# What are the Boltzmann factors?

##### 1 Answer

A general **ratio of the population of states** can be written in statistical mechanics as:

#N_i/N = (g_i e^(-betaepsilon_i))/(q) = (g_i e^(-betaepsilon_i))/(sum_i g_i e^(-betaepsilon_i))# where:

#g_i# is the degeneracy of state#i# with energy#epsilon_i# .#beta = 1/(k_BT)# is a constant containing the Boltzmann constant and temperature.#N_i# is the number of particles in state#i# and#N# is the total number of particles.

If we then consider a single state relative to energy zero, we have two states such that:

#N_1/N_0 = N_1/N cdot N/N_0#

#= (g_1 e^(-betaepsilon_1))/cancel(g_0 e^(-betaepsilon_0) + g_1 e^(-betaepsilon_1)) cdot cancel(g_0 e^(-betaepsilon_0) + g_1 e^(-betaepsilon_1))/(g_0 e^(-betaepsilon_0))#

Since the *energy zero* be

#(N_i)/(N_0) = (g_i)/(g_0) e^(-betaepsilon_i)#

Thus, the **population of state** **with some energy higher than energy zero** is given by

We call **Boltzmann factor** for state