Question

An insulated container containing monatomic gas of molar mass $\mathrm{m}$ moving with a velocity $v_0$. If the container is suddenly stopped. The change in temperature is?
A.$\dfrac{\text{m}{{\text{v}}_{0}}^{2}}{2\text{R}}$
B.$\dfrac{\text{m}{{\text{v}}_{0}}^{2}}{3\text{R}}$
C.$\dfrac{\text{R}}{\text{mv}_{0}^{2}}$
D.$\dfrac{3\text{mv}_{0}^{2}}{2\text{R}}$

Answer 1

Hint:The molecules move faster in a hot gas than in a cold gas; the mass remains the same, but because of the increased velocity of the molecules. Here we have to use the Kinetic theory equation for calculating change in Temperature of a monatomic gas. The particles move faster as the temperature increases, and therefore have higher velocities.
Formula used:
$\dfrac{1}{2}(mn)v_{0}^{2}=n\dfrac{3}{2}R(\Delta T)$

Complete answer:
The absolute temperature of a given gas sample is according to the kinetic temperature interpretation, proportional to the total translational kinetic energy of its molecules. Any change in the absolute gas temperature will therefore contribute to the corresponding translational change of $\mathrm{KE}$ and vice-versa.
When the container stops, its total $\mathrm{KE}$ is transferred to gas molecules in the form of translational $\mathrm{KE}$, thereby increasing the absolute temperature. If $\Delta T=$ change in absolute temperature.
Then $, \mathrm{KE}$ of molecules due to velocity ${{V}_{0}},KE=\dfrac{1}{2}(mn)V_{0}^{2}$ - (I)
increase in translational $KE=n\dfrac{3}{2}R(\Delta T)$- (II)
According to kinetic theory Equation. (i) and (ii) are equal
$\Rightarrow \quad \dfrac{1}{2}(m n) v_{0}^{2}=n \dfrac{3}{2} R(\Delta T)$
$(m n) v_{0}^{2}=n 3 R(\Delta T)$
$\Rightarrow \quad \Delta T=\dfrac{(m n) v_{0}^{2}}{3 n R}$
$\Rightarrow \quad \Delta T=\dfrac{m v_{0}^{2}}{3 R}$

The change in temperature is $\Rightarrow \quad \Delta T=\dfrac{m v_{0}^{2}}{3 R}$
Correct option is (B) .

Note:
The kinetic energy, and hence the temperature, is greater. Turning to the large scale, a gas's temperature is something that with our senses we can qualitatively determine. The particles will gain kinetic energy when a gas is trapped inside a container that has a fixed size (its $v_0$ lume can not change) and the gas is heated, which will make them move faster. If you had a way to increase pressure with no change in $v_0$lume, then yes, by the ideal gas law, the temperature would increase.