Various collision theories, dealing with the frequency of collision between reactant 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 collision density is given by simple collision theory as: \[Z_{\mathrm{AA}}=\frac{\sqrt{2}\ \pi \ \sigma ^{2}\ u\ N_{\text{A}}^{2}}{2}\] Here N A is the number density of molecules and u is the mean molecular speed, given by kinetic theory to be 8kB.Tπm, where m is the molecular mass, and σ = π d 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 collision density Z AB for two unlike molecules A and B, of masses m A and m B is: \[Z_{\mathrm{AB}}=N_{\text{A}}\ N_{\text{B}}\ \sigma ^{2}\ \sqrt{\frac{\pi \ k_{\text{B}}\ T}{\mu }}\] where µ is the reduced mass m A m B m A + m B, and σ = π d AB 2. For the collision frequency 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 Avogadro constant. More advanced collision theories, not involving the assumption that molecules behave as hard spheres, are known as generalized kinetic theories.