Transition state theory

Reaction coordinate, transition state. via Wikipedia

Target

For a reaction $\ce{A + B -> C}$ , the reaction speed is given by

$\mathrm{d}[\ce{C}]/\mathrm{d}t = k[\ce{A}][\ce{B}]$

we want to know the reaction speed constant $k$ as a function of $T$ .

Assumption

Transition state theory assumes at steady state, there is a complex $\ce{AB^\ddag}$ will reach equilibrium with reactant $\ce{A}$ and $\ce{B}$ .

$\ce{A + B <=> AB^\ddag -> C}$

Derivation

From equilibrium condition

$K^\ddag = \frac{[\ce{AB^\ddag}]}{[\ce{A}][\ce{B}]} = \exp\left(-\frac{\Delta G^\ddag}{RT}\right)$

And the rate for product $\ce{C}$ is generation is determined by the speed which $\ce{AB^\ddag}$ decomposes

$\mathrm{d}[\ce{C}]/\mathrm{d}t = \nu[\ce{AB^\ddag}]$

And assume $h\nu = k_BT$ .

Then we have the following derivation:

$\mathrm{d}[\ce{C}]/\mathrm{d}t = \nu[\ce{AB^\ddag}] = \nu K^\ddag[\ce{A}][\ce{B}] = \frac{k_BT}{h} \exp\left(-\frac{\Delta G^\ddag}{RT}\right)[\ce{A}][\ce{B}]$

Then we have

$k = \frac{k_BT}{h} \exp\left(-\frac{\Delta G^\ddag}{RT}\right)$