Radon (Rn) is the heaviest (and only radioactive) member of the noble gases. How much slower is the root-mean-square speed of Rn than He at 300 K?

Verified step by step guidance

1. Identify the formula to calculate the root-mean-square speed (rms) of a gas, which is given by \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the molar mass of the gas in kilograms per mole.
2. Convert the molar masses of Radon (Rn) and Helium (He) from grams per mole to kilograms per mole, as the molar mass in the rms formula needs to be in kilograms per mole.
3. Plug in the values for \( T \), \( k \), and the molar mass of Rn into the rms formula to calculate the rms speed for Radon.
4. Similarly, calculate the rms speed for Helium using its molar mass, along with the same values for \( T \) and \( k \).
5. Compare the rms speeds of Radon and Helium by dividing the rms speed of Radon by the rms speed of Helium to find out how much slower Radon moves compared to Helium at the same temperature.

Key Concepts

Root-Mean-Square Speed

The root-mean-square (RMS) speed is a measure of the average speed of gas particles in a system. It is calculated using the formula v_rms = sqrt(3RT/M), where R is the ideal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas. This concept is crucial for comparing the speeds of different gases at the same temperature.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a key factor in determining the RMS speed of a gas, as heavier gases will generally have lower RMS speeds at a given temperature. Understanding molar mass allows for the comparison of different gases, such as radon (Rn) and helium (He).

Kinetic Molecular Theory

Kinetic molecular theory explains the behavior of gases in terms of particles in constant motion. It posits that the temperature of a gas is directly related to the average kinetic energy of its particles. This theory helps to understand why lighter gases, like helium, move faster than heavier gases, like radon, at the same temperature.