At what temperature is the root mean square speed of an atom...

At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at $-20{ }^{\circ} \mathrm{C} ?$ (atomic mass of Ar $=39.9 \mathrm{u}$, of $\mathrm{He}=4.0 \mathrm{u})$

Key Concepts

Root Mean Square Speed

The root mean square (rms) speed is a statistical measure of the speed of particles in a gas. It is the square root of the average of the squares of the speeds and is expressed by the formula v?rms? = ?(3RT/M), where R is the gas constant, T is the absolute temperature, and M is the molar mass. This concept is central in linking microscopic molecular motion with macroscopic properties like temperature.

Kinetic Theory of Gases

The kinetic theory of gases explains that a gas consists of a large number of small particles moving in random directions, colliding elastically with each other and the container walls. This theory connects the microscopic kinetic energies of gas molecules to macroscopic properties such as pressure and temperature, and underpins the derivation of the rms speed formula.

Temperature and Kinetic Energy Relationship

In kinetic theory, temperature is directly related to the average kinetic energy of the particles in a gas. As temperature increases, the kinetic energy and thus the speed of the particles also increases. This relationship is fundamental for understanding how changes in temperature affect molecular speeds.

Molecular Mass

Molecular mass, or molar mass, influences the speed of gas particles because, for a given temperature, lighter molecules move faster than heavier ones. In the rms speed formula, the molar mass appears in the denominator, indicating that the gas’s speed inversely depends on its mass. This concept is crucial when comparing the speeds of different gases.

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