The rms speed of oxygen molecule in a vessel at particular temmperature is `(1+5/x)^(1/2)v` ,where v is the average speed of the molecule .the value of x will be : Given escape velocity on earth `V_e =11.2times 10^3 m/s)

Options

A. `27`
B. `8`
C. `4`
D. `28`

Answer

To solve the problem, we need to find the value of \( x \) given the relationship between the root mean square (rms) speed of an oxygen molecule and its average speed.

1. **Understanding the given relationship**:
The rms speed of an oxygen molecule is given as:
\[ v_{\text{rms}} = \left(1 + \frac{5}{x}\right)^{1/2} v \]
where \( v \) is the average speed of the molecule.

2. **Using the formulas for rms speed and average speed**:
The formula for the rms speed \( v_{\text{rms}} \) is:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
and the average speed \( v \) is given by:
\[ v = \sqrt{\frac{8RT}{\pi M}} \]

3. **Setting up the equation**:
We can equate the two expressions for rms speed:
\[ \sqrt{\frac{3RT}{M}} = \left(1 + \frac{5}{x}\right)^{1/2} \sqrt{\frac{8RT}{\pi M}} \]

4. **Canceling common terms**:
Since \( \sqrt{\frac{RT}{M}} \) is common on both sides, we can simplify:
\[ \sqrt{3} = \left(1 + \frac{5}{x}\right)^{1/2} \sqrt{\frac{8}{\pi}} \]

5. **Squaring both sides**:
Squaring both sides to eliminate the square root gives:
\[ 3 = \left(1 + \frac{5}{x}\right) \frac{8}{\pi} \]

6. **Rearranging the equation**:
Rearranging this equation, we get:
\[ 1 + \frac{5}{x} = \frac{3\pi}{8} \]

7. **Solving for \( \frac{5}{x} \)**:
Subtracting 1 from both sides:
\[ \frac{5}{x} = \frac{3\pi}{8} - 1 \]
To combine the terms on the right, we convert 1 to a fraction:
\[ 1 = \frac{8}{8} \implies \frac{5}{x} = \frac{3\pi - 8}{8} \]

8. **Finding \( x \)**:
Inverting both sides gives:
\[ x = \frac{5 \cdot 8}{3\pi - 8} = \frac{40}{3\pi - 8} \]

9. **Substituting \( \pi \)**:
Using \( \pi \approx \frac{22}{7} \):
\[ x = \frac{40}{3 \cdot \frac{22}{7} - 8} = \frac{40}{\frac{66}{7} - 8} \]
Converting 8 to a fraction:
\[ 8 = \frac{56}{7} \implies x = \frac{40}{\frac{66 - 56}{7}} = \frac{40 \cdot 7}{10} = 28 \]

Thus, the value of \( x \) is \( 28 \).