Question

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

Short answer

At the same temperature, all gases have the same average kinetic energy despite differences in molecular mass.

Step-by-step solution

Understanding Kinetic Energy

The kinetic energy of a gas is given by the equation \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of a gas molecule and \( v \) is its velocity. The kinetic energy is related to temperature, meaning that all gases at the same temperature have the same average kinetic energy, regardless of their mass.

Kinetic Energy and Temperature Relationship

According to the kinetic molecular theory, the average kinetic energy of gas molecules is proportional to the temperature of the gas in Kelvin. Thus, two different gases at the same temperature will have the same average kinetic energy.

Comparison of Gases

Even though uranium hexafluoride (UF_6) has a much larger molar mass than helium, at a given temperature, the average kinetic energies of both gases are identical due to the temperature dependency. While the mass of UF_6 is greater, its molecules move more slowly compared to the lighter helium molecules, balancing the kinetic energies.

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion. For gas molecules, this is expressed using the formula: \[ KE = \frac{1}{2}mv^2 \] Where \( m \) is the mass of the gas molecule and \( v \) is its velocity. In gases, kinetic energy is distributed among the molecules, making it a crucial concept for understanding how gases behave. It's important to realize that, despite differing masses, all gas molecules at the same temperature have the same average kinetic energy. This may seem counterintuitive as heavier molecules might appear to have more kinetic energy; however, since they move slower, their overall kinetic energy equates to that of a lighter, faster-moving gas.

Temperature Relationship

Temperature plays a vital role in understanding the kinetic energy of gas molecules. According to the kinetic molecular theory, the temperature of a gas directly relates to the average kinetic energy of its molecules. A simple way to remember this is:

• Temperature increase ⇒ Average kinetic energy increase
• Temperature decrease ⇒ Average kinetic energy decrease

This relationship indicates that at a constant temperature, different gases possess the same average kinetic energy, regardless of their molecular weights.

In practical terms, this means that both uranium hexafluoride and helium will have the same average kinetic energy if maintained at the same temperature. However, the massive difference in their weights results in helium molecules moving much faster compared to uranium hexafluoride. Thus, temperature, and not the mass, is the common denominator that determines kinetic energy distribution among gas molecules.

Gas Molecules

Gas molecules, whether light like helium or heavy like uranium hexafluoride, behave in ways that are sometimes not initially obvious. Their behavior is largely dictated by the kinetic molecular theory, and one of its core principles is the relationship between molecular speed, mass, and temperature.
Gas molecules are in constant, random motion. Their velocity and mass dictate their kinetic energy, but due to the temperature relationship, at a fixed temperature, all gases should have the same average kinetic energy. Thus:

• Lighter gases (like helium) move faster to maintain kinetic energy equivalency with heavier gases.
• Heavier gases (like uranium hexafluoride) move slower, balancing out their increased mass with decreased velocity.

Understanding this helps explain why gases with vastly different masses can have the same average kinetic energy at a given temperature. It is a fascinating concept that underscores the unique traits of gases studied in physical chemistry.