Question

Three vessels of equal volume contain gases at the same temperature and pressure. The first vessel contains neon (monoatomic), the second contains chlorine (diatomic) and the third contains uranium hexafluoride (polyatomic). Arrange these on the basis of their root mean square speed (\( V_{\text{rms}} \)) and choose the correct answer from the options given below: 

• A. \( V_{\text{rms}} \text{(mono)} > V_{\text{rms}} \text{(dia)} > V_{\text{rms}} \text{(poly)} \)
• B. \( V_{\text{rms}} \text{(dia)} < V_{\text{rms}} \text{(poly)} < V_{\text{rms}} \text{(mono)} \)
• C. \( V_{\text{rms}} \text{(mono)} < V_{\text{rms}} \text{(dia)} < V_{\text{rms}} \text{(poly)} \)
• D. \( V_{\text{rms}} \text{(mono)} = V_{\text{rms}} \text{(dia)} = V_{\text{rms}} \text{(poly)} \)

Answer 1

The Correct Option is A

To solve this problem, we need to determine the root mean square speed (\(V_{\text{rms}}\)) of the gases in each vessel. The root mean square speed of a gas is given by the formula:

\(V_{\text{rms}} = \sqrt{\frac{3kT}{m}}\)

where:

• \(k\) is the Boltzmann constant.
• \(T\) is the absolute temperature of the gas.
• \(m\) is the molar mass of the gas.

Since the temperature \(T\) is constant for all gases, the \(V_{\text{rms}}\) is inversely proportional to the square root of the molar mass: \(V_{\text{rms}} \propto \frac{1}{\sqrt{m}}\).

Let's consider the gases in the vessels:

1. Neon (monoatomic gas): Atomic mass = 20 g/mol
2. Chlorine (diatomic gas): Molecular mass = 71 g/mol (approximate)
3. Uranium hexafluoride (polyatomic gas): Molecular mass = 352 g/mol

To find the order of \(V_{\text{rms}}\) based on their masses:

• Neon (\(m = 20\)): Has the smallest molar mass, so it will have the highest \(V_{\text{rms}}\).
• Chlorine (\(m = 71\)): Has a greater molar mass than neon but less than uranium hexafluoride, resulting in a moderate \(V_{\text{rms}}\).
• Uranium hexafluoride (\(m = 352\)): Has the largest molar mass, so it will have the lowest \(V_{\text{rms}}\).

Therefore, the correct order based on the \(V_{\text{rms}}\) is: \(V_{\text{rms}} \text{(mono)} > V_{\text{rms}} \text{(dia)} > V_{\text{rms}} \text{(poly)}\).

The correct answer is:

\(V_{\text{rms}} \text{(mono)} > V_{\text{rms}} \text{(dia)} > V_{\text{rms}} \text{(poly)}\)