Question

Find the average kinetic energy of a gas molecule of (i) monoatomic gas, (ii) diatomic gas at ${{20}^{0}}C$

Answer 1

Hint: Find the formula of average kinetic energy of monoatomic and diatomic molecules. These can be found if we know the number of degrees of freedom for a monatomic and diatomic gas. Next, we know the temperature and the Boltzmann constant, we can easily find out.
Formulas used:
$\begin{align}
  & K.E=\dfrac{5}{2}kT \\
 & K.E=\dfrac{3}{2}kT \\
\end{align}$

Complete answer:
Let us find the degrees of freedom of a monatomic gas. A monatomic gas consists of three degrees of freedom. As each degree of freedom contribute an energy of $\dfrac{1}{2}kT$, the energy of monatomic gas is
 $\begin{align}
  & K.E=\dfrac{3}{2}kT \\
 & \Rightarrow K.E=\dfrac{3}{2}\times 1.38\times {{10}^{-23}}\times 293 \\
 & \Rightarrow K.E=6.12\times {{10}^{-21}}J \\
\end{align}$
Similarly, the energy of diatomic gas will be,
$\begin{align}
  & K.E=\dfrac{5}{2}\times 1.38\times {{10}^{23}}\times 293 \\
 & \Rightarrow K.E=10.22\times {{10}^{-21}}J \\
\end{align}$
Therefore, we have found the kinetic energies in the following ways.

Additional information:
The average kinetic energy for molecules is the same for all gases at the same temperature, and its value is directly proportional to the absolute temperature. Kinetic molecular theory states that gas particles are in constant motion and exhibit perfectly elastic collisions. This theory is used to explain both Charles and Boyle's laws. The average kinetic energy of a collection of gas particles is directly proportional to absolute temperature. An ideal gas is a hypothetical gas whose molecules exhibit no interaction and undergo elastic collision with each other and the walls of the container. For a collection of gas particles at a given temperature, Maxwell Boltzmann distribution describes average molecular speeds.

Note:
The degrees of freedom play an important role in figuring out the formula for the kinetic energy. For a monatomic gas the molecules have 3 degrees of freedom, so the formula is taken in a different way. Similarly, for diatomic gases there will be 5 degrees of freedoms which makes the kinetic energy of those molecules a different one.