If nitrogen gas molecule goes straight up with its rms speed at `0^@ C` from the surface of the earth and there are no collisions with other molecules, then it will rise to an approximate height of.

Options

A. `8 km`
B. `12 km`
C. `12 m`
D. `8 m`

Explanation

To solve the problem of how high a nitrogen gas molecule can rise if it goes straight up with its root mean square (RMS) speed at 0°C, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the RMS Speed (Vrms) of Nitrogen Gas:**
The formula for the RMS speed is given by:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]
Where:
- \( R \) = Universal gas constant = \( 8.314 \, \text{J/(mol K)} \)
- \( T \) = Temperature in Kelvin = \( 0°C + 273 = 273 \, \text{K} \)
- \( M \) = Molar mass of nitrogen (N₂) = \( 28 \, \text{g/mol} = 28 \times 10^{-3} \, \text{kg/mol} \)

Plugging in the values:
\[ V_{rms} = \sqrt{\frac{3 \times 8.314 \times 273}{28 \times 10^{-3}}} \]

2. **Calculate the RMS Speed:**
- Calculate the numerator:
\[ 3 \times 8.314 \times 273 \approx 6810.642 \, \text{J/(kg)} \]
- Calculate the denominator:
\[ 28 \times 10^{-3} \approx 0.028 \, \text{kg} \]
- Now calculate:
\[ V_{rms} = \sqrt{\frac{6810.642}{0.028}} \approx \sqrt{243,000} \approx 493 \, \text{m/s} \]

3. **Use the Kinematic Equation to Find Maximum Height (H):**
The maximum height can be calculated using the formula:
\[ H = \frac{V^2}{2g} \]
Where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).

Plugging in the values:
\[ H = \frac{(493)^2}{2 \times 9.81} \]

4. **Calculate the Maximum Height:**
- Calculate \( (493)^2 \):
\[ 493^2 \approx 243,009 \]
- Now calculate:
\[ H = \frac{243,009}{19.62} \approx 12,394 \, \text{m} \approx 12.4 \, \text{km} \]

5. **Consider the Effect of Decreasing g with Height:**
Since the height is significant, the value of \( g \) will decrease slightly with height. Therefore, we can conclude that the approximate height the nitrogen molecule can rise is around \( 12 \, \text{km} \).

### Final Answer:
The nitrogen gas molecule will rise to an approximate height of **12 km**.