Two gases-argon (atomic radius `0.07` nm, atomic weight 40) and xenon (atomic radius `0.1` nm, atomic weight 140) have the same number density and are at the same temperature. The ratio of their respective mean free times is closest to :

Options

A. `1.83`
B. `4.67`
C. `3.67`
D. `2.3`

Answer

To find the ratio of the mean free times of argon and xenon, we can follow these steps:

### Step 1: Understand the Mean Free Time Formula
The mean free time (τ) is given by the formula:
\[ \tau = \frac{\lambda}{v} \]
where \( \lambda \) is the mean free path and \( v \) is the average speed of the gas molecules.

### Step 2: Mean Free Path Calculation
The mean free path \( \lambda \) can be expressed as:
\[ \lambda = \frac{1}{\sqrt{2} \pi n d^2} \]
where \( n \) is the number density and \( d \) is the diameter of the gas molecules. Since both gases have the same number density, we can simplify our calculations.

### Step 3: Average Speed Calculation
The average speed \( v \) of the gas is given by:
\[ v = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

### Step 4: Ratio of Mean Free Times
From the formulas, we can express the ratio of mean free times for argon (τₐ) and xenon (τₓ):
\[ \frac{\tau_a}{\tau_x} = \frac{\lambda_a / v_a}{\lambda_x / v_x} = \frac{\lambda_a}{\lambda_x} \cdot \frac{v_x}{v_a} \]

### Step 5: Substitute Mean Free Path and Speed
Since the number density is the same for both gases, we can express the mean free path ratio as:
\[ \frac{\lambda_a}{\lambda_x} = \frac{d_x^2}{d_a^2} \]
And for the speeds:
\[ \frac{v_a}{v_x} = \sqrt{\frac{M_x}{M_a}} \]

### Step 6: Combine the Ratios
Thus, we can write:
\[ \frac{\tau_a}{\tau_x} = \frac{d_x^2}{d_a^2} \cdot \sqrt{\frac{M_x}{M_a}} \]

### Step 7: Insert Values
Now, we substitute the values:
- For argon: \( d_a = 0.07 \, \text{nm} \), \( M_a = 40 \)
- For xenon: \( d_x = 0.1 \, \text{nm} \), \( M_x = 140 \)

Calculating the diameter ratio:
\[ \frac{d_x^2}{d_a^2} = \frac{(0.1)^2}{(0.07)^2} = \frac{0.01}{0.0049} \approx 2.04 \]

Calculating the mass ratio:
\[ \sqrt{\frac{M_x}{M_a}} = \sqrt{\frac{140}{40}} = \sqrt{3.5} \approx 1.87 \]

### Step 8: Final Calculation
Now we combine these results:
\[ \frac{\tau_a}{\tau_x} \approx 2.04 \cdot 1.87 \approx 3.82 \]

### Conclusion
The ratio of their respective mean free times is closest to:
\[ \frac{\tau_a}{\tau_x} \approx 3.82 \]