Problem 30 - University Physics with Modern Physics

Question

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare (a) the average kinetic energies of the three types of atoms and (b) the root-mean-square speeds. (Hint: The periodic table in Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

Step-by-Step Solution

Answer

(a) All gases have the same average kinetic energy; (b) Ne > Kr > Rn for root-mean-square speeds.

Step 1: Understanding Average Kinetic Energy

The average kinetic energy (\( KE_{avg} \)) of gases can be given by the formula \( KE_{avg} = \frac{3}{2} k_B T \), where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature. Since all gases in the flask are at the same temperature, \( KE_{avg} \) is the same for neon, krypton, and radon.

Step 2: Calculate Root-Mean-Square Speed

The root-mean-square (rms) speed (\( v_{rms} \)) of a gas is given by \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the ideal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass in kg/mol. The molar masses from the periodic table are: Ne = 20.18 g/mol, Kr = 83.8 g/mol, Rn = 222 g/mol, which we convert to kg/mol (e.g. Ne: 20.18 g/mol = 0.02018 kg/mol).

Step 3: Comparing Root-Mean-Square Speeds

Since \( v_{rms} \) is inversely proportional to the square root of the molar mass, we expect neon to have the highest \( v_{rms} \) (due to its lowest molar mass), followed by krypton, and finally radon which will have the lowest root-mean-square speed.

Key Concepts

Average Kinetic Energy

The concept of average kinetic energy is fundamental in understanding the behavior of gases. It provides insight into the energy each particle within a gas possesses due to its motion. The average kinetic energy of a gas is expressed using the formula \( KE_{avg} = \frac{3}{2} k_B T \), where \( k_B \) is the Boltzmann constant and \( T \) represents the absolute temperature in Kelvin.
This formula reveals an important aspect: the average kinetic energy is dependent solely on the temperature of the gas and is independent of the type of gas (or identity of the particle).

• Because all gases in the flask share the same temperature, the average kinetic energy is identical for neon, krypton, and radon. This highlights the principle that temperature is a measure of the average kinetic energy of particles.
• No matter what the gas composition, as long as they are maintained at the same temperature, their average kinetic energies will be the same.

Understanding this concept helps solidify the idea that kinetic energy is strictly linked to temperature, not the mass or type of gas involved.

Root-Mean-Square Speed

Root-mean-square speed is a concept that transforms the understanding of kinetic energy into motion by calculating how fast gas particles are moving. The root-mean-square (rms) speed is derived from the formula \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the ideal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas in kg/mol.
Here is how rms speed is interpreted:

• Since \( v_{rms} \) is inversely proportional to the square root of the molar mass of the gas, lighter gases will move faster compared to heavier gases at the same temperature.
• Using the periodic table, one can convert the molar mass from grams per mole to kilograms per mole for accurate calculations (e.g., Ne: 20.18 g/mol = 0.02018 kg/mol).

This means in the given scenario, neon, having the smallest molar mass, would have the highest rms speed, followed by krypton, and radon with the largest molar mass, would have the slowest movement.

Molar Mass

Molar mass is an essential factor in calculating various properties of a gas, such as root-mean-square speed. It is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). For scientific calculations involving kinetic theory, it's often converted to kilograms per mole (kg/mol).
In the context of kinetic theory and particle motion:

• Molar mass directly affects the root-mean-square speed of gas particles; lighter gases (such as neon which is 20.18 g/mol) move faster than heavier gases (such as krypton at 83.80 g/mol and radon at 222 g/mol) at a given temperature.
• This inverse relationship between molar mass and speed emphasizes the impact of atomic weight on motion: heavier atoms require more energy to attain the same speed as lighter ones.

Knowing the molar mass allows predictions about the comparative speeds of different gases, which aids in understanding how gas mixtures behave under various conditions. The periodic table is a valuable tool in these calculations, offering the necessary molar mass data for each element.