Animated Solution for Physics - Thermodynamics: An insulated box containing a monoatomic gas of molar mass $M$ moving with a speed $v_0$ is suddenly stopped. Find the increment in gas temperature as a result of stopping the box.

Imagine a thermally insulated box filled with a monoatomic gas, cruising along at a constant speed $v_0$. Suddenly, the box is brought to a dead stop. What happens to all that kinetic energy?

In physics, energy never simply vanishes. Because the box is perfectly insulated, no heat can escape to the surroundings ($Q=0$). The macroscopic kinetic energy of the gas—the energy it possessed because the entire box was moving—must be conserved. It has nowhere to go but inward, converting entirely into the microscopic internal energy of the gas molecules. This sudden influx of internal energy manifests as a spike in the gas's temperature.

Let's quantify this energy transfer. If the total mass of the gas is $m$, its initial macroscopic kinetic energy is simply:

When the box stops, this entire kinetic energy transforms into internal energy, $\Delta U$. For an ideal gas, the change in internal energy is given by:

Here, $n$ is the number of moles, $C_V$ is the molar heat capacity at constant volume, and $\Delta T$ is the temperature increment we want to find. We use $C_V$ because the volume of the rigid box remains constant during the sudden stop.

To solve the equation, we need to express $n$ and $C_V$ in terms of the given variables. The number of moles $n$ is the total mass $m$ divided by the molar mass $M$ ($n=\frac{m}{M}$). Furthermore, for a monoatomic gas, the molar heat capacity at constant volume is $C_V=\frac{3}{2}R$.

Now, we equate the lost kinetic energy to the gained internal energy:

Notice how the total mass $m$ beautifully cancels out from both sides! This tells us that the temperature rise depends only on the nature of the gas (its molar mass) and how fast it was moving, not on how much gas is actually in the box. Rearranging the equation to solve for $\Delta T$, we get:

And there we have it—an elegant result demonstrating the seamless bridge between classical mechanics and thermodynamics.