Question

Exactly equal amounts (in moles) of gas A and gas B are combined in a 1 -L container at room temperature. Gas \(\mathrm{B}\) has a molar mass that is twice that of gas A. Which statement is true for the mixture of gases and why? a. The molecules of gas \(\mathrm{B}\) have greater kinetic energy than those of gas A. b. Gas \(\mathrm{B}\) has a greater partial pressure than gas \(\mathrm{A}\). c. The molecules of gas \(\mathrm{B}\) have a greater average velocity than those of gas \(\mathrm{A}\). d. Gas B makes a greater contribution to the average density of the mixture than gas A.

Short answer

d. Gas B makes a greater contribution to the average density of the mixture than gas A because density is directly proportional to molar mass, and gas B has twice the molar mass of gas A.

Step-by-step solution

Recall the Kinetic Molecular Theory of Gases

According to the Kinetic Molecular Theory, all gases at the same temperature have the same average kinetic energy. This means that for two different gases at the same temperature, their kinetic energies will be equal, regardless of the molar mass.

Apply the Ideal Gas Law

The Ideal Gas Law states that for a gas at constant temperature and volume, the pressure is directly proportional to the number of moles. Since gases A and B have equal moles, they will exert equal pressures if all other conditions are the same.

Understand the Relationship Between Mass, Velocity, and Kinetic Energy

Kinetic energy can be expressed by the equation KE = (1/2)mv^2, where m is the mass and v is the velocity of the gas molecules. Since gas B has a molar mass twice that of gas A and kinetic energy is equal for both gases at the same temperature, gas B's molecules must be moving with a lower average velocity to have the same kinetic energy as those of gas A.

Determine the Gas that Contributes More to Average Density

Density is directly proportional to the molar mass, given that moles and volume are constant for both gases. Since gas B has a higher molar mass, it will contribute more to the overall density of the mixture.

Evaluate Each Statement

Now evaluate each of the given statements:a. This is incorrect because all gases at the same temperature have the same average kinetic energy.b. This is incorrect because both gases have the same number of moles and are under the same conditions, therefore they will have the same partial pressure.c. This is incorrect because with greater molar mass at the same kinetic energy, gas B will have a lower average velocity.d. This statement is correct because gas B has a higher molar mass and density is directly proportional to molar mass.

Key concepts

Gas Laws

The behavior of gases under different conditions is often described by the Gas Laws. These fundamental rules illustrate how gases respond to changes in temperature, pressure, and volume. One of the most basic gas laws is Boyle's Law, which states that pressure and volume are inversely proportional when temperature is held constant. Charles’s Law follows that volume and temperature are directly proportional at constant pressure.

Another important concept is the partial pressure, which tells us that in a mixture of gases, each gas exerts a pressure independently. If we have equal moles of two gases in the same volume, according to Dalton’s Law, they will have equal partial pressures. This key understanding helps us unravel puzzles like our textbook problem — determining properties of a gas mixture.

Kinetic Energy of Gases

Capturing the essence of gas behavior calls for a closer look at the kinetic energy of gases. Kinetic Molecular Theory provides a framework explaining that gas particles are in random constant motion and exert pressure by colliding with the walls of their container. A critical takeaway from this theory is that the average kinetic energy of gas particles is determined by the system's temperature.

Mathematically, the kinetic energy (KE) can be expressed as \(KE = \frac{1}{2}mv^2\), where \(m\) represents the mass of the particles, and \(v\) their velocity. This equation highlights an inverse relationship between mass and velocity, ensuring that gases with different masses have the same kinetic energy at a fixed temperature. This concept refutes the notion that heavier gas molecules naturally have higher kinetic energies.

Ideal Gas Law

One cornerstone of understanding gases is the Ideal Gas Law, represented by the equation \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for moles of gas, \(R\) the ideal gas constant, and \(T\) for temperature. The law suggests that the state of an ideal gas is determined by its pressure, volume, and temperature, thus connecting all the gas laws into a single expression.

In our textbook example, both gases A and B are in equal moles and the same container, and thus under the guidelines of the Ideal Gas Law, they should possess the same pressure if the temperature is constant - the law does not discriminate between different gas species if all other conditions are equivalent.

Molecular Velocities in Gases

The speed at which gas molecules move is not a fixed quantity but varies following a distribution, with some molecules moving slowly and others zipping at high velocities. But we can define an average velocity or root mean square velocity, which is closely tied to the gas's kinetic energy and temperature.

As we learned, equal kinetic energy does not mean equal velocities. In fact, heavier gas molecules will move more slowly compared to lighter ones when all have the same kinetic energy. This is critical in explaining why gas B, with twice the molar mass of gas A, would have slower average velocity, assuming they are at the same temperature.

Gas Density

Gas density refers to the mass of gas per unit volume and is a significant property, particularly when we deal with gas mixtures. It is heavily influenced by the molar mass of the gas — a higher molar mass corresponds to a higher density when the number of moles and the volume are constant.

In our problem, since both gases A and B occupy the same volume and are present in equal moles, the denser gas is the one with the greater molar mass — which is gas B. Understanding this allows students to determine which gas contributes more to the overall density in a mixture, a practical interpretation of the Gas Laws.