Import Question JSON

$\text{Hint: } V_{\text{rms}} = \sqrt{\frac{3kT}{m}}$ $\text{Step 1: Find the correct ratio of the pressures.}$ $\text{Let } P_1, N_1 \text{ be the pressure and the number of molecules of the gas in the first vessel, and } P_2, N_2 \text{ be the pressure and the number of molecules of the gas in the second vessel.}$ $\text{Given the ratio of the number of molecules;} \Rightarrow \frac{N_1}{N_2} = \frac{1}{4} \Rightarrow N_2 = 4N_1$ $\text{The ideal gas law states that;} PV = NkT \quad \ldots (1)$ $\text{where } P \text{ is the absolute pressure of a gas, } V \text{ is the volume it occupies,}$ $N \text{ is the number of atoms and molecules in the gas, and } T \text{ is its absolute temperature,}$ $k \text{ is the Boltzmann constant.}$ $\text{Using (1), the ratio of the pressures is given by;}$ $\Rightarrow \frac{P_1V}{P_2V} = \frac{N_1kT}{N_2kT} \Rightarrow \frac{P_1}{P_2} = \frac{N_1}{N_2} = \frac{1}{4}$ $\text{Step 2: Find the correct ratio of the } V_{\text{rms}}.$ $\text{Let } V_1, V_2 \text{ be the RMS velocity of the gas in the first and the second vessel.}$ $\text{For an ideal gas, the RMS velocity is given by;}$ $V_{\text{rms}} = \sqrt{\frac{3kT}{m}} \quad \ldots (2)$ $\text{where } T \text{ is its absolute temperature, } k \text{ is the Boltzmann constant and } m \text{ the mass of one gas molecule.}$ $\text{Using (2), the ratio of the } V_{\text{rms}} \text{ is given by;}$ $\Rightarrow \frac{V_1}{V_2} = \frac{\sqrt{\frac{3kT}{m}}}{\sqrt{\frac{3kT}{m}}} = \frac{1}{1}$ $\text{Therefore, (A) and (B) are the valid statements.}$ $\text{Hence, option (3) is the correct answer.}$