Dependence of RMS Speed and Kinetic Energy on Temperature and Mass

To determine which statements are correct, we analyze the expressions derived from the kinetic theory of gases for a gas molecule at thermal equilibrium.

1. Root Mean Square Speed ($u_{\text{rms}}$)

The root mean square speed of a gas molecule is given by the formula: $u_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}}$ where:

• $R$ is the universal gas constant,

• $k_B$ is the Boltzmann constant,

• $T$ is the absolute temperature,

• $M$ is the molar (molecular) mass of the gas,

• $m$ is the mass of a single gas molecule.

• Analysis of Option A: From the formula, $u_{\text{rms}} \propto \sqrt{T}$. If the absolute temperature $T$ is increased four times ($T' = 4T$): $u_{\text{rms}}' = \sqrt{\frac{3R(4T)}{M}} = 2\sqrt{\frac{3RT}{M}} = 2u_{\text{rms}}$ Thus, $u_{\text{rms}}$ doubles when the absolute temperature is increased four times. Hence, Option A is correct.

• Analysis of Option D: At a constant temperature $T$, the root mean square speed is related to the molecular mass $M$ by: $u_{\text{rms}} \propto \frac{1}{\sqrt{M}}$ Thus, $u_{\text{rms}}$ is inversely proportional to the square root of its molecular mass. Hence, Option D is correct.

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2. Average Translational Kinetic Energy ($\mathcal{E}_{\text{av}}$)

The average translational kinetic energy per molecule of a gas is given by: $\mathcal{E}_{\text{av}} = \frac{1}{2} m u_{\text{rms}}^2 = \frac{1}{2} m \left(\frac{3k_B T}{m}\right) = \frac{3}{2} k_B T$

• Analysis of Option B: From the expression, $\mathcal{E}_{\text{av}} \propto T$. If the temperature $T$ is increased four times ($T' = 4T$): $\mathcal{E}_{\text{av}}' = \frac{3}{2} k_B (4T) = 4\mathcal{E}_{\text{av}}$ Thus, $\mathcal{E}_{\text{av}}$ quadruples (increases four times), rather than doubling. Hence, Option B is incorrect.

• Analysis of Option C: The equation $\mathcal{E}_{\text{av}} = \frac{3}{2} k_B T$ depends solely on the absolute temperature $T$ and the Boltzmann constant $k_B$. It is independent of the mass $m$ or molecular mass $M$ of the gas. Thus, at a given temperature, $\mathcal{E}_{\text{av}}$ does not depend on the molecular mass. Hence, Option C is correct.

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Conclusion

The correct statements are A, C, and D.