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https://doi.org/10.1351/goldbook.C01170
Various collision theories, dealing with the frequency of collision between @R05163@ 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 @C01162@ 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 @N04262@ 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 @C01162@ $$Z_{\mathrm{AB}}$$ 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 @R05214@ $$\frac{m_{\text{A}}\ m_{\text{B}}}{m_{\text{A}}+m_{\text{B}}}$$, and $$\sigma =\pi \ d_{\mathrm{AB}}^{2}$$. For the @C01166@ 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 @A00543@. More advanced collision theories, not involving the assumption that molecules behave as hard spheres, are known as generalized kinetic theories.
Source:
PAC, 1996, 68, 149. (A glossary of terms used in chemical kinetics, including reaction dynamics (IUPAC Recommendations 1996)) on page 160 [Terms] [Paper]