Animated Solution for Physics - Thermodynamics: Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as $V^q$, where $V$ is the volume of the gas. The value of $q$ is $\left(\gamma =\frac{C_p}{C_V}\right)$

Collision Time in Adiabatic Expansion

Imagine you are shrunk down to the size of an atom, standing inside an isolated chamber filled with an ideal gas. All around you, gas molecules are zipping past, colliding with each other in a chaotic, never-ending dance. The average time a molecule spends flying freely before it crashes into another is called the mean free time or average collision time, denoted by $\tau$.

But what happens to this collision time if the chamber suddenly starts to expand adiabatically? Let's break down the physics and the math behind this fascinating phenomenon.

To understand how $\tau$ changes, we first need to look at its mathematical definition derived from the Kinetic Theory of Gases:

Here, $d$ is the diameter of the molecule, $n$ is the number density (number of molecules per unit volume), and $v_{\text{rms}}$ is the root mean square velocity of the molecules.

We need to express this microscopic collision time in terms of macroscopic thermodynamic variables like Volume ($V$) and Temperature ($T$).

First, let's look at the number density $n$. Since the chamber is closed, the total number of molecules $N$ is constant. Therefore, $n=\frac{N}{V}$, which means $n$ is inversely proportional to the volume:

Next, the kinetic energy of the gas tells us that the rms velocity depends purely on the temperature. Specifically, $v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$, which gives us:

Substituting these proportionalities back into our formula for $\tau$, we get a beautiful relationship:

The problem states that the gas undergoes an adiabatic expansion. This means no heat enters or leaves the system. As the gas expands and does work, its internal energy drops, causing the temperature to fall.

For an adiabatic process, the relationship between temperature and volume is strictly governed by the equation:

This allows us to express the temperature entirely in terms of the volume:

Now, we substitute this temperature dependence back into our proportionality for $\tau$. We replace $T$ with $V^{1-\gamma }$:

To simplify this, we use the basic laws of exponents. When dividing terms with the same base, we subtract the exponents:

The question tells us that $\tau$ increases as $V^q$. By comparing our derived expression with $V^q$, the answer reveals itself elegantly:

Through the lens of thermodynamics, we can perfectly predict the microscopic behavior of billions of colliding molecules just by knowing how the macroscopic volume changes. That is the true power of physics!