Benesi-Hildebrand method

The Benesi-Hildebrand method is a mathematical approach in physical chemistry to determine the equilibrium constant K and the stoichiometry of non-binding interactions. It is typically used in chemical reaction equilibria of 1: 1 complexes . Examples of this are charge transfer complexes and host-guest molecule complexations ( English host – guest complexation ). The theoretical basis of the Benesi-Hildebrand method is the assumption that the characteristic electronic absorption spectrum of one reactant is transparent in the collective absorption / emission bandwidth of the system when the other reactant is in excess. By measuring the absorption spectrum before and after the formation of the product and its equilibrium, the association constant of the reaction can therefore be determined.

history

The method was developed in 1949 by HA Benesi and JH Hildebrand to explain a phenomenon in which iodine changes its color in different aromatic solvents. This phenomenon was ascribed to the formation of an iodine-solvent complex through acid-base interactions, which led to a shift in the absorption spectrum. In this system, the solvent is considered to be an excess reactant. Today the Benesi-Hildebrand method is one of the standard procedures for determining association constants by means of absorption spectra.

Derivation

The Benesi-Hildebrand method is used for reaction equilibria of 1: 1 complexes, here between reactants H and G under UV / VIS absorption. The equilibrium reaction of such complexes can be simplified as:

${\displaystyle {\ce {H + G <=> HG}}}$

It is assumed that the measured absorption A is composed of the absorptions of the reactants and the complex.

${\displaystyle A=A^{{\ce {H}}}+A^{{\ce {G}}}+A^{{\ce {HG}}}}$

With the assumption that the initial concentration [G ₀ ] of reactant G is significantly greater than the initial concentration [H ₀ ] of reactant H, the absorption of H can be regarded as negligible.

${\displaystyle A=A^{{\ce {G}}}+A^{{\ce {HG}}}}$

The absorption can be measured both before and after the formation of the complex. The change in absorption Δ A can thus be determined experimentally, where A _{0 represents} the original absorption before the interaction and A the absorption during any point in time of the reaction.

${\displaystyle \Delta A=A-A_{0}\,}$

By applying Beer's Law , the equation can be transformed to use the absorption coefficients ε ^X and concentrations [X] of each component X.

${\displaystyle \Delta A=\varepsilon ^{\ce {HG}}[{\ce {HG}}]\,b+\varepsilon ^{\ce {G}}[{\ce {G}}]\,b-\varepsilon ^{\ce {G}}[{\ce {G}}_{0}]\,b}$

Based on the assumption that the initial concentration [G ₀ ] is significantly greater than the initial concentration [H ₀ ], it can be assumed that [G] = [G ₀ ]. .DELTA ε representing the difference between ε ^HG and ε ^G . It follows:

${\displaystyle \Delta A=\Delta \varepsilon [{\ce {HG}}]\,b}$

A binding isotherm can be described as "the theoretical difference between the concentration of one component as a function of the concentration of another component at constant temperature". This is represented by the following equation:

${\displaystyle [{\ce {HG}}]={\frac {[{\ce {H}}_{0}]K_{a}[{\ce {G}}]}{1+K_{a}[{\ce {G}}]}}}$

If the equation of the binding isotherm is now inserted into the previous equation, the equilibrium constant K _{a can} now be correlated to the change in absorption due to the complex formation.

${\displaystyle \Delta A=b\,\Delta \varepsilon {\frac {[{\ce {H}}_{0}]K_{a}[{\ce {G}}_{0}]}{1+K_{a}[{\ce {G}}_{0}]}}}$

Further optimization leads to the Benesi-Hildebrand equation , which represents 1 / Δ A as a function of 1 / [G ₀ ] (double reciprocal representation). Δ ε can now be calculated from the intercept and K _a from the slope.

${\displaystyle {\frac {1}{\Delta A}}={\frac {1}{b\,\Delta \varepsilon [{\ce {G}}_{0}][{\ce {H}}_{0}]K_{a}}}+{\frac {1}{b\,\Delta \varepsilon [{\ce {H}}_{0}]}}}$

Limitations

In most cases the Benesi-Hildebrand method gives excellent linear representations and gives reasonably satisfactory results for K and ε . However, the collection of experimental data is also associated with problems, including the calculation of different results for ε with other concentrations, a lack of consistency between the results of Benesi-Hildebrand compared to other methods (e.g. partition measurements) and the wrong values ​​in the calculation of intercepts of zero or less.

Additional concerns have also been raised about the accuracy of the Benesi-Hildebrand method, as certain conditions cause the calculations made to be invalid. The reactant concentrations must always meet the assumption that the initial concentration of one reactant is significantly greater than the initial concentration of the other reactant. If this condition is no longer met, the Benesi-Hildebrand representation no longer runs linearly, but scatters. Furthermore, in the case of the determination of equilibrium constants of weakly bound complexes, it is common for 2: 1 complex formation to occur in solution. It has been observed that these 2: 1 complexes generate inappropriate parameters that significantly interfere with the accurate determination of association constants. Due to this fact, one of the points of criticism of this method is the inflexibility to investigate only reactions with 1: 1 product complexes.

Another limitation, both of the Benesi-Hildebrand method and other methods, is the dependence on K and ε . Typically there is always only one parameter ( K or ε ) and its product Kε is a proportion of the slope or axis intercept, consequently only one parameter can be determined independently of the other.

