Which graph correctly represents variation between relaxation time (t) of gas molecules with absolute temperature of gas T.

Options

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Answer

To solve the problem of determining the correct graph that represents the variation of relaxation time (τ) of gas molecules with the absolute temperature (T) of the gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Relaxation Time (τ)**:
- Relaxation time (τ) is defined as the average time between two successive collisions of gas molecules.

2. **Mean Free Path (λ)**:
- The mean free path (λ) can be expressed as:
\[ \lambda = \frac{1}{\sqrt{2 \pi b^2 n}} \]
- Here, \(b\) is the effective diameter of the gas molecules and \(n\) is the number of molecules per unit volume.

3. **Relationship Between Relaxation Time and Mean Free Path**:
- The relaxation time can be expressed in terms of mean free path and velocity (v):
\[ \tau = \frac{\lambda}{v} \]

4. **Velocity of Gas Molecules**:
- The average velocity of gas molecules is related to temperature (T) by the equation:
\[ v \propto \sqrt{T} \]
- Therefore, we can express the velocity as:
\[ v = k \sqrt{T} \]
- where \(k\) is a constant.

5. **Substituting Velocity in Relaxation Time Equation**:
- Substituting the expression for velocity into the relaxation time equation gives:
\[ \tau = \frac{\lambda}{k \sqrt{T}} \]

6. **Expressing Relaxation Time in Terms of Temperature**:
- From the above equation, we can see that:
\[ \tau \propto \frac{1}{\sqrt{T}} \]
- This indicates that relaxation time is inversely proportional to the square root of the absolute temperature.

7. **Graphical Representation**:
- The relationship τ ∝ 1/√T suggests that if we plot τ on the y-axis and 1/√T on the x-axis, we would expect a linear graph.
- Therefore, the correct graph would be a straight line that decreases as 1/√T increases.

### Conclusion:
The graph that correctly represents the variation of relaxation time (τ) of gas molecules with the absolute temperature (T) is a decreasing curve, indicating that as the temperature increases, the relaxation time decreases.