Question

Consider separate \(1.0-\mathrm{L}\) gaseous samples of \(\mathrm{H}_{2}, \mathrm{Xe}, \mathrm{Cl}_{2},\) and \(\mathbf{O}_{2}\) all at STP. a. Rank the gases in order of increasing average kinetic energy. b. Rank the gases in order of increasing average velocity. c. How can separate \(1.0-\mathrm{L}\) samples of \(\mathrm{O}_{2}\) and \(\mathrm{H}_{2}\) each have the same average velocity?

Short answer

(a) Since all the gases are at STP, their average kinetic energies are the same: H₂ = Xe = Cl₂ = O₂. (b) Based on root-mean-square velocity, which is inversely proportional to the square root of the molar mass: Xe < Cl₂ < O₂ < H₂. (c) Separate 1.0-L samples of O₂ and H₂ can have the same average velocity if the temperature of H₂ is 16 times greater than that of O₂.

Step-by-step solution

Understand Key Concepts

For a given gas at a fixed temperature and pressure, the average kinetic energy (KE) of a gas molecule depends only on the temperature, as given by the expression: \(KE = \frac{3}{2}kT\), where k is the Boltzmann constant and T is the absolute temperature (in Kelvin). The root-mean-square (rms) velocity (\(v_{rms}\)) depends on both the temperature and the molar mass of the gas, and it can be expressed by the equation: \(v_{rms} = \sqrt{\frac{3RT}{M}}\), where R is the ideal gas constant, T is the absolute temperature, and M is the molar mass of the gas. Since all gases are at STP (Standard Temperature and Pressure), they will have the same temperature (273.15K) and pressure (1 atm).

Rank Gases Based on Average Kinetic Energy

At a fixed temperature, the average kinetic energy for all gases is the same, as given by the equation: \(KE = \frac{3}{2}kT\). Since all gases are at STP, their average kinetic energy will be the same. Ranking (a): H2 = Xe = Cl2 = O2

Rank Gases Based on Root-Mean-Square (RMS) Velocity

Using the equation for rms velocity, \(v_{rms} = \sqrt{\frac{3RT}{M}}\), we know that the velocity is inversely proportional to the square root of the molar mass. Therefore, the gas with the lower molar mass will have higher rms velocity. Based on molar masses: H2(2 g/mol) < O2(32 g/mol) < Cl2(71 g/mol) < Xe(131 g/mol) Ranking (b): Xe < Cl2 < O₂ < H₂

Determine Conditions for Equal Average Velocity

In order for separate 1.0-L samples of O₂ and H₂ to have the same average velocity, we need to find the conditions that would make their rms velocities equal. We know that \(v_{rms\_O2} = \sqrt{\frac{3R}{M_{O2}}T_{O2}}\) and \(v_{rms\_H2} = \sqrt{\frac{3R}{M_{H2}}T_{H2}}\). Solving for the ratio of rms velocities: \(\frac{v_{rms\_O2}}{v_{rms\_H2}} = \sqrt{\frac{M_{H2}T_{O2}}{T_{H2}M_{O2}}}\) In order for the velocities to be equal, the ratio of rms velocities must be 1: \(1 = \sqrt{\frac{2T_{O2}}{32T_{H2}}}\) This implies that \(\frac{T_{O2}}{T_{H2}} = 16\), meaning that the temperature of H2 must be 16 times greater than that of the O₂ for their velocities to be equal. In conclusion, by increasing the temperature of H₂ to 16 times the temperature of O₂, the 1.0-L samples of O₂ and H₂ can have the same average velocity.

Key concepts

Average Kinetic Energy

The average kinetic energy of gas molecules is a vital concept in understanding how gases behave under different conditions. It's the energy associated with the motion of particles, and in the context of gases, it refers to the motion of countless tiny molecules zipping around in all directions.

When we keep the temperature constant, as in the case of the gases at STP in the exercise, the average kinetic energy can be determined using the equation \( KE = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) represents the temperature in Kelvin. What this means for our gaseous samples—\( \mathrm{H}_{2}, \mathrm{Xe}, \mathrm{Cl}_{2}, \) and \( \mathrm{O}_{2} \)—is that despite having drastically different molar masses, at STP they all share the same average kinetic energy. This concept is fundamental in kinetic molecular theory, emphasizing that the temperature is the measure of average kinetic energy for a gas sample.

Root-Mean-Square Velocity

Moving on to root-mean-square (rms) velocity, this concept helps us understand the speed at which gas particles are moving in a sample. The rms velocity (\( v_{rms} \) ) is the square root of the average of the squares of the velocities of all the particles in a gas.

Calculated using the equation \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) represents the molar mass of the gas in kilograms per mole, it becomes evident why lighter gases move faster than heavier ones. As we've seen in the exercise, hydrogen (\( \mathrm{H}_{2} \) ) with the lowest molar mass has a higher rms velocity compared to the heavier xenon (\( \mathrm{Xe} \) ).

Molar Mass

Molar mass is essentially the weight of one mole of a substance, typically expressed in grams per mole (g/mol). It's crucial when discussing any gas's properties because it directly affects both the rms velocity and the behavior under different conditions, such as temperature and pressure changes.

In the exercise solution, we used the molar mass to help rank the gases based on rms velocity. Heavier gases like xenon have a larger molar mass than lighter gases like hydrogen. This greater molar mass means a lower rms velocity because, for a given temperature, it takes more energy to move a heavier particle at the same speed as a lighter particle.

Standard Temperature and Pressure (STP)

Understanding STP is crucial for comparing different gases under equal conditions. Standard Temperature and Pressure refer to a set of conditions for experimental measurements, which standardize at 0 degrees Celsius (273.15K) for temperature and 1 atmosphere for pressure.

At STP, all gases have the same average kinetic energy, as shown in our exercise solution. This uniformity allows scientists and students alike to compare different gases in a controlled and meaningful way. Additionally, STP provides a reference point to calculate changes in gas behavior using the ideal gas law when either the temperature, the pressure, or both change from this standard.