Question

The rms speed of oxygen molecule in a vessel at particular temperature is \((1+\frac{5}{x})^{\frac{1}{2}} v\), where \(v\) is the average speed of the molecule. The value of x will be : (Take \(π = \frac{22}{7}\))

• A. 4
• B. 8
• C. 27
• D. 28

Hint:

Remember that the relationship between rms speed and average speed for molecules can involve constants such as \( \pi \), and care should be taken when substituting numerical values.

Answer 1

The Correct Option is D

To find the value of \( x \) such that the root mean square (rms) speed of an oxygen molecule in a vessel at a particular temperature is given by \((1+\frac{5}{x})^{\frac{1}{2}} v\), where \( v \) is the average speed, let's follow these steps:

1. We know the relation between the rms speed \((v_{\text{rms}})\), the average speed \((v_{\text{avg}})\), and the most probable speed \((v_{\text{mp}})\) for any gas:

   • \(v_{\text{rms}} = \sqrt{\frac{3kT}{m}}\)
   • \(v_{\text{avg}} = \sqrt{\frac{8kT}{\pi m}}\)

   where \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of a molecule.

2. Given the rms speed of an oxygen molecule is \((1+\frac{5}{x})^{\frac{1}{2}} v\).

3. Express the given relation in terms of known speed ratios:

   \(v_{\text{rms}} = \sqrt{\frac{3}{8/\pi}} v_{\text{avg}}\) because the ratio \(\frac{v_{\text{rms}}}{v_{\text{avg}}} = \sqrt{\frac{3\pi}{8}}\).

   This implies:

   \((1+\frac{5}{x})^{\frac{1}{2}} v = \sqrt{\frac{3\pi}{8}} v\)

4. Since \(v\) cancels from both sides, equate and solve:

   \((1+\frac{5}{x})^{\frac{1}{2}} = \sqrt{\frac{3\pi}{8}}\)

   By squaring both sides, we get:

   \(1+\frac{5}{x} = \frac{3\pi}{8}\)

5. Using \(\pi = \frac{22}{7}\), substitute into the equation:

   \(1+\frac{5}{x} = \frac{3 \times \frac{22}{7}}{8}\)

   \(1+\frac{5}{x} = \frac{66}{56}\)

   \(1+\frac{5}{x} = \frac{33}{28}\)

6. Rearrange the equation for \(x\):

   \(\frac{5}{x} = \frac{33}{28} - 1\)

   \(\frac{5}{x} = \frac{33}{28} - \frac{28}{28}\)

   \(\frac{5}{x} = \frac{5}{28}\)

   Solving for \(x\), we get:

   \(x = 28\)

7. Therefore, the value of \( x \) is 28, which is the correct option.