# RRKM theory

The Rice–Ramsperger–Kassel–Marcus (RRKM) theory is a theory of chemical reactivity.[1][2][3] It was developed by Rice and Ramsperger in 1927 [4] and Kassel in 1928[5] (RRK theory[6]) and generalized (into the RRKM theory) in 1952 by Marcus[7] who took the transition state theory developed by Eyring in 1935 into account. These methods enable the computation of simple estimates of the unimolecular reaction rates from a few characteristics of the potential energy surface.

## Assumption

Assume that the molecule consists of harmonic oscillators, which are connected and can exchange energy with each other.

• Assume the possible excitation energy of the molecule to be E, which enables the reaction to occur.
• The rate of intra-molecular energy distribution is much faster than that of reaction itself.

## Derivation

Assume that A* is an excited molecule:

${\displaystyle A^{*}{\xrightarrow {k(E)}}A^{\ddagger }\rightarrow P}$

where P stands for product, and A for the critical atomic configuration with the minimum energy E0 along the reaction coordinate.

Unimolecular rate constant ${\displaystyle k_{\mathrm {uni} }}$ is obtained as follows.[8]

${\displaystyle k_{\mathrm {uni} }={\frac {1}{hQ_{r}Q_{v}}}\int \limits _{E_{0}}^{\infty }\mathrm {d} E\sum _{J=0}^{\infty }{\frac {(2J+1)G^{\ddagger }(E,J)\exp \!\left({\frac {-E}{k_{b}T}}\right)}{1+{\frac {k(E,J)}{\omega }}}},}$

where ${\displaystyle k(E,J)}$ is the microcanonical transition state theory rate constant, ${\displaystyle G^{\ddagger }}$ is the sum of states for the active degrees of freedom in the transition state, ${\displaystyle J}$ is the quantum number of angular momentum, ${\displaystyle \omega }$ is the collision frequency between ${\displaystyle A^{*}}$ molecule and bath molecules, ${\displaystyle Q_{r}}$ and ${\displaystyle Q_{v}}$ are the molecular vibrational and external rotational partition functions.