What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature T?

Question:
What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature T?

$\frac{2}{3} \mathrm{k}_{\mathrm{B}} \mathrm{T}$
$\mathrm{k}_{\mathrm{B}} \mathrm{T}$
$\frac{3}{2} \mathrm{k}_{\mathrm{B}} \mathrm{T}$
$\frac{1}{2} \mathrm{k}_{\mathrm{B}} \mathrm{T}$

Correct Option: , 3

Solution:

As per Equi-partition law :

Each degree of freedom contributes

$\frac{1}{2} \mathrm{k}_{\mathrm{B}} \mathrm{T}$ Average Energy

In monoatomic gas D.O.F. $=3$

$\Rightarrow$ Average energy $=3 \times \frac{1}{2} k_{B} T=\frac{3}{2} k_{B} T$

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