Animated Solution for Physics - Thermodynamics: In a dilute gas at pressure $p$ and temperature $T$, the mean time between successive collisions of a molecule varies with $T$ as

Imagine you are a gas molecule trapped inside a container, zipping around at breakneck speeds. You don't travel in a straight line for long before you crash into another molecule.

This chaotic, zigzag motion is the essence of the kinetic theory of gases.

The average time you spend flying freely between these crashes is called the mean time between collisions, denoted by $\tau$.

Logically, this time depends on two things: how far you can travel on average before hitting something (the mean free path, $\lambda$), and how fast you are moving (the average speed, $v_{\text{avg}}$).

Mathematically, this is expressed as:

Let's break down the mean free path, $\lambda$. It represents the average distance a molecule covers between successive collisions.

The formula for the mean free path is:

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

If we assume the gas is kept in a rigid, closed container, the volume $V$ remains constant. Since the total number of molecules $N$ is also constant, the number density $n_V$ does not change.

Therefore, under constant volume conditions, the mean free path $\lambda$ is completely independent of temperature. We can write this proportionality as:

Now, what happens when we increase the temperature $T$ of the gas?

According to the kinetic theory, temperature is a direct measure of the average kinetic energy of the molecules. As the gas heats up, the molecules become more energetic and move faster.

The average speed of a gas molecule is given by:

From this equation, it is clear that the average speed is directly proportional to the square root of the absolute temperature:

Now, let's bring our two proportionalities back to the original equation for the mean collision time.

We know that $\tau =\frac{\lambda }{v_{\text{avg}}}$.

Substituting the temperature dependencies we just found:

Simplifying this, we get our final relationship:

This tells us that as the temperature increases, the molecules move faster, covering the same mean free path in less time. Hence, the time between collisions decreases.

Pressure vs. Volume

There is a subtle but critical trap in this specific problem that has sparked much debate among physicists and students alike.

The problem statement explicitly mentions "at pressure $p$ and temperature $T$". In thermodynamics, mentioning a variable like pressure often implies it is being held constant.

What if the pressure $p$ was actually constant instead of the volume?

From the ideal gas law, $p=n_V{k}_{B}T$, we can see that if $p$ is constant, the number density $n_V$ must be inversely proportional to temperature ($n_V\propto \frac{1}{T}$).

If $n_V\propto \frac{1}{T}$, then the mean free path $\lambda$ would be directly proportional to temperature ($\lambda \propto T$).

Substituting this into our time equation would yield:

In this scenario, the mean collision time would actually increase with temperature!

However, the standard accepted solution for this specific JEE Main question assumes a closed container where volume is constant, leading to the official answer of $\frac{1}{\sqrt{T}}$. Always be prepared to analyze the context of the question!