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

a. Pressure: \(Ne_{small} = Ar_{small} < Ne_{large} = Ar_{large}\) b. Average kinetic energy: \(Ne_{small} = Ne_{large} = Ar_{small} = Ar_{large}\) c. Density: \(Ne_{small} < Ne_{large} < Ar_{small} < Ar_{large}\) d. Root mean square velocity: \(Ne_{small} = Ne_{large} > Ar_{small} = Ar_{large}\)

Step-by-step solution

Define the known variables and constants

Let's start by defining the known variables and constants given in the problem. We have two gases, argon and neon, with different masses and volumes. Also, the mass of an argon atom is twice the mass of a neon atom. m(Ar) = 2 * m(Ne) - Mass of an argon atom is twice the mass of a neon atom V(Ar_large) = 2 * V(Ne_large) = V(Ar_small) = V(Ne_small) - Volumes of containers Now, let's analyze each part of the exercise.

Comparing pressures

The ideal gas law relates pressure, volume, the number of moles, and temperature: P * V = n * R * T Since all the samples are at the same temperature, we can compare the pressures using their volumes and number of moles. We know that the volumes of containers are different, but since we assume an equal number of moles in each container, the pressure can be ranked based on their volumes: P(Ne_small) = P(Ar_small) < P(Ne_large) = P(Ar_large) The order of pressures is: Ne_small = Ar_small < Ne_large = Ar_large

Comparing average kinetic energies

The average kinetic energy of a molecule in an ideal gas can be calculated using the equation: KE(avg) = (3/2) * k * T Since the temperature is constant, the average kinetic energy is the same for all samples of gases. The order of average kinetic energy is: Ne_small = Ne_large = Ar_small = Ar_large

Comparing densities

The density can be calculated using the ideal gas law: ρ = (n * M) / V where M is the molar mass of the gas and n is the number of moles. Since we know the atomic masses of argon and neon are different, let's compare the densities using their volumes and masses: ρ(Ne_small) < ρ(Ne_large) < ρ(Ar_small) < ρ(Ar_large) The order of densities is: Ne_small < Ne_large < Ar_small < Ar_large

Comparing root mean square velocities

The root mean square velocity can be calculated using the kinetic theory of gases: v(rms) = sqrt((3 * R * T) / M) Since the temperature is constant, we can compare the root mean square velocities using their atomic masses: v(rms)(Ne) > v(rms)(Ar) The order of root mean square velocities is: Ne_small = Ne_large > Ar_small = Ar_large #Summary#: a. Pressure: Ne_small = Ar_small < Ne_large = Ar_large b. Average kinetic energy: Ne_small = Ne_large = Ar_small = Ar_large c. Density: Ne_small < Ne_large < Ar_small < Ar_large d. Root mean square velocity: Ne_small = Ne_large > Ar_small = Ar_large

Key concepts

Gas Density

When we talk about gas density, we're referring to the mass of gas per unit volume. In the context of the ideal gas law, represented by the equation \( PV = nRT \), density (\( \rho \) can be derived as \( \rho = \frac{m}{V} \), where \( m \) is the mass of the gas and \( V \) is the volume it occupies. Since gases can compress and expand, their density is inversely proportional to their volume under constant temperature and mass.

Considering the exercise, where we compare the densities of argon and neon in containers of different sizes, the mass of argon is greater due to its larger atomic mass. Hence, even when neon and argon are in containers of the same size, argon's density is higher. When the neon gas is placed in a larger container, its density decreases due to the increase in volume, maintaining the same mass. This is critical in understanding how gas density varies with changes in volume and mass.

Kinetic Energy of Gases

The kinetic energy of gases plays a central role in understanding their temperature and the kinetic molecular theory. It's the energy that particles possess due to their motion. The equation \( KE_{avg} = \frac{3}{2}kT \) helps us calculate the average kinetic energy (\( KE_{avg} \) where \(*k\) is Boltzmann's constant and \( T \) is the absolute temperature.

In our exercise scenario, because the temperature is constant for all gas samples, they all share the same average kinetic energy. It's important to note that while the individual speeds of gas molecules vary, the temperature is a measure of their average kinetic energy. This concept indicates that under the same conditions of temperature, different gases have the same average kinetic energy regardless of their mass.

Root Mean Square Velocity

Root mean square velocity (\(*v_{rms}\)) is a way to express the average speed of particles in a gas. The formula to calculate it is \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the ideal gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas. It provides a useful measure of the speed of gas molecules that factors in their mass and the temperature.

In the exercise, the root mean square velocity is higher for neon than argon because neon has a lower molar mass. Even when comparing larger and smaller containers, as long as the conditions remain consistent, the root mean square velocity is not affected by the volume of the gas but rather by its molar mass and the temperature.

Gas Pressure

Gas pressure is the force exerted by the gas molecules colliding with the surfaces of their container, and it’s one of the critical variables in the ideal gas law equation. In a more mathematical sense, pressure (\( P \)) in the ideal gas law is directly proportional to both the number of moles (\( n \)) and the temperature (\( T \)), and inversely proportional to the volume (\( V \)).

With respect to our problem, it's essential to recognize that the pressure of a gas is independent of its type when the number of moles and temperature are constant. For the smaller and larger containers holding an equal number of moles of gas at the same temperature, the pressure in the smaller container is higher because there is less volume for the gas particles to move, leading to more frequent collisions with the container's walls. This is why both neon and argon exhibit the same pressure in containers of the same size, and this concept helps us understand how gas pressure is influenced by volume changes.