Problem 77 - Chemistry The Central Science In Si Units

Question

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

Step-by-Step Solution

Answer

The RMS speed of Rn is approximately 0.134 times slower than that of He.

Step 1: Understand the Concept

The root-mean-square speed (RMS speed) of a gas is given by the formula \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the ideal gas constant, \( T \) is temperature in Kelvin, and \( M \) is the molar mass of the gas in kilograms per mole.

Step 2: Identify Given Information

You are given that both Radon (Rn) and Helium (He) gases are at \( 300\, \text{K} \). The molar mass of Helium is \( 4 \times 10^{-3} \text{ kg/mol} \) and the molar mass of Radon is \( 222 \times 10^{-3} \text{ kg/mol} \).

Step 3: Apply RMS Speed Formula

For Helium, \( v_{\text{He}} = \sqrt{\frac{3RT}{M_{\text{He}}}} \). For Radon, \( v_{\text{Rn}} = \sqrt{\frac{3RT}{M_{\text{Rn}}}} \).

Step 4: Compute the RMS Speed for Each Gas

Insert the values into the formula: For Helium: \( v_{\text{He}} = \sqrt{\frac{3 \cdot R \cdot 300}{4 \cdot 10^{-3}}} \).For Radon: \( v_{\text{Rn}} = \sqrt{\frac{3 \cdot R \cdot 300}{222 \cdot 10^{-3}}} \).

Step 5: Calculate the Ratio of RMS Speeds

The ratio of the speeds is given by \( \frac{v_{\text{Rn}}}{v_{\text{He}}} = \frac{\sqrt{\frac{3RT}{M_{\text{Rn}}}}}{\sqrt{\frac{3RT}{M_{\text{He}}}}} = \sqrt{\frac{M_{\text{He}}}{M_{\text{Rn}}}} \). Insert the molar masses: \( \sqrt{\frac{4}{222}} \).

Step 6: Solve the Final Expression

Calculate \( \sqrt{\frac{4}{222}} = \sqrt{\frac{2}{111}} \). This approximately equals \( \sqrt{0.018} \), or \( 0.134 \). Thus, the RMS speed of Radon is about \( 0.134 \) times that of Helium.

Key Concepts

Root-Mean-Square Speed

The root-mean-square speed (often abbreviated as RMS speed) is a measure of the average speed of particles in a gas. It gives us an idea of how fast the gas molecules are moving on average. The RMS speed can be calculated using the formula: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \]Here, \( R \) represents the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) stands for the molar mass of the gas in kilograms per mole. This formula is derived from the kinetic molecular theory which connects macroscopic properties of gases with their microscopic characteristics. Understanding this formula reveals that:

• The RMS speed is directly proportional to the square root of the temperature. This means that as the temperature increases, the RMS speed of the molecules also increases.
• It is inversely proportional to the square root of the molar mass. Gases with a higher molar mass will have lower RMS speeds at a given temperature, compared to lighter gases.

In the context of the given problem, Radon molecules are moving more slowly than Helium because Radon has a higher molar mass.

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. Its units are typically grams per mole or kilograms per mole. For molecules, the molar mass is calculated by adding up the atomic masses of all the atoms constituting the molecule.Why does molar mass matter in our problem?

• In gases, the molar mass determines how many particles (or molecules) are packed into a mole of the gas.
• This becomes crucial when calculating RMS speed because the RMS speed includes the square root of the molar mass in its denominator.

For instance, the molar mass of Helium is much lower than that of Radon, specifically:

• Helium: \( 4 \times 10^{-3} \) kg/mol
• Radon: \( 222 \times 10^{-3} \) kg/mol

Thus, at the same temperature, Helium molecules move significantly faster than Radon molecules due to their lower molar mass.

Ideal Gas Law

The ideal gas law is a critical guideline for understanding the behavior of gases. It combines several simpler gas laws, such as Boyle's, Charles', and Avogadro's laws, into one comprehensive equation:\[ PV = nRT \]Where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature of the gas in Kelvin.

While the ideal gas law itself does not appear explicitly in the calculation of RMS speed, it helps understand how different parameters such as pressure, volume, temperature, and amount of gas interrelate.In our specific problem, the ideal gas law provides context. Understanding that temperature uniformly affects the behavior of gases helps us see why Radon, despite being heavier, is compared at the same temperature with Helium to determine their comparative speeds.