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Step-by-step Solution
Step 1: Understand the Physical Situation
An ideal gas of molecular mass $M$ is in a thermally insulated vessel moving with speed $v$. When the vessel is suddenly brought to rest, its kinetic energy is converted into the internal energy of the gas (since no heat is exchanged with the surroundings and no external work is done).
Step 2: Write the Expression for the Lost Kinetic Energy
The total mass of the gas is $m$ (we will relate $m$ to the number of moles later). The initial kinetic energy of the vessel (and thus the gas) is:
$ \frac{1}{2} m v^2. $
Step 3: Relate Kinetic Energy to Change in Internal Energy
Because the vessel is thermally insulated, the decrease in kinetic energy must equal the increase in internal energy of the gas. For an ideal gas with molar heat capacity at constant volume $C_v$, the change in internal energy is:
$ n C_v \,\Delta T, $
where $n$ is the number of moles of the gas, and $\Delta T$ is the increase in its temperature. Also, for an ideal gas,
$ C_v = \frac{R}{\gamma - 1}, $
where $R$ is the universal gas constant and $\gamma$ is the ratio of specific heats.
Step 4: Express $n$ in Terms of $m$ and $M$
The number of moles $n$ of the gas is related to the total mass $m$ and the molecular mass $M$ by:
$ n = \frac{m}{M}. $
Step 5: Form the Energy Balance Equation
Equating the lost kinetic energy to the gain in internal energy:
$ \frac{1}{2} m v^2 \;=\; n \, C_v \,\Delta T \;=\; \left(\frac{m}{M}\right) \cdot \frac{R}{\gamma - 1} \,\Delta T. $
Step 6: Solve for the Change in Temperature $ \Delta T $
Rearranging the above equation to find $ \Delta T $:
$ \Delta T \;=\; \frac{\frac{1}{2} m v^2 (\gamma - 1)}{\left(\frac{m}{M}\right) R} \;=\; \frac{M v^2 (\gamma - 1)}{2 R}. $
Step 7: State the Final Answer
Therefore, the increase in temperature of the gas is:
$ \Delta T = \frac{(\gamma - 1)}{2 R} \,M\, v^2 \, K. $
This matches the given correct option.
