Various collision theories, dealing with the frequency of collision between @[email protected] molecules, have been put forward. In the earliest theories reactant molecules were regarded as hard spheres, and a collision was considered to occur when the distance \(d\) between the centres of two molecules was equal to the sum of their radii. For a gas containing only one type of molecule, A, the @[email protected] is given by simple collision theory as: \[Z_{\mathrm{AA}}=\frac{\sqrt{2}\ \pi \ \sigma ^{2}\ u\ N_{\text{A}}^{2}}{2}\] Here \(N_{\text{A}}\) is the @[email protected] of molecules and \(u\) is the mean molecular speed, given by kinetic theory to be \(\sqrt{\frac{8\ k_{\text{B}}\ T}{\pi \ m}}\), where \(m\) is the molecular mass, and \(\sigma =\pi \ d_{\mathrm{AA}}^{2}\). Thus: \[Z_{\mathrm{AA}}=2\ N_{\text{A}}^{2}\ \sigma ^{2}\ \sqrt{\frac{\pi \ k_{\text{B}}\ T}{m}}\] The corresponding expression for the @[email protected] \(Z_{\mathrm{AB}}\) for two unlike molecules A and B, of masses \(m_{\text{A}}\) and \(m_{\text{B}}\) is: \[Z_{\mathrm{AB}}=N_{\text{A}}\ N_{\text{B}}\ \sigma ^{2}\ \sqrt{\frac{\pi \ k_{\text{B}}\ T}{\mu }}\] where \(\mu \) is the @[email protected] \(\frac{m_{\text{A}}\ m_{\text{B}}}{m_{\text{A}}+m_{\text{B}}}\), and \(\sigma =\pi \ d_{\mathrm{AB}}^{2}\). For the @[email protected] factor these formulations lead to the following expression: \[z_{\mathrm{AA}}\quad \text{or}\quad z_{\mathrm{AB}}=L\ \sigma ^{2}\ \sqrt{\frac{8\ \pi \ k_{\text{B}}\ T}{\mu }}\] where \(L\) is the @[email protected] More advanced collision theories, not involving the assumption that molecules behave as hard spheres, are known as generalized kinetic theories.
PAC, 1996, 68, 149. (A glossary of terms used in chemical kinetics, including reaction dynamics (IUPAC Recommendations 1996)) on page 160 [Terms] [Paper]