Question

Equal volumes of two gases, having their densities in the ratio of $1 : 16$ exert equal pressures on the walls of two containers. The ratio of their rms speeds ($C_1 : C_2$) is

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

Hint:

Whenever pressure is constant, remember the handy shortcut $C \propto \frac{1}{\sqrt{\rho}}$. Because Gas 2 is $16$ times denser than Gas 1, its molecules are heavier and move slower. Taking the square root of $16$ immediately reveals that Gas 1 must move $4$ times faster than Gas 2.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Two different gases are kept in separate containers under identical volumes and identical pressures. Given the ratio of their mass densities, we need to calculate the ratio of their root-mean-square ($v_{\text{rms}}$ or $C$) molecular speeds.

Step 2: Key Formula or Approach:
The kinetic theory of gases relates the pressure $P$ of an ideal gas directly to its density $\rho$ and root-mean-square velocity $C$:
$$P = \frac{1}{3}\rho C^2$$ Isolating the rms speed expression from this equation yields:
$$C = \sqrt{\frac{3P}{\rho}}$$ Since the problem states that both gases exert equal pressures ($P_1 = P_2 = P$), the rms velocity is inversely proportional to the square root of the gas density:
$$C \propto \frac{1}{\sqrt{\rho}}$$

Step 3: Detailed Explanation:
We are given the density ratio of the two gases:
$$\frac{\rho_1}{\rho_2} = \frac{1}{16}$$ Setting up the inverse proportionality ratio for their respective rms speeds:
$$\frac{C_1}{C_2} = \sqrt{\frac{\rho_2}{\rho_1}}$$ Substitute the given values into the radical expression:
$$\frac{C_1}{C_2} = \sqrt{\frac{16}{1}} = \frac{4}{1}$$ This gives an explicit ratio of $4 : 1$.

Step 4: Final Answer:
The ratio of their rms speeds is $4 : 1$, which perfectly matches option (B).