The Boltzmann distribution function is one of the fundamental results of statistical thermodynamics and provides a relationship between the population of particles among different energy states and the energies of those states at a given temperature. It is particularly important in physical chemistry because it connects microscopic molecular behaviour with measurable macroscopic properties. The distribution is based on the idea that, at thermal equilibrium, particles do not occupy all available energy states equally. States of lower energy are more highly populated than states of higher energy, and the relative populations depend upon temperature. Consider a system containing (N) distinguishable particles distributed among a large number of possible energy levels. Let the (i)-th energy level have energy (E_i) and contain (n_i) particles. If the total number of particles is (N), then the number constraint is [ \sum_i n_i=N ] The total energy of the system is fixed at (U), so the energy constraint is [ \sum_i n_iE_i=U ] According to Boltzmann's statistical approach, the probability or population of a particular energy state is proportional to the exponential of its energy divided by thermal energy. Thus, [ n_i\propto e^{-E_i/kT} ] where (k) is the Boltzmann constant and (T) is the absolute temperature. Introducing a normalization constant gives [ n_i=Ce^{-E_i/kT} ] The constant (C) is determined from the condition that the total number of particles is (N). Therefore, [ N=\sum_i n_i=C\sum_i e^{-E_i/kT} ] and hence [ C=\frac{N}{\sum_i e^{-E_i/kT}} ] Thus, the Boltzmann distribution function is [ \boxed{n_i=N\frac{e^{-E_i/kT}}{\sum_j e^{-E_j/kT}}} ] The denominator is known as the partition function: [