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

To arrange the gas samples by the attributes given: a. Pressure: Compare by the number of moles and the container volume; more moles and smaller volume lead to higher pressure. b. Average Kinetic Energy: Since the temperature is the same for all samples, their average kinetic energy will be equal. c. Density: Rank based on the product of moles and molar mass divided by the container volume; higher product and smaller volume lead to increased density. d. Root Mean Square Velocity: Compare using \( v_{rms} = \sqrt{\frac{3kT}{m}} \), considering the mass ratio between argon and neon atoms (m_argon = 2 * m_neon).

Step-by-step solution

Define variables and constants

Let's assign symbols to the variables representing pressure (P), average kinetic energy (KE), density (ρ), and root mean square velocity (v_rms). Let's also assign symbols representing the number of moles (n), volume (V), Boltzmann's constant (k), and temperature (T) for each gas sample. Assume mass (m) of the argon atom is twice the mass of a neon atom.

Use the ideal gas law to compare gas pressures

The ideal gas law is given by the equation: PV = nRT We want to compare the pressures of the gas samples. We assume that the temperature and the gas constant R are the same for all samples giving us: P = nR * (T/V) We know that the larger containers have a volume twice the volume of the smaller containers. Therefore, in comparing the pressures, we find that the gas with the more moles and smaller volume will have the highest pressure and vice versa.

Compare the average kinetic energies of gas samples

The average kinetic energy of a gas molecule is given by: KE = (3/2) * kT Since the temperature is the same for all gas samples, their average kinetic energy will also be the same.

Compare the densities of gas samples

Density (ρ) can be defined as mass (m) per unit volume (V). The mass of a gas sample can be calculated using the equation: m = n * M where M is the molar mass and n is the number of moles for each of the gases. Using the given information, we know that the mass of an argon atom is twice the mass of a neon atom. Therefore, the ranking of densities would depend on the product of moles and molar mass for each gas sample divided by the volume of their container.

Compare the root mean square velocities of gas samples

The root mean square (rms) velocity of a gas molecule is given by the equation: v_rms = \(\sqrt{(\frac{3kT}{m})}\) The temperature is the same for all gas samples, and we know the mass ratio between argon and neon atoms (m_argon = 2 * m_neon). We can now compare the samples based on their root mean square velocities. In summary, these are the key steps to ordering gas samples by various attributes. Remember to consider molar mass, molecular mass, and container volume when comparing the samples, as these factors will influence the ranking of these attributes.

Key concepts

Pressure

Pressure is a crucial concept in understanding how gases behave. It is defined as the force exerted by the gas molecules per unit area. Imagine gas molecules in a container swirling around, colliding with the walls. These collisions exert a force, creating pressure.

Using the Ideal Gas Law, pressure (\( P \)) can be related to the number of gas moles (\( n \)), the gas constant (\( R \)), temperature (\( T \)), and volume (\( V \)):

• \( P = \frac{nRT}{V} \)

This equation tells us that pressure is directly proportional to the amount of gas and the temperature, but inversely proportional to volume. So, more gas at a higher temperature in a smaller container means higher pressure.

Why It Matters

Understanding how pressure works helps explain weather patterns, how we breathe, and how engines work. It's all about those little gas molecules bouncing around!

Root Mean Square Velocity

Root mean square velocity (\( v_{rms} \)) measures how fast gas molecules move on average. Picture tiny invisible ping-pong balls moving at different speeds inside a container.

The formula for calculating \( v_{rms} \)is:

• \( v_{rms} = \sqrt{\frac{3kT}{m}} \)

where \( k \)is the Boltzmann constant, \( T \)is the temperature, and \( m \)is the mass of a single molecule.

Impact on Different Gases

Although gases might be at the same temperature, their molecular masses vary. Heavier molecules like argon will have a lower \( v_{rms} \)compared to lighter ones such as neon. Hence, lighter gases tend to move faster!

Ideal Gas Law

The Ideal Gas Law is a foundational equation that helps us understand the behavior of gases. Represented as:

• \( PV = nRT \)

it connects pressure (\( P \)), volume (\( V \)), number of moles (\( n \)), the gas constant (\( R \)), and temperature (\( T \)) in a single relationship.

The beauty of the Ideal Gas Law is its simplicity and versatility, making it a go-to tool for chemists and physicists.

Using the Law

By manipulating this equation, you can predict how changing one variable (like temperature) affects another (like volume), assuming other factors stay constant. It's like a mathematical Swiss Army knife for gas calculations!

Kinetic Energy

Kinetic energy in gases is all about motion. Every gas particle has kinetic energy due to its motion, and this energy plays a crucial role in determining other gas properties like temperature. The average kinetic energy (\( KE \)) of a gas is expressed by:

• \( KE = \frac{3}{2}kT \)

where \( k \)is the Boltzmann constant and \( T \)is the temperature.

Consistency Across Gases

At a given temperature, all gases have the same average kinetic energy. This is why temperature is a measure of the average kinetic energy of particles in a substance. Whenever temperature increases, so does kinetic energy, causing molecules to move faster.

Density

Density is a measure of how much mass is contained in a unit volume of a material. For gases, it’s slightly more nuanced because their volume can change easily with pressure and temperature.

To find the density (\( \rho \)) of a gas, you use the formula:

• \( \rho = \frac{m}{V} \)

where \( m \)is the mass and \( V \)is the volume.

Influencing Factors

A gas can be more dense if it has a larger mass or if it's in a smaller volume. Comparing gases like argon and neon, knowing that argon has atoms with more mass, it would generally be denser if both gases are at the same conditions.