Question

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 \(300 \mathrm{K?}\)

Short answer

The root-mean-square speed of Radon (Rn) is approximately 1.17 x 10^3 m/s slower than the root-mean-square speed of Helium (He) at 300 K.

Step-by-step solution

Identify the given values and constants

Given values: Temperature (T) = 300 K Constants: Boltzmann constant (k_B) = 1.38 × 10^-23 J/K

Find the molar mass of both Helium and Radon

Molar mass of He = 4 g/mol Molar mass of Rn = 222 g/mol

Convert molar mass to molecular mass

To convert molar mass to molecular mass, we need to multiply molar mass by the unified atomic mass unit (u). 1 u = 1.66 × 10^-27 kg Molecular mass of He = (4 g/mol) * (1.66 × 10^-27 kg/u) = 6.64 × 10^-27 kg Molecular mass of Rn = (222 g/mol) * (1.66 × 10^-27 kg/u) = 3.68 × 10^-25 kg

Calculate the root-mean-square speeds for both Helium and Radon

Now we can plug these values into the root-mean-square formula: \(v_{rms, He} = \sqrt{\frac{3 * 1.38 * 10^{-23} * 300}{6.64 * 10^{-27}}}\) \(v_{rms, He} ≈ 1.38 * 10^3 m/s\) \(v_{rms, Rn} = \sqrt{\frac{3 * 1.38 * 10^{-23} * 300}{3.68 * 10^{-25}}}\) \(v_{rms, Rn} ≈ 2.14 * 10^2 m/s\)

Find the difference in root mean square speeds

Now we will find the difference in root mean square speeds of He and Rn: Difference = Speed of He - Speed of Rn Difference = \(1.38 * 10^3 - 2.14 * 10^2 ≈ 1.17 * 10^3 m/s\) The root-mean-square speed of Radon (Rn) is approximately 1.17 x 10^3 m/s slower than the root-mean-square speed of Helium (He) at 300 K.

Key concepts

Noble Gases

Noble gases are a fascinating group of elements in the periodic table, known for their unique set of properties. These gases include Helium (He), Neon (Ne), Argon (Ar), Krypton (Kr), Xenon (Xe), and Radon (Rn). They reside in Group 18 and are characterized by their low reactivity due to having a complete valence electron shell. This makes them very stable. Radon is unique among them as it is the only radioactive noble gas. It is heavier and denser than other noble gases, making it an interesting subject for studies, particularly in examining their behaviors under various physical conditions such as speed and diffusion.

• They lack color, odor, and taste, making them hard to detect without special equipment.
• Due to their stable electronic configurations, they generally do not form compounds easily.
• Each noble gas has unique applications, like Helium in balloons and Neon in signs.

Molar Mass Conversion

Molar mass conversion is an essential part of chemistry, which allows us to relate gram measurements to atomic or molecular scales. The molar mass of an element or compound is the mass of one mole of that substance, expressed in grams per mole (g/mol). This connects directly to the molecular mass, which is often used in thermodynamic calculations. For calculations involving molecular speeds, converting molar mass to molecular mass is crucial, as it allows the use of the kinetic molecular theory on a practical level. Multiplied by the unified atomic mass unit, which is approximately 1.66 x 10^-27 kg, the molar mass gives us the mass of a single molecule or atom in kilograms.
This conversion facilitates calculations of particle speeds and energies based on their masses, especially when dealing with root-mean-square speed calculations.

• Helium has a molar mass of 4 g/mol, representing its light atomic structure.
• Radon's molar mass is 222 g/mol, reflecting its heavier and more massive nature.

Radioactivity

Radioactivity refers to the process by which an unstable atomic nucleus loses energy by emitting radiation. Among the noble gases, only Radon is radioactive, which significantly affects its physical properties and practical considerations.
As a radioactive element, Radon undergoes radioactive decay, a spontaneous transformation where the nucleus changes into another element, emitting particles or electromagnetic radiation. This property of Radon makes it notable for being a health hazard when accumulated in confined spaces like basements, as inhalation of its isotopes may lead to radiation exposure risks.

• Radon's decay products are also radioactive and can attach to dust and other particles in the air.
• Careful residential monitoring and ventilation can mitigate radon exposure.
• This property distinguishes Radon from non-radioactive noble gases, which do not pose such threats.

Molecular Speed Calculation

The molecular speed calculation is a key concept in understanding the physical behavior of gases, often employing the root-mean-square (RMS) speed. This speed represents the square root of the average of the squared velocities of the gas molecules. This calculation embodies the kinetic molecular theory of gases, which indicates that temperature affects molecular speed. The formula for calculating RMS speed is:\[v_{rms} = \sqrt{\frac{3k_BT}{m}}\] where:

• \( v_{rms} \): root-mean-square speed
• \( k_B \): Boltzmann constant
• \( T \): temperature in Kelvin
• \( m \): molecular mass

Molecules move faster at higher temperatures and a gas's molecular speed is contingent upon its molecular mass. In the context of noble gases like Helium and Radon, Helium atoms, being lighter, move faster than the heavier Radon atoms at the same temperature. This difference in speed impacts properties like diffusion rates and mean free paths within a gaseous system. Understanding RMS speed assists in many practical applications, including the separation, transport, and analysis of gases.