Question: 43: Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains helium (monoatomic), the second contains fluorine (diatomic) and the third contains sulfur hexafluoride (polyatomic). The correct statement, among the following is:
(1) All vessels contain unequal number of respective molecules
(2) The root mean square speed of molecules is same in all three cases
(3) The root mean square speed of helium is the largest
(4) The root mean square speed of sulfur hexafluoride is the largest
Answer: Option (3)
Explanation:
To determine the root mean square (rms) speed of gas molecules, we use the formula:
v_\text{rms} = \sqrt{\frac{3 k_B T}{m}}where k_B is the Boltzmann constant, T is the absolute temperature,
and m is the mass of one molecule of the gas.
Given that all vessels have the same temperature T and pressure P,
the rms speed depends inversely on the square root of molecular mass:
v_\text{rms} \propto \frac{1}{\sqrt{m}}Helium (He) is monoatomic with very small molecular mass
(m_\text{He} \approx 4 \, \text{u}).
Fluorine (F2) is diatomic with molecular mass
(m_\text{F2} \approx 38 \, \text{u}).
Sulfur hexafluoride (SF6) is polyatomic with molecular mass
(m_\text{SF6} \approx 146 \, \text{u}).
Since helium has the smallest molecular mass, its molecules move the fastest.
Therefore, the rms speed of helium is the largest among the three gases.
Hence, the correct statement is: The root mean square speed of helium is the largest.