Question

Uranium hexafluoride \(\left(\mathrm{UF}_{6}\right)\) is a much heavier gas than hydrogen, yet at a given temperature, the average kinetic energies of these two gases are the same. Explain.

Short answer

At the same temperature, all gases share the same average kinetic energy despite having different masses, resulting in varying velocities.

Step-by-step solution

Kinetic Energy Formula

According to the kinetic molecular theory, the average kinetic energy for a gas molecule is given by the equation: \( KE = \frac{3}{2}kT \), where \( k \) is Boltzmann's constant and \( T \) is the absolute temperature in Kelvin. This formula indicates that the average kinetic energy is solely dependent on the temperature of the gas.

Temperature Dependence

Since the average kinetic energy of gas molecules depends only on temperature, different gases at the same temperature have identical average kinetic energies, regardless of their molecular weights or identities.

Mass and Velocity Relationship

Kinetic energy can also be expressed as \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the molecule and \( v \) is its velocity. For gases at the same temperature, even though the average kinetic energy is the same, heavier gas molecules (like \( \mathrm{UF}_{6} \)) will move more slowly compared to lighter molecules (like \( \mathrm{H}_2 \)) because of their larger mass.

Conclusion

Both Uranium hexafluoride \( \mathrm{UF}_{6} \) and hydrogen \( \mathrm{H}_2 \) gases have the same average kinetic energy at a given temperature. However, due to the significantly higher mass of \( \mathrm{UF}_{6} \), its molecules move slower than those of \( \mathrm{H}_2 \) to maintain equal average kinetic energies.

Key concepts

Average Kinetic Energy

Understanding the concept of average kinetic energy is key to explaining how gases behave at a molecular level. Kinetic energy is the energy an object possesses due to its motion. For gas molecules, this is determined by their speed and mass. The formula used in kinetic molecular theory to describe average kinetic energy is \( KE = \frac{3}{2}kT \), where \( k \) is Boltzmann's constant and \( T \) is the absolute temperature.

The key insight here is that average kinetic energy depends solely on the temperature and not directly on the type or mass of the gas. This means that gases at the same temperature will have the same average kinetic energy. It's an equalizer across different gases, allowing them to have the same energy despite differing in masses or identities.

Temperature Dependence

Temperature plays a crucial role in determining the kinetic energy of gas molecules. In the kinetic molecular theory, temperature is a measure of the average kinetic energy of the particles in a substance. Since the average kinetic energy of a gas is directly proportional to the absolute temperature, when two gases are at the same temperature, their average kinetic energies are the same.

This temperature dependence means that if you heat a gas, its molecules move faster, increasing the average kinetic energy. Conversely, cooling the gas slows them down, decreasing the energy. Therefore, irrespective of their size or mass, gas molecules at the same temperature have the same kinetic energy. This is why uranium hexafluoride \( (\mathrm{UF}_{6}) \) and hydrogen \( (\mathrm{H}_2) \) can have identical average kinetic energies at the same temperature, despite \( \mathrm{UF}_{6} \) being significantly heavier.

Mass and Velocity Relationship

The relationship between mass and velocity is fundamental to understanding how different gases can have the same kinetic energy. The formula \( KE = \frac{1}{2}mv^2 \) shows that kinetic energy also depends on a gas molecule’s mass \( m \) and velocity \( v \). To have the same average kinetic energy at the same temperature, heavy molecules must move slower than lighter ones.

For instance, while uranium hexafluoride \( (\mathrm{UF}_{6}) \) is much heavier than hydrogen \( (\mathrm{H}_2) \), its molecules travel more slowly to balance out the mass in maintaining kinetic energy equality. Light gases like hydrogen, on the other hand, move at higher speeds. This inverse relationship between mass and velocity ensures that the kinetic energy can remain consistent across different gases at a given temperature.

Gas Molecules

In gases, molecules are in constant random motion, colliding with each other and the walls of the container. The behavior of gas molecules is well explained by the kinetic molecular theory, which helps predict and understand gas properties.

According to this theory, gas molecules are assumed to be in non-stop motion and the energy of their movement relates to temperature. It explains that gases have no definite shape or volume, and they expand to fill their containers.

• Gas molecules are tiny and the space between them is large relative to their size.
• They move in straight lines until they collide with another molecule or the container walls.
• The collisions are perfectly elastic, meaning there's no loss in energy from the motion of particles in each collision.

Understanding these properties is essential to explain how gases, regardless of type or mass, can exhibit similar average kinetic energies when at the same temperature.