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

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which graph correctly represents the variation between the relaxation time (τ) of gas molecules and the absolute temperature (T) of the gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Relaxation Time**:
- Relaxation time (τ) is defined as the mean free path (λ) divided by the velocity (v) of the gas molecules.
\[ \tau = \frac{\lambda}{v} \]

2. **Mean Free Path Relation**:
- The mean free path (λ) can be expressed as:
\[ \lambda = \frac{1}{\sqrt{2 \pi n d^2}} \]
- Here, \(n\) is the number density of gas molecules, and \(d\) is the diameter of the molecules. This equation does not directly relate λ to temperature (T).

3. **Velocity Relation**:
- The velocity (v) of gas molecules is related to the absolute temperature (T) by the equation:
\[ v \propto \sqrt{T} \]
- This means that as the temperature increases, the velocity of the gas molecules also increases.

4. **Substituting Velocity into Relaxation Time**:
- Since τ is inversely proportional to velocity, we can write:
\[ \tau \propto \frac{1}{v} \]
- Substituting the relation of velocity with temperature, we get:
\[ \tau \propto \frac{1}{\sqrt{T}} \]

5. **Graphical Representation**:
- The relationship \(τ \propto \frac{1}{\sqrt{T}}\) indicates that as the absolute temperature (T) increases, the relaxation time (τ) decreases. This suggests that the graph of τ versus T will be a decreasing curve.
- Specifically, if we plot τ against \(1/\sqrt{T}\), we expect a linear relationship.

6. **Conclusion**:
- Based on the analysis, the graph that correctly represents the variation of relaxation time (τ) with absolute temperature (T) is the one where τ decreases as T increases, which corresponds to the graph labeled as A.

### Final Answer:
The correct graph is **Graph A**, which represents the variation between relaxation time (τ) of gas molecules and absolute temperature (T) of the gas.