Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increase as `V^q`, where V is the volume of the gas. The value of q is : `(gamma=(C_p)/(C_v))`

Options

A. `(3gamma - 5)/(6)`
B. `(gamma + 1)/(2)`
C. `(gamma - 1)/(2)`
D. `(3 gamma + 5)/(6)`

Answer

To solve the problem, we need to determine the value of \( q \) in the expression for the average time of collision \( \tau \) between molecules of an ideal gas undergoing adiabatic expansion. The average time of collision increases as \( V^q \), where \( V \) is the volume of the gas.

### Step-by-step Solution:

1. **Understanding Adiabatic Process**:
In an adiabatic process, the relationship between pressure \( P \), volume \( V \), and the heat capacities \( C_p \) and \( C_v \) is given by:
\[ PV^\gamma = \text{constant} \]
where \( \gamma = \frac{C_p}{C_v} \).

2. **Mean Free Path and RMS Speed**:
The average time of collision \( \tau \) can be expressed in terms of the mean free path \( \lambda \) and the root mean square (RMS) speed \( V_{RMS} \):
\[ \tau = \frac{\lambda}{V_{RMS}} \]

3. **Mean Free Path**:
The mean free path \( \lambda \) is proportional to the volume \( V \):
\[ \lambda \propto V \]

4. **RMS Speed**:
The RMS speed \( V_{RMS} \) can be derived from the ideal gas law. We know:
\[ V_{RMS} = \sqrt{\frac{3RT}{M}} \]
Using the ideal gas law \( PV = nRT \), we can express \( RT \) as:
\[ RT = \frac{PV}{n} \]
Therefore, we can rewrite \( V_{RMS} \):
\[ V_{RMS} = \sqrt{\frac{3PV}{M}} \]

5. **Substituting into the Time of Collision**:
Now substituting \( \lambda \) and \( V_{RMS} \) into the expression for \( \tau \):
\[ \tau \propto \frac{V}{\sqrt{3PV}} \]
Simplifying this gives:
\[ \tau \propto \frac{V}{\sqrt{3P}} \cdot \frac{1}{\sqrt{V}} = \frac{V^{1/2}}{\sqrt{3P}} \]

6. **Expressing Pressure in Terms of Volume**:
From the adiabatic condition \( PV^\gamma = C \), we can express pressure \( P \) in terms of volume \( V \):
\[ P = \frac{C}{V^\gamma} \]
Substituting this into the expression for \( \tau \):
\[ \tau \propto \frac{V^{1/2}}{\sqrt{3 \cdot \frac{C}{V^\gamma}}} = \frac{V^{1/2} \cdot V^{\gamma/2}}{\sqrt{3C}} = \frac{V^{(1+\gamma)/2}}{\sqrt{3C}} \]

7. **Identifying the Value of \( q \)**:
From the final expression, we can see that:
\[ \tau \propto V^{(1+\gamma)/2} \]
Thus, we can identify \( q \) as:
\[ q = \frac{1 + \gamma}{2} \]

### Final Answer:
The value of \( q \) is:
\[ q = \frac{1 + \gamma}{2} \]