The '''Benesi-Hildebrand method''' is a mathematical approach used in the determination of the equilibrium constant K and stoichiometry of nonbonding interactions This method has been shown to be useful for observing 1:1 complexes but typically generates inappropriate stoichiometric parameters for 1:2 complexes

## Derivation

To observe one-to-one binding between a single host (H) and guest (G) using UV/Vis absorbance the Benesi-Hildebrand method can be employed The basis behind this method is that the acquired absorbance should be a mixture of the host guest and the host-guest complex

$A=A^\{HG\}+A^G+A^H$With the assumption that the inital

concentration of the guest (G

_{0}) is much larger than the initial

concentration of the host (H

_{0}) then the absorbance from H

_{0} should be negligible

$A=A^\{HG\}+A^G$The absorbance can be collected before and following the formation of the HG complex This change in absorbance (ΔA) is what is experimentally acquired with A

_{0} being the initial absorbance before the interaction of HG and A being the absorbance taken at any point of the reaction

$\{Delta\}A=A-A\_0$Using the Beer-Lambert law the

equation can be rewritten with the absorption coefficients and concentrations of each component

$\{Delta\}A=epsilon^\{HG\}[1]b+epsilon^\{G\}[2]b-epsilon^\{G\}[3]\_0b$Due to the previous assumption that

[4]_{0} >>

[5]_{0} one can expect that

[6] =

[7]_{0} Δε represents the change in value between ε

^{HG} and ε

^{G}$\{Delta\}A=\{Delta\}epsilon[8]b$A binding isotherm can be described as "the theoretical change in the

concentration of one component as a function of the

concentration of another component at constant temperature" This can be described by the following equation:

$[9]=\; frac\{[10]\_0K\_a[11]\}\{1+K\_a[12]\}$By substituting the binding isotherm

equation into the previous

equation the equilibrium constant K

_{a} can now be correlated to the change in absorbance due to the formation of the HG complex

$\{Delta\}A=b\{Delta\}epsilon\{frac\{[13]\_0K\_a[14]\_0\}\{1+K\_a[15]\_0\}\}$Further modications results in an

equation where a double reciprocal plot can be made with 1/ΔA as a function of 1/

[16]_{0} Δε can be derived from the intercept while K

_{a} can be calculated from the

slope$frac\{1\}\{\{Delta\}A\}=frac\{1\}\{b\{Delta\}epsilon[17]\_0[18]\_0K\_a\}\; +frac\{1\}\{b\{Delta\}epsilon[19]\_0\}$