Question

\begin{equation}\begin{array}{l}{\text { A flask at room temperature contains exactly equal amounts (in moles) }} \\ {\text { of nitrogen and xenon. }} \\\ {\text { a. Which of the two gases exerts the greater partial pressure? }} \\\ {\text { b. The molecules or atoms of which gas will have the gressure? }} \\\ {\text { velocity? }}\end{array}\end{equation} \begin{equation}\begin{array}{l}{\text { c. The molecules of which gas will have the greater average kinetic }} \\ {\text { energy? }} \\ {\text { d. If a small hole were opened in the flask, which gas would effuse }} \\ {\text { more quickly? }}\end{array}\end{equation}

Short answer

a. Both gases exert the same partial pressure. b. Nitrogen molecules will have a greater velocity. c. Both gases have the same average kinetic energy. d. Nitrogen will effuse more quickly.

Step-by-step solution

Understand Dalton's Law of Partial Pressures

According to Dalton's Law, in a mixture of non-reacting gases, the total pressure is equal to the sum of the partial pressures of the individual gases. The partial pressure of a gas is directly proportional to the number of moles of that gas. Since the number of moles of nitrogen (N2) and xenon (Xe) are the same, their partial pressures are equal if the gases have the same volume, temperature, and the ideal gas behavior can be assumed.

Apply Graham's Law of Effusion

Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. Since xenon atoms are much heavier than nitrogen molecules, nitrogen will effuse more quickly through a small hole.

Relate Kinetic Molecular Theory to Gas Behavior

The Kinetic Molecular Theory states that the average kinetic energy of gas particles is directly proportional to the absolute temperature, and it is the same for all gases at a given temperature. This means that both nitrogen and xenon have the same average kinetic energy since they are at the same temperature.

Determine the Velocity of Gas Molecules

While the average kinetic energy is the same for both gases, their velocities differ due to their masses. The formula for kinetic energy is \( E_k = \frac{1}{2} m v^2 \), where \( m \) is mass and \( v \) is velocity. Since nitrogen has a lower mass compared to xenon, for the same kinetic energy, nitrogen gas molecules will have a greater velocity.

Key concepts

Dalton's Law of Partial Pressures

When studying the behavior of gases in a mixture, Dalton's Law of Partial Pressures becomes crucial. It tells us that in a blend of non-reactive gases, each gas acts independently and contributes to the total pressure as if the other gases were not present. The partial pressure of each gas is a measure of this contribution and is proportional to its mole fraction within the mixture. To put it simply, if two gases are in equal amount and under the same conditions, they exert an equal force on the walls of their container. This understanding is fundamental when analyzing gas mixtures and predicting how they will behave under different conditions.

For example, if nitrogen and xenon are in a flask in equal molar amounts at a constant temperature and volume, according to Dalton's Law, their partial pressures would be the same. This applies provided that the behavior of the gas is ideal, which is a good approximation under many conditions.

Graham's Law of Effusion

Another fascinating aspect of gas behavior is observed when gas particles escape through a tiny opening, a process known as effusion. Graham's Law of Effusion provides insight into the rate at which different gases effuse. It tells us that lighter gas particles will effuse faster than heavier ones. Specifically, the rate of effusion is inversely proportional to the square root of the particles' mass.

In practical terms, if we take two gases such as nitrogen and xenon, nitrogen gas molecules will effuse more quickly due to their lighter mass. This effect has practical implications, for instance, in industrial processes where separation of gases is required or in the analysis of gas leaks.

Kinetic Molecular Theory

To truly grasp gas behavior, it's important to delve into the Kinetic Molecular Theory. This theory states that gas particles are in constant rapid motion and that the temperature of a gas reflects the average kinetic energy of its molecules. One of the most important tenets of this theory is that at a given temperature, all gases have the same average kinetic energy regardless of their chemical identity or molecular mass.

This means that for our nitrogen and xenon in the flask, while their velocities may differ due to difference in mass, their average kinetic energies are identical. This concept is a cornerstone in understanding how temperature affects gas behavior and can be used to predict how gases will act under varying thermal conditions.

Average Kinetic Energy of Gases

Building on the Kinetic Molecular Theory, the average kinetic energy of gas particles is central to understanding gas dynamics. At a given temperature, this energy remains constant across different types of gases, which is why this average is directly related to temperature and independent of other factors like molarity or volume of the gas.

Applying the equation for kinetic energy, which is \( E_k = \frac{1}{2} m v^2 \), we can infer that even though nitrogen and xenon have the same average kinetic energy in our scenario, their velocities must differ. The lighter nitrogen molecules move faster, compensating for their lower mass. This is why, despite the same average kinetic energy, the experienced effects such as gas pressure and effusion rates can vary markedly between different gases.