If the volume of a gas is doubled either isothermally or adiabatically, in which case change in mean free path is more.

Options

A. Adiabatic
B. Isothermal
C. Same in both
D. No change

Explanation

To solve the problem of determining in which case the change in mean free path is more when the volume of a gas is doubled either isothermally or adiabatically, we can follow these steps:

### Step 1: Understand Mean Free Path
The mean free path (λ) is defined as the average distance traveled by a gas molecule between collisions. It is given by the formula:
\[ \lambda = \frac{V}{\sqrt{2} \pi d^2 n} \]
where:
- \(V\) = volume of the gas
- \(d\) = diameter of the gas molecules
- \(n\) = number density of the gas molecules (number of molecules per unit volume)

### Step 2: Analyze the Isothermal Process
In an isothermal process, the temperature of the gas remains constant. When the volume is doubled (from \(V\) to \(2V\)), the number density \(n\) will change because the number of molecules remains constant while the volume increases.

The number density \(n\) is given by:
\[ n = \frac{N}{V} \]
where \(N\) is the number of molecules.

When the volume doubles:
\[ n_{\text{final}} = \frac{N}{2V} = \frac{n_{\text{initial}}}{2} \]

Substituting this into the mean free path formula:
\[ \lambda_{\text{isothermal}} = \frac{2V}{\sqrt{2} \pi d^2 \left(\frac{n_{\text{initial}}}{2}\right)} = \frac{4V}{\sqrt{2} \pi d^2 n_{\text{initial}}} \]

### Step 3: Analyze the Adiabatic Process
In an adiabatic process, the gas expands without heat exchange. When the volume is doubled, the temperature of the gas decreases. The relationship between volume and temperature in an ideal gas during adiabatic expansion can be expressed as:
\[ TV^{\gamma-1} = \text{constant} \]
where \(\gamma\) is the heat capacity ratio.

As the volume doubles, the temperature drops, which will affect the mean free path. However, the number of molecules remains constant, so the number density \(n\) will also decrease in a similar manner as in the isothermal case.

### Step 4: Compare Changes in Mean Free Path
For both processes, we can see that the mean free path increases when the volume is doubled. However, in the isothermal case, the number density decreases more significantly due to the constant temperature condition, while in the adiabatic case, the temperature drop also influences the mean free path.

### Conclusion
The mean free path increases in both cases, but the rate of increase is more pronounced in the isothermal case due to the more significant reduction in number density. Therefore, the change in mean free path is more in the isothermal process.

### Final Answer
The change in mean free path is more in the isothermal process.

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