Animated Solution for Physics - Thermodynamics: The plot that depicts the behaviour of the mean free time $\tau$ (time between two successive collisions) for the molecules of an ideal gas, as a function of temperature ($T$), qualitatively is (graphs are schematic and not drawn to scale)

Imagine you are inside a sealed container filled with an ideal gas. It is absolute chaos! Millions of molecules are zipping around at breakneck speeds, constantly crashing into one another.

In this microscopic demolition derby, the distance a molecule travels between two consecutive crashes is called the mean free path ($\lambda$).

But how long does it take, on average, for a molecule to experience its next collision? This crucial time interval is known as the mean free time ($\tau$).

To find the mean free time, we rely on a simple kinematic principle: time equals distance divided by speed.

Here, $v_{\text{av}}$ represents the average speed of the gas molecules. To understand how $\tau$ changes with temperature ($T$), we need to analyze how both $\lambda$ and $v_{\text{av}}$ depend on $T$.

The formula for the mean free path is given by:

where $d$ is the diameter of the molecule and $n$ is the number density (number of molecules per unit volume, $N/V$).

Here is the critical catch: In a standard closed container, the volume $V$ is fixed. Since the total number of molecules $N$ is also fixed, the number density $n$ remains perfectly constant. Therefore, the mean free path $\lambda$ is completely independent of temperature!

Next, let's look at the average speed of the molecules. According to the kinetic theory of gases, the average speed is directly tied to the thermal energy:

This tells us that the average speed is directly proportional to the square root of the absolute temperature ($v_{\text{av}}\propto \sqrt{T}$). As the gas gets hotter, the molecules move faster.

Now, let's substitute our findings back into the master equation for mean free time. Since $\lambda$ is a constant and $v_{\text{av}}$ is proportional to $\sqrt{T}$, we get:

This is a beautiful inverse square-root relationship! Mathematically, if we plot $\tau$ on the y-axis and $\frac{1}{\sqrt{T}}$ on the x-axis, the relationship takes the form of $y=mx$, which is the equation of a straight line passing through the origin.

Looking at our options, the graph that perfectly depicts this linear relationship is option (c).

Pro-Tip: Always check the constraints! If the problem had stated that the gas was expanding at a constant pressure instead of a constant volume, the density would decrease as temperature increased. This would cause the mean free path to increase, completely changing the final graph. Always read the fine print!