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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 moleculesA andB, 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.