Question

Consider the following samples of gases at the same temperature. Arrange each of these samples in order from lowest to highest: a. pressure b. average kinetic energy c. density d. root mean square velocity Note: Some samples of gases may have equal values for these attributes. Assume the larger containers have a volume twice the volume of the smaller containers, and assume the mass of an argon atom is twice the mass of a neon atom.

Short answer

In summary, for the given samples of gases: a. Pressure: All samples (A, B, C, and D) have the same pressure. b. Average Kinetic Energy: All samples (A, B, C, and D) have the same average kinetic energy. c. Density: The order by increasing density is A = C < B = D. d. Root Mean Square Velocity: The order by increasing root mean square velocity is B = D < A = C.

Step-by-step solution

a. Pressure

To determine the pressure of each sample, we can use the ideal gas law formula: PV = nRT, where P is pressure, V is volume, n is the amount of substance, R is the ideal gas constant, and T is temperature (which we know is the same for all samples). Since the temperature is constant across all samples, we can compare the pressure by comparing the ratio of the amount of substance and its volume (n/V). 1. Sample A: X / V_small. 2. Sample B: X / V_small. 3. Sample C: 2X / (2V_small) -> X / V_small. 4. Sample D: 2X / (2V_small) -> X / V_small. All samples have the same pressure, since they have the same ratio of amount and volume.

b. Average Kinetic Energy

Average kinetic energy of a gas is given by the formula (3/2) * kT, where k is the Boltzmann's constant and T is temperature. Since all samples have the same temperature, they also have the same average kinetic energy.

c. Density

Density can be calculated as mass/volume. Let's compare the density of each sample: 1. Sample A: X * Mass_neon / V_small. 2. Sample B: X * Mass_argon (which is 2Mass_neon) / V_small -> 2X * Mass_neon / V_small. 3. Sample C: 2X * Mass_neon / (2V_small) -> X * Mass_neon / V_small. 4. Sample D: 2X * Mass_argon / (2V_small) -> X * Mass_argon / V_small -> 2X * Mass_neon / V_small. The order by increasing density is A = C < B = D.

d. Root Mean Square Velocity

The root mean square velocity (v_rms) is given by the formula: v_rms = \(\sqrt{\frac{3RT}{M}}\), where R is the ideal gas constant, T is temperature, and M is the molar mass of the gas. Temperature is constant across all samples, so the main factor to consider for comparison is the molar mass of the gas. 1. Sample A: v_rms_Neon, as it's Neon gas. 2. Sample B: v_rms_Argon, as it's Argon gas, and it will be slower than v_rms_Neon because of the larger molar mass of Argon. 3. Sample C: Again, it's Neon gas, so v_rms_Neon. 4. Sample D: It's Argon gas, so v_rms_Argon, slower than Neon. The order by increasing root mean square velocity is B = D < A = C. In summary: a. Pressure: A = B = C = D. b. Average Kinetic Energy: A = B = C = D. c. Density: A = C < B = D. d. Root Mean Square Velocity: B = D < A = C.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure (P), volume (V), temperature (T), and amount of substance (n) for an ideal gas. Presented as the formula \( PV = nRT \) where R represents the ideal gas constant. This law assumes that gases consist of tiny particles moving in random motion with no intermolecular forces affecting them.

When analyzing different samples of gases at constant temperature, like in our exercise, we can deduce that the pressure of the samples is not dependent on the type of gas, provided the number of moles and volume are kept constant. Understanding the ideal gas law is crucial for gauging how gases will behave under different conditions and is the foundation for many chemistry and physics calculations.

Kinetic Molecular Theory

Kinetic Molecular Theory (KMT) explains the properties and behavior of gases by considering them as collections of fast-moving particles in random motion. This theory states that the pressure exerted by a gas results from collisions of gas particles with the walls of the container.

KMT also implies that at a given temperature, all gases have the same average kinetic energy, explaining why the average kinetic energy for each gas sample in our problem is equal. It's the inherent energy due to the motion of particles, represented by the equation \( (3/2)kT \) where k is Boltzmann's constant. The theory helps to elucidate not just the macroscopic gas laws but also the microscopic origins of temperature and pressure.

Root Mean Square Velocity

Root mean square velocity (v_rms) takes into account the individual speeds of all the particles in a gas sample, squaring them, averaging them, and then taking the square root of this average. This velocity is essential as it's directly linked to the kinetic energy of the gas.

The formula is given by \( v_\text{rms} = \sqrt{\frac{3RT}{M}} \) where R is the ideal gas constant, T is temperature, and M is the molar mass. In practical terms, heavier gas molecules will move slower than lighter ones at the same temperature, which is why in our sample comparison, argon has a lower root mean square velocity than neon due to its higher molar mass.

Density of Gases

The density of a gas is determined by its mass per unit volume. Mathematically put, density (\( \rho \)) is mass (m) divided by volume (V), or \( \rho = \frac{m}{V} \).

In a scenario like our textbook problem, density is directly proportionate to the molar mass of the gas and the number of moles, while inversely proportionate to its volume. This explains why, even though samples B and D contain argon gas with twice the mass of neon in samples A and C, their relative densities are equal because the sample sizes are proportionally larger. The concept of density is pivotal when comparing how different gases will stratify or respond to changes in pressure or temperature.

Molar Mass

Molar mass is a physical property defined as the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. It is expressed in units of grams per mole (g/mol). It's critical in converting between grams of a substance and moles, hence relating a gas's molecular weight to its physical properties.

In our exercise, the molar mass comes into play when comparing the density and root mean square velocities of neon and argon. While neon, with a lower molar mass, moves faster (represented by higher v_rms), argon, with double the molar mass, moves slower and contributes to higher density assuming equal mole quantities and volumes as neon.