Question

Consider three identical flasks filled with different gases. Flask A: \(\mathrm{CO}\) at 760 torr and \(0^{\circ} \mathrm{C}\) Flask B: \(\mathrm{N}_{2}\) at 250 torr and \(0^{\circ} \mathrm{C}\) Flask C: \(\mathrm{H}_{2}\) at 100 torr and \(0^{\circ} \mathrm{C}\) a. In which flask will the molecules have the greatest average kinetic energy? b. In which flask will the molecules have the greatest average velocity?

Short answer

a. All flasks A, B, and C have equal average kinetic energy because they are at the same temperature (\(0^{\circ} \mathrm{C}\)). b. Flask C (Hydrogen gas) has the greatest average velocity among the molecules of all flasks due to its lower molecular mass as compared to flasks A and B, which contain Carbon monoxide and Nitrogen, respectively.

Step-by-step solution

Determine if temperature differs among flasks

The flasks are all at \(0^{\circ} \mathrm{C}\). Since temperature is the same for all three flasks, the molecules are sharing the same average kinetic energy. To find the difference in velocity, let's look at molecular mass.

Determine molecular mass

The molecular masses for the gases in each flask are as follows: Flask A: Carbon monoxide (CO) has a molecular mass of 12 + 16 = 28 g/mol. Flask B: Nitrogen (N2) has a molecular mass of 28 g/mol (14 x 2 = 28 g/mol). Flask C: Hydrogen (H2) has a molecular mass of 2 g/mol (1 x 2 = 2 g/mol).

Calculate the average kinetic energy

Since temperature is the same for all three flasks, the molecules in each flask have the same average kinetic energy. Therefore, the answer for part (a) is that all flasks A, B, and C have equal average kinetic energy.

Determine the average velocity of molecules in each flask

The relationship between molecular mass and molecular velocity is given by the equation: \(v_{rms} = \sqrt{\dfrac{3RT}{M}}\) where \(v_{rms}\) is the root mean square velocity, R is the gas constant (8.314 J/mol K), T is the temperature (0°C = 273.15 K), and M is the molar mass in kg/mol. For Flask A, the average velocity is: \[ v_{A} = \sqrt{\dfrac{3(8.314)(273.15)}{0.028}}\] For Flask B, the average velocity is: \[ v_{B} = \sqrt{\dfrac{3(8.314)(273.15)}{0.028}}\] For Flask C, the average velocity is: \[ v_{C} = \sqrt{\dfrac{3(8.314)(273.15)}{0.002}}\] Now, comparing the average velocities: Since 0.028 > 0.002, \(v_{C} > v_{A} = v_{B}\). Therefore, the answer for part (b) is that Flask C (Hydrogen gas) has the greatest average velocity among the molecules of all flasks.

Key concepts

Average Molecular Velocity

When we talk about gases, each molecule is in constant motion, randomly bouncing around and occasionally colliding with other molecules. The concept of average molecular velocity refers to the speed at which these molecules move on average. The velocity is not uniform because individual molecules travel at different speeds depending on collisions or the type of gas involved. However, factors such as molecular mass and temperature are key in determining the average velocity of the gas molecules. In our flask problem, even though the temperature is the same, differing molecular masses result in different average velocities for gases. This is why, among Carbon monoxide, Nitrogen, and Hydrogen gases—although they all share the same temperature—their varying molecular weights lead to differences in their average velocities.

Gas Laws

Gas laws help us understand and predict the behavior of gases under different conditions. They establish relationships among pressure, volume, temperature, and number of molecules in a gas. In the context of our exercise, we can observe these laws at work. All gases in the flasks are at the same temperature but at different pressures, which are given in torr. Although gases like CO and N extsubscript{2} have similar molecular velocities due to their identical molecular masses, differences in pressure and molecular mass influence properties such as density and velocity, as per the gas laws.

Molecular Mass

Molecular mass is a crucial factor when discussing the behavior of gases, especially concerning their velocity. Molecular mass, often measured in grams per mole (g/mol), represents the mass of a single molecule of a compound or element. To find it, we sum the atomic masses of each atom in a molecule’s structure. For instance, Carbon monoxide (CO) has a molecular mass of 28 g/mol—carbon being 12 g/mol and oxygen 16 g/mol. When temperatures are equal among gas samples, lighter molecules (like H extsubscript{2}) naturally travel faster than heavier ones. This is part of why, in our flasks, the light hydrogen molecules achieve greater velocities than those in the flasks containing heavier carbon monoxide and nitrogen.

Root Mean Square Velocity

Root mean square velocity (\(v_{rms}\)) is a specific measure of the average speed of gas molecules, calculated using the formula:\[v_{rms} = \sqrt{\dfrac{3RT}{M}}\]In this equation, \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas in kg/mol. This equation provides a glimpse into how kinetic energy impacts molecular speeds and further solidifies the connection between molecular mass and velocity.In our example, since the flasks have different gases but are kept at the same temperature, the differences in \(v_{rms}\) arise due to their distinct molecular masses. Flask C, containing hydrogen with the lowest molecular mass, displays the highest root mean square velocity, demonstrating the inverse relationship between mass and speed in gas molecules.