At what temperature is the rms velocity of hydrogen molecule equal to that of an oxygen molecule

step 1 In this question we will learn about some basic concept of the kinetic theory. The question is a

At what temperature is the rms velocity of hydrogen molecule...

At what temperature is the $\mathrm{rms}$ velocity of hydrogen molecule equal to that of an oxygen molecule at $27^{\circ} \mathrm{C} ?$
(a) $10 \mathrm{~K}$
(b) $20 \mathrm{~K}$
(c) $18.75 \mathrm{~K}$
(d) $19.75 \mathrm{~K}$

Key Concepts

Molecular Mass and Temperature Relationship

In the context of gases, the relationship between molecular mass and temperature is crucial since, for a given kinetic energy, lighter molecules will have higher speeds relative to heavier ones. When equating the rms velocities of different gases, the differences in molecular masses and the corresponding adjustments in temperature are central considerations.

Equipartition of Energy

The equipartition theorem states that energy in a thermodynamic system at equilibrium is distributed equally among all available degrees of freedom. For translational motion in gases, this means that every molecule has, on average, the same kinetic energy regardless of its mass if they are at the same temperature.

Root Mean Square (rms) Velocity

The rms velocity is a measure of the average speed of particles in a gas, derived from their kinetic energy. It represents the square root of the average of the squares of the individual speeds and establishes a bridge between the microscopic motion of molecules and the macroscopic temperature.

Kinetic Theory of Gases

This theory connects the macroscopic properties of gases such as pressure, volume, and temperature with the microscopic motions of gas molecules. It explains that gas pressure results from collisions of molecules with the walls of a container and that temperature is a measure of the average kinetic energy of the gas molecules.

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