Assertion Reason on Kinetic Energy and RMS Speed of Gas

To determine the correctness of the given Assertion and Reason, let us analyze them individually:

1. Analysis of Assertion A:

The average kinetic energy per molecule of an ideal gas is directly proportional to its absolute temperature $T$.

• The average translational kinetic energy per molecule is given by: $E_{\text{trans}} = \frac{3}{2} k_B T$ where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.

• The total average kinetic energy per molecule considering degrees of freedom $f$ is given by: $E_{\text{total}} = \frac{f}{2} k_B T$ Since both hydrogen ($\text{H}_2$) and oxygen ($\text{O}_2$) are diatomic gases, they both have the same number of degrees of freedom ($f = 5$ at room temperature).

In either case, the average kinetic energy per molecule depends only on the temperature $T$ and is independent of the size of the container. Therefore, if the average kinetic energy of $\text{H}_2$ and $\text{O}_2$ molecules is the same, their temperatures must also be equal.

Thus, Assertion A is TRUE.

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2. Analysis of Reason R:

The root-mean-square (r.m.s.) speed of molecules of an ideal gas is given by the formula: $v_{\text{rms}} = \sqrt{\frac{3 R T}{M}}$ where:

• $R$ is the universal gas constant,
• $T$ is the absolute temperature,
• $M$ is the molar mass of the gas.

For hydrogen gas ($\text{H}_2$), the molar mass is $M_{\text{H}_2} = 2\text{ g/mol}$, and for oxygen gas ($\text{O}_2$), the molar mass is $M_{\text{O}_2} = 32\text{ g/mol}$.

At the same temperature $T$: $\frac{v_{\text{rms}}(\text{H}_2)}{v_{\text{rms}}(\text{O}_2)} = \sqrt{\frac{M_{\text{O}_2}}{M_{\text{H}_2}}} = \sqrt{\frac{32}{2}} = 4$

Since $M_{\text{H}_2} \neq M_{\text{O}_2}$, the r.m.s. speeds of $\text{H}_2$ and $\text{O}_2$ molecules are not the same at the same temperature.

Thus, Reason R is FALSE.

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Conclusion:

Assertion A is true, but Reason R is false.

Correct Option: C