The plot that depicts the behavior 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)

Minimum Required Theory

• Mean Free Path ($$\lambda$$): The average distance a molecule travels between two successive collisions. For an ideal gas at constant volume/density ($$n$$), $$\lambda$$ is constant and independent of temperature:
• Average Speed ($$v_{avg}$$): The average speed of gas molecules depends on the absolute temperature ($$T$$):
• Mean Free Time ($$\tau$$): The average time interval between two successive collisions is given by the ratio of the mean free path to the average speed:
• As $$T \to 0$$: The value of $$\tau \to \infty$$. When the temperature is close to absolute zero, molecular motion slows down drastically, meaning collisions happen very infrequently.
• As $$T$$ increases: The denominator $$\sqrt{T}$$ grows, causing the mean free time $$\tau$$ to decrease rapidly.
• As $$T \to \infty$$: The value of $$\tau \to 0$$. At extremely high temperatures, molecules move exceptionally fast, causing collisions to occur almost instantaneously.
• Y-axis: Mean free time ($$\tau$$)
• X-axis: Temperature ($$T$$)
• Curve: A downward-sloping curve starting from a high value at low temperatures and asymptotically approaching the horizontal axis at high temperatures.

$$\lambda = \frac{1}{\sqrt{2}\pi d^2 n}$$

$$v_{avg} = \sqrt{\frac{8RT}{\pi M}} \implies v_{avg} \propto \sqrt{T}$$

$$\tau = \frac{\lambda}{v_{avg}}$$

Step-by-Step Solution

Step 1: Express $$\tau$$ as a function of Temperature ($$T$$)

Since the mean free path $$\lambda$$ remains independent of temperature for a fixed sample of gas, substitute the temperature dependence of $$v_{avg}$$ into the mean free time equation:

$$\tau \propto \frac{1}{\sqrt{T}}$$

Step 2: Analyze the behavior of the curve

Step 3: Identify the shape of the graph

The relation $$\tau \propto \frac{1}{\sqrt{T}}$$ produces a smoothly decreasing, non-linear asymptotic curve.

Unlike a standard rectangular hyperbola ($$y \propto \frac{1}{x}$$), this curve drops steeply at first and gradually flattens out along the temperature axis (X-axis) as $$T$$ increases.

The Correct Graph

The correct schematic plot features: