Mean Classical Thermal Energy and the Equipartition Theorem

Consider a system of particles, each of mass m but with differing values of speed v; the kinetic energy of each particle is mv2/2, which may be resolved along mutually perpendicular x, y and z axes. The mean value of any distribution of the form ϕ(X) is given by: where the integration is taken over the range of the variable. Assuming that the energies of the particles follow a Boltzmann distribution, then from Appendix A4, the mean value for the kinetic energy uK for a single particle is: Since positive and negative directions of v are equally probable, introducing spherical coordinates from Appendix A6 and replacing r of that discussion by v, Eq. (A5.2) may be now written as: which simplifies to: Making the substitution t=pv2,wherep=m/2kT, and following the argument of Example A7.1 in Appendix A7, Eq....