The temperature at which oxygen molecules have the same root...

The temperature at which oxygen molecules have the same root mean square speed as helium atoms have at $300 \mathrm{~K}$ is: Atomic masses: He $=4 \mathrm{u}, \mathrm{O}=16 \mathrm{u})$
(a) $300 \mathrm{~K}$
(b) $600 \mathrm{~K}$
(c) $1200 \mathrm{~K}$
(d) $2400 \mathrm{~K}$

Key Concepts

Root Mean Square Speed

This concept defines the average speed of gas particles based on their kinetic energy and is given by the formula v_rms = ?(3kT/m) (or ?(3RT/M) when using molar quantities). It connects the microscopic motion of particles to macroscopic temperature and is fundamental to understanding molecular speeds in different gases.

Relationship Between Temperature and Kinetic Energy

Temperature is a measure of the average kinetic energy of gas particles. Since the kinetic energy of a gas particle is proportional to the temperature, comparing speeds of gases at different temperatures involves understanding how T influences the energy and hence the speed of the particles.

Molecular Mass Considerations

The mass of the particles plays a crucial role in determining their speed at a given temperature, as indicated by the inverse relationship between speed and the square root of mass. When comparing different gases, such as helium atoms and oxygen molecules, the difference in their masses directly affects their root mean square speeds.

Proportionality in Kinetic Theory

In kinetic theory, the square of the rms speed is directly proportional to the temperature divided by the mass of the particle. This proportionality allows for setting up relationships between two different gases by equating their rms speeds and solving for the temperature or mass factor, highlighting how differences in molecular mass affect the relationship between temperature and speed.

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