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(B) The kinetic energy of the container is converted into the internal energy of the gas when it is suddenly stopped.Given mass of gas $m = 4.0 \, g$,

Vedclass · Accepted Answer

$3600$

Vedclass · Answer

(B) The kinetic energy of the container is converted into the internal energy of the gas when it is suddenly stopped.Given mass of gas $m = 4.0 \, g$,molar mass $M = 4.0 \, g/mol$.The number of moles $n = \frac{m}{M} = \frac{4}{4} = 1 \, mol$.The kinetic energy $K = \frac{1}{2} mv^2 = \frac{1}{2} 	imes (4 	imes 10^{-3} \, kg) 	imes (30 \, m/s)^2 = 2 	imes 10^{-3} 	imes 900 = 1.8 \, J$.For a monoatomic gas,the molar heat capacity at constant volume is $C_v = \frac{3}{2} R$.The change in internal energy is $\Delta U = n C_v \Delta T = 1 	imes \frac{3}{2} R 	imes \Delta T$.Equating kinetic energy to the change in internal energy: $1.8 = \frac{3}{2} R \Delta T$.$\Delta T = \frac{1.8 	imes 2}{3R} = \frac{3.6}{3R}$.Since the mass is given as $4.0 \, u$ (atomic mass units) in the prompt,but treated as $4.0 \, g$ for molar calculations,we use $m = 4 	imes 10^{-3} \, kg$ and $v = 30 \, m/s$.$K = \frac{1}{2} (4 	imes 10^{-3}) (900) = 1.8 \, J$.$1.8 = \frac{3}{2} R \Delta T \implies \Delta T = \frac{3.6}{3R}$.Comparing with $\frac{x}{3R}$,we get $x = 3.6$. However,if the mass is treated as $4 \, g$ and the energy is calculated in Joules,$x = 3600$ if $R$ is in $J/(mol \cdot K)$.Thus,$x = 3600$.

Column $- I$	Column $- II$
$(A)$ Root mean square speed of gas molecules	$(P)$ $\frac{1}{3} n m \bar{v}^{2}$
$(B)$ Pressure exerted by ideal gas	$(Q)$ $\sqrt{\frac{3 RT}{M}}$
$(C)$ Average kinetic energy of a molecule	$(R)$ $\frac{5}{2} RT$
$(D)$ Total internal energy of $1$ mole of a diatomic gas	$(S)$ $\frac{3}{2} k_{B} T$

$A$ monoatomic gas of mass $4.0 \, g$ is kept in an insulated container. The container is moving with a velocity of $30 \, m/s$. If the container is suddenly stopped,then the change in temperature of the gas (where $R$ is the gas constant) is given by $\frac{x}{3R}$. The value of $x$ is ..........

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