analyte present, but a relationship for the selectivity between two analytes [A] and [B] can easily be determined.
First the equilibrium between the two analytes is determined as Equation \ref{8}
\[ K_{A,B} \ = \frac{[A^{x-}_{S}]^{z} [B^{z-}_{M}]^{x}}{[A^{x-}_{M}]^{z} [B^{z-}_{S}]^{x}} \label{8} \]
The equilibrium constant can be written as Equation \ref{9} (ignoring activity):
\[ \alpha _{A,B} \ = \frac{[A^{x-}_{S}][B^{z-}_{M}]}{[A^{x-}_{M}][B^{z-}_{S}]} \label{9} \]
The selectivity can then be determined to be Equation \ref{10}
\[ \alpha _{A,B} \ = \frac{[A^{x-}_{S}][B^{z-}_{M}]}{[A^{x-}_{M}][B^{z-}_{S}]} \label{10} \]
Equation \ref{10} can then be simplified into a logarithmic form as the following two equations:
\[ \log \alpha _{A,B} = \frac{1}{z} log K_{A,B} \ + \frac{x-z}{z} log \frac{ k_{A}^{1} V_{M}}{V_{S}} \label{11} \]
\[ \log \alpha _{A,B} = \frac{1}{x} log K_{A,B} \ + \frac{x-z}{z} log \frac{ k_{A}^{1} V_{M}}{V_{S}} \label{12} \]
When the two charges are the same, it can be seen that the selectivity is only a factor of the selectivity coefficients and the charges. When the two charges are different, it can be seen that the two retention factors are dependent upon each other.
In situations with a polyatomic eluent, three models are used to account for the multiple anions in the eluent. The first is the dominant equilibrium model, in which one anion is so dominant in concentration; the other eluent anions are ignored. The dominant equilibrium model works best for multivalence analytes. The second is the effective charge model, where an effective charge of the eluent anions is found, and a relationship similar to EQ is found with the effective charge. The effective charge models works best with monovalent analytes. The third is the multiple eluent species model, where Equation \ref{13} describes the retention factor:
\[ \log K_{A}^{1} \ =\ C_{3} - (\frac{X_{1}}{a} + \frac{X_{2}}{b} + \frac{X_{3}}{c}) -\ log C_{P} \label{13} \]
For more information, please visit IC Ion Chromatography.