Question

Consider separate \(1.0-\mathrm{L}\) gaseous samples of \(\mathrm{He}, \mathrm{N}_{2},\) and \(\mathrm{O}_{2}\) all at \(\mathrm{STP}\) and all acting ideally. Rank the gases in order of increasing average kinetic energy and in order of increasing average velocity.

Short answer

The average kinetic energy for all three gases (He, N2, and O2) at STP is the same. For the average velocity, the gases can be ranked in order of increasing average velocity as: \(O_2 < N_2 < He\).

Step-by-step solution

Define Relevant Equations

To rank the gases based on average kinetic energy and average velocity, we need two equations. For average kinetic energy (KE), we can use the following equation: \( KE = \dfrac{3}{2}nRT \) where n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature (in Kelvin). For the average velocity (root-mean-square velocity, v_rms), we can use the following equation: \( v_{rms} = \sqrt{\dfrac{3RT}{M}} \) where M is the molar mass of the gas (in kg/mol).

Calculate the Average Kinetic Energy

Since all gases are at the same temperature (STP, which corresponds to 273.15 K) and contain the same number of moles (1.0 L), the average kinetic energy will be the same for all gases as KE only depends on \(n\) and \(T\) in the equation. So, He, N2, and O2 will have the same average kinetic energy at STP.

Calculate the Average Velocity (v_rms) for Each Gas

Using the equation for the average velocity (v_rms), we need to know the molar mass (M) of each gas. Here are the molar masses: He: 4.00 g/mol (0.004 kg/mol) N2: 28.02 g/mol (0.028 kg/mol) O2: 32.00 g/mol (0.032 kg/mol) Now, we will calculate the average velocity (v_rms) for each gas at STP: He: \( v_{rms} = \sqrt{\dfrac{3 \times 8.314 \times 273.15}{0.004}} \approx 1446.89 \, m/s \) N2: \( v_{rms} = \sqrt{\dfrac{3 \times 8.314 \times 273.15}{0.028}} \approx 517.37 \, m/s \) O2: \( v_{rms} = \sqrt{\dfrac{3 \times 8.314 \times 273.15}{0.032}} \approx 482.52 \, m/s \)

Rank the Gases

Now that we have calculated the average velocity (v_rms) for each gas, we can rank them in order of increasing average velocity: O2 < N2 < He As for the average kinetic energy, we already determined that they are the same for all gases at STP. #Conclusion# Thus, the gases can be ranked in order of increasing average kinetic energy as "Same for all gases" since all are at STP and contain the same number of moles (1.0 L). In terms of increasing average velocity, the order is O2 < N2 < He.

Key concepts

Average Velocity

The concept of average velocity is central to understanding the motion of particles in a gas. When gases are at a specific temperature and pressure, each gas molecule moves with a particular speed. The term 'average velocity' refers to the mean speed of all these gas molecules.

In the context of gases, average velocity is not merely about the average of speeds, but it is actually linked to the root-mean-square velocity, denoted as \( v_{rms} \). This reflects the fact that, mathematically, the velocities of gas molecules are squared, averaged, and then square-rooted. It's more accurate than a simple arithmetic mean as it accounts for variations in speed, especially in ideal gas conditions. Hence, \( v_{rms} \) gives us a robust measurement of average speed in the system.

• The average velocity of a gas can tell us about the energy dynamics within the gas.
• All gases have molecules that are constantly moving and colliding, leading to an average velocity measurement.

For the ranking of gases like He, N\(_2\), and O\(_2\) at STP, the \( v_{rms} \) allows us to compare their speeds accurately, where a higher value indicates a higher average velocity of the gas particles.

Root-Mean-Square Velocity

When considering gases and their motion, the root-mean-square (rms) velocity is a critical metric. It provides a more comprehensive view of how fast molecules in a gas are moving on average. The rms velocity combines the effects of all random molecular motions into a single average value.

The formula for root-mean-square velocity \( v_{rms} = \sqrt{\dfrac{3RT}{M}} \) is derived from kinetic theory. Here, \( R \) is the ideal gas constant, \( T \) is the absolute temperature in Kelvin, and \( M \) is the molar mass. This equation shows that rms velocity is directly related to temperature and inversely related to molar mass:

• Higher temperatures lead to faster rms velocities.
• Lighter gases (with lower molar mass) will have higher rms velocities than heavier gases at the same temperature.

In practice, this means that helium (He) gas molecules, which have a smaller molar mass, exhibit a higher root-mean-square velocity than nitrogen (N\(_2\)) or oxygen (O\(_2\)) molecules at standard temperature and pressure (STP). Understanding rms velocity helps in predicting how gases will behave under various conditions.

Molar Mass

Molar mass is a fundamental property of gases that significantly influences their physical behaviors, including their velocity at a given temperature. Molar mass refers to the mass of one mole of a substance, expressed in grams per mole (g/mol).

In terms of gas dynamics, molar mass is used in the equation for root-mean-square velocity \( v_{rms} = \sqrt{\dfrac{3RT}{M}} \). This equation reveals how the speed of gas molecules depends on both temperature and their molecular weight:

• Gases with lower molar mass, such as helium (4.00 g/mol), have faster molecules.
• Heavier gas molecules, like those of nitrogen (28.02 g/mol) or oxygen (32.00 g/mol), move more slowly.

This means that under standard conditions, helium molecules will move more quickly than nitrogen or oxygen molecules due to their lower molar mass. Hence, molar mass is a key parameter in understanding gas behavior and making predictions about their movement and energy under various conditions.