Question

Consider two gases, \(A\) and \(B\), each in a \(1.0\) -\(\mathrm{L}\) container with both gases at the same temperature and pressure. The mass of gas \(A\) in the container is \(0.34\) \(\mathrm{g}\) and the mass of gas \(B\) in the container is \(0.48 \mathrm{g}\). a. Which gas sample has the most molecules present? Explain. b. Which gas sample has the largest average kinetic energy? Explain. c. Which gas sample has the fastest average velocity? Explain. d. How can the pressure in the two containers be equal to each other since the larger gas \(B\) molecules collide with the container walls more forcefully?

Short answer

a. We cannot determine which gas sample has the most molecules present without knowing the molar masses of the gases A and B. b. Both gas samples have the same average kinetic energy as they have the same temperature, according to the Kinetic Molecular Theory. c. Without information about the molar masses or molecular masses of gases A and B, we cannot compare their average velocities. d. The pressure in the two containers can be equal despite differences in molecule size because the product of the number of moles and the specific gas constant for each gas is equal. The balance between more forceful collisions and fewer collisions occurring in the larger gas molecules can result in equal pressure in both containers.

Step-by-step solution

a. Determining the gas sample with most molecules present

To determine the gas sample with the most molecules present, we need to calculate the number of moles of each gas. We can use the masses and molar masses of each gas to do this. Let's represent the molar masses of A and B as \(M_A\) and \(M_B\). The number of moles can be calculated using: n = mass / molar mass Let's calculate the number of moles for gas A: \(n_A = m_A / M_A\) And for gas B: \(n_B = m_B / M_B\) We cannot determine the actual number of moles without knowing the molar mass of each gas, but we can compare the ratios of the gas masses to their molar masses: \(n_A / n_B = (m_A / M_A) / (m_B / M_B)\) If this ratio is greater than 1, there are more moles present in gas A. If it's less than 1, then gas B has more moles present. Since we don't know the molar masses of A and B, we cannot determine which gas has more molecules present. We would need more information about the gases.

b. Determining the gas sample with the largest average kinetic energy

As per the Kinetic Molecular Theory, the average kinetic energy of a gas depends only on its temperature and is given by the formula: \(KE_{avg} = (3/2) k_B T\) where \(k_B\) is Boltzmann's constant and T is the temperature in Kelvin. Since both gases are at the same temperature, their average kinetic energy is the same. Therefore, neither of the gas samples has a larger average kinetic energy.

c. Determining the gas sample with the fastest average velocity

To compare the average velocities of the two gases, we need to use the equation for the root-mean-square (rms) speed, which is given by: \(v_{rms} = \sqrt{(3k_BT)/(m)}\) where m is the mass of a single molecule and k_B and T are Boltzmann's constant and temperature, respectively. As the temperature is the same for both gases, we can write: \(v_{rms,A}/v_{rms,B} = \sqrt{(m_B)/(m_A)}\) Since we do not know the molar masses or molecular masses of A and B, we cannot compare their average velocities without this information. We would need more information about the gases to determine which has the fastest average velocity.

d. Explaining equal pressure in the containers

The pressure of a gas inside a container can be described by the Ideal Gas Law equation: \(PV = nRT\) Where P is the pressure, V is the volume, n is the number of moles of the gas, R is the universal gas constant, and T is the temperature. Both containers have the same pressure, volume, and temperature. Therefore: \(n_AR_A = n_BR_B\) This equality implies that the product of the number of moles and the specific gas constant for each gas is equal. It's important to note that the collisions of gas molecules with the container walls might be more forceful for larger molecules, but there could be fewer collisions occurring. These two factors can balance out, resulting in equal pressure in both containers.

Key concepts

Molecular Mass

Understanding the concept of molecular mass is fundamental when comparing different gas samples. In essence, molecular mass represents the combined mass of all the atoms in a molecule, with units typically in atomic mass units (amu) or grams per mole (g/mol).

When dealing with gases, knowing the molecular mass is crucial for calculations involving the molar mass and, consequently, the number of moles of gas present, since the number of moles (n) is given by the ratio of the sample's mass (m) to its molar mass (M):
\[n = \frac{m}{M}\]
For gases A and B, we cannot specifically compare the number of molecules without knowing their molecular masses. However, if one had this information, the gas with the larger molar mass would have fewer molecules per gram, since each molecule weighs more. This is why, when comparing gas samples, identifying the molecular mass allows us to determine which sample contains more molecules — a key step in analyzing gas behavior under the ideal gas law.

Average Kinetic Energy

Next up is the concept of average kinetic energy, a term often linked with the temperature of the gas. The Kinetic Molecular Theory teaches us that the average kinetic energy of gas molecules is directly proportional to the absolute temperature (T) of the gas. It follows the equation:
\[KE_{avg} = \frac{3}{2} k_B T\]
where \(k_B\) is the Boltzmann constant. Important to note here is that this relationship does not depend on the identity of the gas or its molecular mass; all ideal gas molecules at the same temperature have the same average kinetic energy.

In the comparison between gases A and B, since they are at the same temperature and under the same conditions, they share identical average kinetic energies. This highlights a fundamental principle in the study of gases—their kinetic energy is a thermal property, rather than a chemical one.

Root-Mean-Square Velocity

To comprehend the differences in the behavior of gas molecules, we also look at their root-mean-square (rms) velocity. This measurement is indicative of the speed at which gas molecules move on average and is calculated through the equation:
\[v_{rms} = \sqrt{\frac{3k_BT}{m}}\]
Here, \(m\) stands for the mass of a single molecule, not the molar mass. In situations where multiple gases are at the same temperature, the average kinetic energy levels out, but their velocities may differ due to variations in molecular mass.

For gases A and B, one cannot determine which has a faster average velocity without knowing their respective molecular masses, but in general, the gas with the lower molecular mass will have a higher rms velocity. This is because the smaller mass at the same energy allows for faster movement, just as a lighter sports ball can be thrown faster than a heavier one if both are given the same amount of kinetic energy.

Ideal Gas Law

Lastly, let's delve into the Ideal Gas Law, encapsulated in the iconic equation:
\[PV = nRT\]
This equation defines the relationship between the pressure (P), volume (V), number of moles (n), temperature (T), and the ideal (universal) gas constant (R). It is a cornerstone of gas law calculations and is useful for answering questions like how gases exert pressure and behave under varying conditions.

In our textbook problem, we explored why gases A and B, despite potentially having different types of molecules, have the same pressure. The ideal gas law elucidates that if volume and temperature are kept constant, any differences in the number of moles or types of gases are accounted for in the pressure variable. Larger gas B molecules may hit the walls of the container more forcefully, which would suggest higher pressure, but this can be counterbalanced by fewer gas B molecules being present (due to a larger molecular mass and thus fewer molecules per mole) compared to gas A, resulting in an equilibrium where both gases exert an equal pressure on their respective containers.