The volume of a gas and the number of gas molecules within that volume for four situations are
(1) `2V_(0)andN_(0)` (2) `3V_(0)and3N_(0)`
(3) `8V_(0)and4N_(0)` (4) `3V_(0)and9N_(0)`
Which of them has mean free path greatest?

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To determine which situation has the greatest mean free path, we need to analyze the mean free path formula and the given conditions.

### Step-by-Step Solution:

1. **Understand the Mean Free Path Formula**:
The mean free path (λ) can be expressed as:
\[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n} \]
where:
- \(d\) is the diameter of the gas molecules (assumed constant for all cases),
- \(n\) is the number of molecules per unit volume.

2. **Calculate the Number of Molecules per Unit Volume (n)**:
For each case, we will calculate \(n\) using the formula:
\[ n = \frac{N}{V} \]
where \(N\) is the number of molecules and \(V\) is the volume.

- **Case 1**: \(N = N_0\), \(V = 2V_0\)
\[ n_1 = \frac{N_0}{2V_0} = \frac{1}{2} \frac{N_0}{V_0} \]

- **Case 2**: \(N = 3N_0\), \(V = 3V_0\)
\[ n_2 = \frac{3N_0}{3V_0} = \frac{N_0}{V_0} \]

- **Case 3**: \(N = 4N_0\), \(V = 8V_0\)
\[ n_3 = \frac{4N_0}{8V_0} = \frac{1}{2} \frac{N_0}{V_0} \]

- **Case 4**: \(N = 9N_0\), \(V = 3V_0\)
\[ n_4 = \frac{9N_0}{3V_0} = 3 \frac{N_0}{V_0} \]

3. **Compare the Values of n**:
- \(n_1 = \frac{1}{2} \frac{N_0}{V_0}\)
- \(n_2 = \frac{N_0}{V_0}\)
- \(n_3 = \frac{1}{2} \frac{N_0}{V_0}\)
- \(n_4 = 3 \frac{N_0}{V_0}\)

From this, we can see:
- Cases 1 and 3 have the smallest \(n\) values (\(\frac{1}{2} \frac{N_0}{V_0}\)).
- Case 2 has a medium \(n\) value (\(\frac{N_0}{V_0}\)).
- Case 4 has the largest \(n\) value (\(3 \frac{N_0}{V_0}\)).

4. **Determine the Mean Free Path**:
Since the mean free path is inversely proportional to \(n\):
- The smaller the value of \(n\), the larger the mean free path.
- Therefore, cases 1 and 3, which have the smallest \(n\), will have the largest mean free path.

5. **Conclusion**:
The situations with the greatest mean free path are Case 1 and Case 3.

### Final Answer:
The situations with the greatest mean free path are **Case 1** and **Case 3**.