Modifications

Although it was initially used in connection with UV / VIS spectroscopy , many modifications have been made since then, which means that the Benesi-Hildebrand method can also be used with other spectroscopic methods, including fluorescence, infrared or NMR.

Modifications were also made to improve the precision in the determination of K and ε . One such modification is that of N. Rose and R. Drago; the Benesi-Hildebrand equation was adapted to:

${\displaystyle K^{-1}={\frac {A}{\varepsilon _{{\ce {HG}}}}}-[{\ce {H}}_{0}]-[{\ce {G}}_{0}]+{\frac {C_{{\ce {H}}}C_{{\ce {G}}}}{A}}\,\varepsilon _{{\ce {HG}}}}$

This modification is based on the selection of values ​​for ε and the measurement of absorption data and initial concentrations of both reactants. This allows the calculation of K ⁻¹ . The plot of ε _HG against K ⁻¹ results in a linear relationship. Repeating this procedure for different concentrations and plotted on the same graph provides a common point of intersection for all straight lines. This is the optimal value for ε _HG and K ⁻¹ . In this method, however, the biggest problem is that the straight lines do not always intersect at the same point or, in the worst case, not at all.

In 1982 B. Seal, H. Sil and D. Mukherjee developed another graphical procedure to evaluate K and ε independently of one another. This approach relies on a more complex mathematical rearrangement of the Benesi-Hildebrand method and a graphical approach, but has proven to be fairly accurate compared to standard values.

Individual evidence

1. ↑ E. Anslyn: Modern Physical Organic Chemistry . 2006, ISBN 1-891389-31-9 , pp. 221 .
2. ↑ HA Benesi, JH Hildebrand: A Spectrophotometric Investigation of the Interaction of Iodine with Aromatic Hydrocarbons . In: J. Am. Chem. Soc. tape 71 , 1949, pp. 2703-2707 , doi : 10.1021 / ja01176a030 .
3. ^ R. Scott: Some Comments on the Benesi-Hildebrand Equation . In: Rec. Trav. Chim. tape 75 , 1956, pp. 787-789 , doi : 10.1002 / recl.19560750711 .
4. ^ S. McGlynn: Energetics of Molecular Complexes . In: Chem. Rev. Band 58 , 1958, pp. 1113-1156 , doi : 10.1021 / cr50024a004 .
5. ↑ M. Hanna, A. Ashbaugh: Nuclear Magnetic Resonance Study of Molecular Complexes of 7,7,8,8-Tetracyanoquinodimethane and Aromatic Donors . In: J. Phys. Chem. Band 68 , 1964, pp. 811-816 , doi : 10.1021 / j100786a018 .
6. ↑ P. Qureshi, R. Varshney, S. Singh: Evaluation of є for the p-dinitrobenzene-aniline complexes by the Scott equation. Failure of the Benesi-Hildebrand equation . In: Spectrochim. Acta A. Band 50 , 1994, pp. 1789-1790 , doi : 10.1016 / 0584-8539 (94) 80184-3 .
7. ^ B. Arnold, A. Euler, K. Fields, R. Zaini: Association constants for 1,2,4,5-tetracyanobenzene and tetracyanoethylene charge-transfer complexes with methyl-substituted benzenes revisited . In: J. Phys. Org. Chem. Band 13 , 2000, pp. 729-734 , doi : 10.1002 / 1099-1395 (200011) 13:11 <729 :: AID-POC311> 3.0.CO; 2-L .
8. ↑ M. Mukhopadhyay, D. Banerjee, A. Koll, A. Mandal, A. Filarowski, D. Fitzmaurice, R. Das, S. Mukherjee: Excited state intermolecular proton transfer and caging of salicylidine-3,4,7-methyl amines in cyclodexrins . In: J. Photochem. Photobiol. A. Band 175 , 2005, pp. 94-99 , doi : 10.1016 / j.jphotochem.2005.04.025 .
9. ↑ K. Wong, S. Ng: On the use of the modified Benesi-Hildebrand equation to process NMR hydrogen bonding data . In: Spectrochimica Acta A . tape 32 , 1975, pp. 455-456 , doi : 10.1016 / 0584-8539 (76) 80101-8 .
10. ↑ N. Rose, R. Drago: Molecular Addition Compounds of Iodine. I. An Absolute Method for the Spectroscopic Determination of Equilibrium Constants . In: J. Am. Chem. Soc. tape 81 , 1959, pp. 6138-6141 , doi : 10.1021 / ja01532a009 .
11. ^ R. Drago, N. Rose: Molecular Addition Compounds of Iodine. II. Recalculation of Thermodynamic Data on Lewis Base-Iodine Systems Using an Absolute Equation . In: J. Am. Chem. Soc. tape 81 , 1959, pp. 6141-6145 , doi : 10.1021 / ja01532a010 .
12. ↑ B. Seal, A. Mukherjee, D. Mukherjee: An Alternative Method of Solving the Rose-Drago Equation for the Determination of Equilibrium Constants of Molecular Complexes . In: Bull. Chem. Soc. Jpn. tape 52 , 1979, pp. 2088-2090 , doi : 10.1246 / bcsj.52.2088 .
13. ↑ B. Seal, H. Sil, D. Mukherjee: Independent determination of equilibrium constant and molar extinction coefficient of molecular complexes from spectrophotometric data by a graphical method . In: Spectrochim. Acta A. Band 38 , 1982, pp. 289-292 , doi : 10.1016 / 0584-8539 (82) 80210-9 .