Question

The mean kinetic energy of a gas molecule is proportional to
A) \[\sqrt T \]
B) ${T^3}$
C) $T$
D) None of the above

Answer 1

Hint: The macroscopic properties of gases such as volume, pressure, and the temperature is known as the kinetic theory of gases explains. Extremely small particles present in every gas are known as molecules. They are identical, spherical, rigid, and perfectly elastic.

Complete step by step answer:
The equipartition theorem states that in thermal equilibrium, the average energy of each degree of freedom is $\dfrac{{{k_b}T}}{2}$, Where $T$ is temperature and $k_b$ is the Boltzmann constant. In other words, the energy is shared equally among all degrees of freedom of the system.
The mean kinetic energy of a gas molecule is proportional to the absolute temperature $T$.

$\therefore $ We can say that option (C) is correct.

Additional information:
Some important points regarding the kinetic theory of gases are the following:
- The speed of gas molecules lies between zero and infinity.
- No attractive and repulsive forces act between the molecules.
- The volume of molecules is negligible in comparison to the volume of gas.
- Time spent in a collision between molecules is negligible in comparison to time between two successive collisions.
- The number of collisions per unit volume in gas remains constant.
- Molecules continuously collide with the walls of the container due to which their momentum changes. - The change in momentum is transferred to the walls of the container. Hence, the pressure is exerted by the gas molecules on the walls of the container.
- The density of the gas is constant at all points of the container.

Note:
The kinetic energy per degree of freedom is proportional to $\dfrac{1}{2}$ times Boltzmann constant multiplied by temperature. Also, the temperature will decrease when the pressure drops to a certain point.