Population of energy level and it thermal equilibrium condition

Population of energy level:

The number of atoms per unit volume in any energy level is called the population of that energy level.

The population $N$ of any energy level $E$ depends on the temperature $T$ which can be described by

$N=e^{-\left(\frac{E}{kT}\right)}$

Where $k \rightarrow$ Boltzmann's Constant

The above equation is called the Boltzmann equation.

Population of energy level at thermal equilibrium condition:

At thermal equilibrium, the number of atoms (Population) at each energy level decreases exponentially with increasing energy level, as shown in the figure below.

Let us consider, two energy levels $E_{1}$ and $E_{2}$. The population of these energy levels can be calculated by

$N_{1}=e^{\left(-\frac{E_{1}}{kT} \right)} \quad (1)$

$N_{2}=e^{\left(-\frac{E_{2}}{kT} \right)} \quad (2)$

The ratio of the population in these two levels is called the relative population.

$\frac{N_{2}}{N_{1}}= \frac{e^{\left(-\frac{E_{2}}{kT} \right)}}{e^{\left(-\frac{E_{1}}{kT} \right)}}$

$\frac{N_{2}}{N_{1}}= e^{\left(-\frac{(E_{2}-E_{1})}{kT} \right)} $

$\frac{N_{2}}{N_{1}}= e^{-\frac{\Delta E}{kT} } $

This equation is known as Boltzmann's distribution. The above equation suggests the relative population is dependent on two factors.

1.) The energy difference $(\Delta E)$

2.) The absolute temperature $T$