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Benesi-Hildebrand method

    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 (G0) is much larger than the initial concentration of the host (H0) then the absorbance from H0 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 A0 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 εHG and εG
    {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 Ka 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 Ka 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}