Consider an ideal gas confined in an isolated closed chamber . As gas undergoes an adiabatic expansion the average time of collision between the molecule increases as v^q where v is volume of the gas . The value of q is :

In the scenario of an ideal gas undergoing adiabatic expansion in an isolated closed chamber, the relationship between the average time of collision between molecules and the volume of the gas is quite interesting. The question you've posed is about determining the value of $q$ in the relationship where the average time of collision increases as $v^q$. Let's break this down step by step.

The Basics of Adiabatic Expansion

First, it's essential to understand what adiabatic expansion means. In an adiabatic process, there is no heat exchange with the surroundings. This means that any work done by the gas during expansion comes at the expense of its internal energy, leading to a drop in temperature. The rapid movement and separation of gas molecules during this process play a significant role in collision dynamics.

Molecular Collisions and Volume

Now, when we talk about the average time between collisions of gas molecules, we need to consider how gas density and volume influence this. In a closed chamber, as the volume $V$ of the gas increases during adiabatic expansion, the number of gas molecules per unit volume decreases. This reduction in density means that the average distance between molecules increases, which in turn affects how often they collide.

Mathematical Relationship

The average time between collisions can be related to the volume of the gas using kinetic theory. The time between collisions $\tau$ is proportional to the average distance between molecules divided by their average speed. As the volume increases, the average distance traveled by a molecule before colliding with another molecule becomes larger. This can be expressed as:

• $\tau \propto \frac{V^{1/3}}{v}$

Here, $V^{1/3}$ represents the increase in the characteristic length scale (or average distance) as the volume expands, and $v$ is the average speed of the gas molecules. In an adiabatic process, the average speed of molecules decreases due to the drop in temperature.

Determining $q$

From the relationship derived above, we can see that the average time between collisions scales with volume. However, the exact value of $q$ in the relationship $\tau \propto V^q$ can be inferred from the dependence of average collision time on volume. Given that the volume affects the average distance between molecules significantly, we can conclude that:

• As volume increases, the time between collisions increases as $V^{1/3}$.
• Thus, we deduce that $q=\frac{1}{3}$.

Final Thoughts

In summary, during adiabatic expansion of an ideal gas, the average time between molecular collisions increases as $V^{1/3}$. Therefore, the value of $q$ is $\frac{1}{3}$. This relationship highlights the fascinating interplay between volume, molecular dynamics, and thermodynamic processes in gases. Understanding these concepts not only deepens our grasp of gas behavior but also illustrates the principles governing molecular interactions in various states of matter.