Question

An insulated container containing monatomic gas of molar mass m is moving with the velocity ${{v}_{o}}$ if the container is suddenly stopped, find the change in temperature.

Answer 1

First of all, find the kinetic energy in terms of molar mass m. After that, find kinetic energy in terms of temperature. Then evaluate the change in temperature by comparing both relations of kinetic energy.
Formula used:
Kinetic energy is given as
$K.E=\dfrac{1}{2}m{{v}^{2}}$
$K.E=\dfrac{3}{2}R\Delta T$

Complete step-by-step solution
We know that m is molar mass and ${{v}_{0}}$ is the velocity of gas.
Change in velocity will be equal to final velocity minus the initial velocity.
So,
$\begin{aligned} & \Delta v=\left| 0-{{v}_{0}} \right| \\\ & \therefore \Delta v={{v}_{0}} \\\ \end{aligned}$, final velocity is zero as the container is stopped suddenly.
Suppose ‘$m$’ is the mass of gas, then using the formula of kinetic energy, we get -
$\begin{aligned} & \Delta K.E=\dfrac{1}{2}m{{\left( \Delta v \right)}^{2}} \\\ & \Delta K.E=\dfrac{1}{2}m{{\left( {{v}_{0}} \right)}^{2}} \\\ \end{aligned}$
For ‘n’ number of moles, the kinetic energy will be:
$\Delta K.E=\dfrac{1}{2}\left( mn \right){{v}_{0}}^{2}......\left( 1 \right)$
Where m is molar mass of gas and ${{v}_{0}}$ is the velocity of gas molecules. Now according to kinetic theory of gases’,
$\Delta K.E=\dfrac{3}{2}R\Delta T$
For ‘n’ number of moles,
$\Delta K.E=\dfrac{3}{2}nR\Delta T....\left( 2 \right)$
By comparing equation (1) and (2) we get,
$\begin{aligned} & \dfrac{3}{2}nR\Delta T=\dfrac{1}{2}mn{{v}_{0}}^{2} \\\ & \Rightarrow \Delta T=\dfrac{mn{{v}_{0}}^{2}}{3nR} \\\ & \therefore \Delta T=\dfrac{m{{v}_{0}}^{2}}{3R} \\\ \end{aligned}$
Therefore, the change in temperature due to sudden stop of the container will be $\dfrac{m{{v}_{0}}^{2}}{3R}$.
Additional Information:
Under the physics of Kinetic theory of gas some facts are assumed for the system under consideration. These assumptions of Kinetic theory of gases are:
1. All gases are made up of molecules moving randomly in all directions.
2. The size of the molecule is much smaller than the average separation between the molecules.
3. The molecules exert no force on each other or the walls of the container except during collisions.
4. All collisions between two molecules or between a molecule and a wall are perfectly elastic. Also, the time spent during a collision is negligibly small.
5. The molecules obey Newton’s laws of motion.
6. When a gas is left for sufficient time, it comes to a steady state. The density and the disruption of molecules with different velocities are independent of position, direction, and time. This assumption may be justified if the number of molecules is very large.

Note: For the solution of this question, first we need to figure out the formula that needs to be used to get the change in temperature, as we have the mass and velocity related information of the container. One major point to note is that kinetic energy and kinetic theory of gas are two different things, that is being used in this question, as one talks about the energy of a molecule and the other thing takes the different element of gas into consideration along with certain assumptions.