Question

As we know, UF \(_{6}\) 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 a given temperature, both UF\(_{6}\) and helium will have the same average kinetic energy, despite UF\(_{6}\) being heavier, because according to the kinetic theory of gases, kinetic energy of gases depends only on temperature and it is same for all gas molecules at a particular temperature.

Step-by-step solution

Understand the Kinetic Theory of Gases

The kinetic theory of gases explains a number of properties of gases by assuming that gases are composed of very small particles in constant, random motion. The key insight from this theory relevant to this problem is the idea that the average kinetic energy of a gas is directly proportional to its temperature, not its mass. This means that for any two gases at the same temperature, the average kinetic energy of the gas particles in each case will be the same. This explains why UF\(_{6}\) and helium, despite the heavier mass of UF\(_{6}\), have the same average kinetic energy at a given temperature.

Discuss how Mass doesn't affect Average Kinetic Energy

Though UF\(_{6}\) is heavier than helium, the average kinetic energy of the two gases at a given temperature is the same because kinetic energy depends on temperature and not on the mass of the gas. The mass of the molecular gas affects the speed of particles, with lighter particles generally moving faster than heavier ones, but does not affect the kinetic energy. So, irrespective of the mass, the kinetic energy of all molecular gases is nearly the same at a particular temperature.

Key concepts

Average Kinetic Energy

The average kinetic energy of gas molecules is an essential concept in understanding the behavior of gases. It is defined by the kinetic theory of gases, which describes how gas particles are constantly in motion. The average kinetic energy of gas particles at a certain temperature represents the energy associated with this motion.

The formula for average kinetic energy is given by:
\[ KE_{avg} = \frac{3}{2} k T \]
where \( KE_{avg} \) is the average kinetic energy, \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin.
- It's important to note that the average kinetic energy depends solely on temperature, not on the type or mass of the gas.
- Therefore, two gases at the same temperature will have the same average kinetic energy, regardless of molecular weight. This is why the relatively heavy UF\(_{6}\) and lighter helium can have the same average kinetic energy when they are both at the same temperature.

Gas Temperature

Temperature is a crucial factor in the discussion about kinetic energy in gases. It is a measure of the average kinetic energy of the particles in a substance. In gases, temperature reflects how fast the particles move.

Effect of Temperature on Gases
- Higher temperature means higher average kinetic energy and faster-moving particles.
- Lower temperature results in less kinetic energy and slower particle movement.This relationship is direct and linear, meaning if you double the temperature, you will double the average kinetic energy. It is this straightforward relationship that explains why different gases, regardless of their mass, can have the same kinetic energy if they share the same temperature. Hence, heavier gases like UF\(_{6}\) and much lighter gases like helium will have the same kinetic energy at an identical temperature level.

Molecular Mass and Speed

Molecular mass and speed are interlinked concepts that influence how we understand gas behavior. While mass doesn't affect the kinetic energy of gas particles, it does affect their speed.

The speed of gas particles is inversely proportional to the square root of their mass. This is expressed in the formula for the root mean square speed:\[\text{v}_{rms} = \sqrt{\frac{3kT}{m}}\]where \( \text{v}_{rms} \) is the root mean square speed, \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of the particles.- **Lighter gas particles move faster**: Lighter gases like helium have higher speeds at a given temperature due to their lower mass.- **Heavier gas particles move slower**: Conversely, heavier gases like UF\(_{6}\) have slower speeds at the same temperature.This means despite having the same average kinetic energy, their speeds can be quite different, illustrating the balancing act between mass and speed in gases.