Animated Solution for Physics - Thermodynamics: A thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and its suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by

A Moving Vessel

Imagine you are observing a thermally insulated vessel filled with an ideal gas. This vessel is hurtling through space with a velocity $v$. The gas inside is just tagging along, meaning every single molecule has a macroscopic kinetic energy due to the bulk motion of the vessel, in addition to its own random microscopic thermal motion.

Now, the problem states that this vessel is suddenly brought to rest.

What happens to all that kinetic energy? It can't just disappear—the law of conservation of energy forbids it. Because the vessel is thermally insulated, no energy can escape to the surroundings as heat ($\Delta Q=0$). Therefore, the macroscopic kinetic energy of the bulk gas must transform entirely into the microscopic kinetic energy of the gas molecules. In thermodynamics, we call this microscopic kinetic energy the internal energy ($U$) of the gas.

When the vessel stops, the molecules keep moving forward until they smash into the front wall of the vessel. They bounce back, colliding with other molecules, and very quickly, this organized directional motion descends into chaotic, random motion.

This increase in random microscopic motion is exactly what we measure as an increase in temperature!

Let's write this down mathematically. The loss in the macroscopic kinetic energy of the gas equals the gain in its internal energy:

If the total mass of the gas is $m$, its initial kinetic energy is simply:

So, the change in internal energy $\Delta U$ is:

Now, we need to express the change in internal energy $\Delta U$ in terms of the temperature change $\Delta T$. For an ideal gas, the change in internal energy is always given by:

Here, $n$ is the number of moles of the gas, and $C_V$ is the molar heat capacity at constant volume.

We are given the molar mass of the gas as $M$. The number of moles $n$ is simply the total mass $m$ divided by the molar mass $M$:

We are also given the ratio of specific heats, $\gamma$. The molar heat capacity at constant volume $C_V$ is related to $\gamma$ and the universal gas constant $R$ by the standard formula:

Let's substitute these expressions back into our energy conservation equation.

Look closely at this equation. Do you see it? The total mass of the gas, $m$, appears on both sides of the equation! We can safely cancel it out:

This is a profound physical insight. The temperature rise $\Delta T$ does not depend on the total amount of gas in the vessel. Whether you have 1 gram or 100 kilograms of gas, the temperature increase will be exactly the same, provided the gas has the same molar mass $M$ and the same $\gamma$.

Finally, we just need to rearrange the equation to solve for the temperature increase $\Delta T$. Multiplying both sides by $M(\gamma -1)$ and dividing by $R$, we get:

This perfectly matches option (c). The $K$ in the options simply denotes Kelvin, the SI unit of temperature.

This problem is a beautiful demonstration of how macroscopic mechanics (kinetic energy) seamlessly bridges into thermodynamics (internal energy and temperature). Always remember, energy is never lost; it just changes its disguise!