Question

Equal volumes of two gases are kept in different containers having densities in the ratio $1 : 16$. They exert equal pressures on the wall of their respective containers. Then the ratio of their r.m.s. velocities is

• A. $16 : 1$
• B. $1 : 8$
• C. $4 : 1$
• D. $1 : 12$

Hint:

Since velocity is inversely proportional to the square root of density, the less dense gas must move faster to generate the exact same pressure. Because its density is 16 times smaller, its velocity must be $\sqrt{16} = 4$ times larger, instantly pointing you to a ratio starting with the larger number.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem presents two different gases contained within identical volumes under uniform pressure conditions. We are given the ratio of their mass densities ($\rho_1 : \rho_2 = 1 : 16$) and need to compute the ratio of their root-mean-square velocities ($v_{\text{rms1}} : v_{\text{rms2}}$).

Step 2: Key Formula or Approach:
According to the Kinetic Theory of Gases, the pressure exerted by an ideal gas is linked to its density and r.m.s. velocity via the formula: $$P = \frac{1}{3}\rho v_{\text{rms}}^2 \implies v_{\text{rms}} = \sqrt{\frac{3P}{\rho}}$$ Given that the pressure $P$ is identical for both gas containers, the root-mean-square velocity scales inversely with the square root of the gas density: $$v_{\text{rms}} \propto \frac{1}{\sqrt{\rho}} \implies \frac{v_{\text{rms1}}}{v_{\text{rms2}}} = \sqrt{\frac{\rho_2}{\rho_1}}$$

Step 3: Detailed Explanation:
Extract the density ratio information provided in the problem statement: $$\frac{\rho_1}{\rho_2} = \frac{1}{16} \implies \frac{\rho_2}{\rho_1} = \frac{16}{1}$$ Substitute this inverted ratio directly into our derived proportionality formula: $$\frac{v_{\text{rms1}}}{v_{\text{rms2}}} = \sqrt{\frac{16}{1}}$$ Evaluate the square root of the numeric terms: $$\frac{v_{\text{rms1}}}{v_{\text{rms2}}} = \frac{4}{1} \implies 4 : 1$$

Step 4: Final Answer:
The ratio of their root-mean-square velocities is $4 : 1$, which perfectly matches option (C).