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: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

Short answer

(a) Average kinetic energies are equal. (b) Ne has the highest \( v_{rms} \), Rn the lowest.

Step-by-step solution

Understanding Kinetic Energy

According to kinetic molecular theory, the average kinetic energy of gas particles is given by \( KE_{avg} = \frac{3}{2} k_B T \), where \( k_B \) is Boltzmann's constant and \( T \) is the temperature in Kelvin. Importantly, the average kinetic energy is dependent only on temperature, not the type of gas. Therefore, all three gases (Ne, Kr, and Rn) have the same average kinetic energy at the same temperature.

Finding Molar Masses

Check Appendix D for the molar masses: Neon (Ne) is approximately 20 g/mol, Krypton (Kr) is approximately 84 g/mol, and Radon (Rn) is approximately 222 g/mol.

Calculating Root-Mean-Square Speed

The root-mean-square speed for a gas is calculated using the formula \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass in kg/mol. Since Neon has the smallest molar mass, it will have the highest root-mean-square speed, followed by Krypton, and then Radon.

Summary

For part (a), the average kinetic energies of Ne, Kr, and Rn are equal at constant temperature. For part (b), the root-mean-square speed varies inversely with the square root of the molar mass, with Ne having the highest speed and Rn the lowest.

Key concepts

Average Kinetic Energy

In the realm of physics and chemistry, kinetic molecular theory plays a vital role in understanding how gas particles behave. One of its fundamental aspects is the concept of average kinetic energy. This is determined by the equation \( KE_{avg} = \frac{3}{2} k_B T \), where \( k_B \) is Boltzmann's constant and \( T \) is the temperature in Kelvin.
This equation tells us that average kinetic energy is influenced solely by temperature, not by the nature of the gas itself.
So, in our example, even though neon, krypton, and radon are different gases, they have the same average kinetic energy if they are at the same temperature. This concept underscores how temperature is a universal measure influencing the energy exhibited by particles in a gas.
Thus, regardless of the gas type, temperature holds the key to determining their kinetic energy dynamics.

Root-Mean-Square Speed

Root-mean-square speed, often abbreviated as \( v_{rms} \), is a mathematical way of representing the speed of gas particles in a container. It accounts for the fact that particles in a gas don't move uniformly but cover a range of speeds.
The formula for calculating the root-mean-square speed is \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass. Of the three gases we discussed, neon, krypton, and radon, the speed differences arise from their varying molar masses.
Since Neon has the smallest molar mass, it ends up having the highest \( v_{rms} \), meaning its particles move the fastest. On the contrary, Radon, with the largest molar mass, has the slowest particles.

• High molar mass (Rn): slower speeds
• Low molar mass (Ne): faster speeds

This shows how inversely the root-mean-square speed of particles relates to the square root of their molar mass, making each gas particle's speed distinct.

Molar Mass

Molar mass is a fundamental concept in understanding the properties of different substances, especially gases. It is defined as the mass of one mole of a given substance, and it is typically expressed in grams per mole (g/mol).
In our example, to find the molar mass of each gas, one should refer to a periodic table or specific appendices like Appendix D in textbooks. The values are as follows:

• Neon (Ne): approximately 20 g/mol
• Krypton (Kr): approximately 84 g/mol
• Radon (Rn): approximately 222 g/mol

These values are crucial for calculating properties such as the root-mean-square speed of gases, as seen earlier. Understanding molar mass helps us comprehend how different gases behave under equivalent temperature and pressure conditions.
Moreover, it reflects how mass factors contribute to the energy and speed of gas particles, linking back to the kinetic molecular theory. This interconnectedness highlights the importance of molar mass in gas dynamics and other areas of chemistry and physics.