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# Curtin-Hammett principle

In chemical kinetics, the Curtin-Hammett principle states that, for a reaction that has a pair of reactive intermediates or reactants that interconvert rapidly (as is usually the case for conformers), each going irreversibly to a different product, the product ratio will depend only on the difference in the free energy of the transition state going to each product, and not on the equilibrium constant between the intermediates .

For example, given species A and B that equilibrate rapidly while A turns irreversibly into C, and B turns irreversibly into D:

K is the equilibrium constant between A and B, and k1 and k2 are the rate constants for the formation of C and D, respectively. When the rate of interconversion between A and B is much faster than either k1 or k2 then the Curtin-Hammett principle tells us that the C:D product ratio will not reflect K, but the relative energy of the transition states.

The reaction coordinate free energy profile can be represented by the following scheme:

The ratio of products will depend only on the value labeled ΔΔG in the figure: C will be the major product, because the energy of TS1 is lower than the energy of TS2. It doesn't matter whether A is more stable than B or not, or by how much. This can be understood qualitatively by thinking what would happen if the free energy of A were increased, while keeping everything else constant. On one hand, ΔG1 would become smaller, which would make k1 larger, therefore favoring the formation of C. But on the other hand, the amount of A in equilibrium would decrease, because the change in ΔG would increase the value of K, favoring B. These two effects cancel out, leading to the conclusion that the relative energies of A and B don't matter. This can also be proved algebraically:

The rate of formation for compound C from A is given as $\frac{d[C]}{dt} = k_1[A]$

and that of D from B as: $\frac{d[D]}{dt} = k_2[B] = k_2K[A]$

with Kc the equilibrium constant. The ratio of the rates is then: $\frac{\frac{d[D]}{dt}}{\frac{d[C]}{dt}} = \frac{k_2K[A]}{k_1[A]} = \frac{k_2K}{k_1} = \frac{e^{-\Delta G_2^{\ddagger}/RT} e^{-\Delta G/RT}}{e^{-\Delta G_1^{\ddagger}/RT}} = e^{ - \frac{\Delta \Delta G^{\ddagger}}{RT} }$

The product ratio can also be written as: $\frac{[D]}{[C]} = e^{ - \frac{\Delta \Delta G^{\ddagger}}{RT} }$

## Application to stereoselective reactions

The Curtin-Hammett principle is used to explain the selectivity ratios for stereoselective reactions, such as in kinetic resolution. A typical example is the following: a prochiral molecule binds to a chiral catalyst, forming a pair of diastereomeric intermediates, depending on which face of the substrate was bound to the catalyst. These intermediates equilibrate rapidly (like A and B in the diagram above), and each one then leads to a different enantiomer of the product through the rate-determining step.