Which has a higher average speed, a Ferrari at 145 mph or a gaseous UF6 molecule at 145 °C?

Verified step by step guidance

1. Identify the type of speeds to compare: the speed of a Ferrari (a macroscopic object) and the average speed of a gaseous UF6 molecule (a microscopic particle).
2. Understand that the speed of the Ferrari is given directly as 145 mph. Convert this speed into meters per second (m/s) for consistency in units when comparing with molecular speeds.
3. Use the formula for the average speed of a gas molecule, which is given by \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant (1.38 x 10^{-23} J/K), \( T \) is the temperature in Kelvin, and \( m \) is the mass of the molecule in kilograms.
4. Convert the temperature of the UF6 gas from degrees Celsius to Kelvin by adding 273.15 to the Celsius temperature.
5. Calculate the mass of a UF6 molecule by using its molar mass (352 g/mol) and converting it to kilograms per molecule, considering Avogadro's number (6.022 x 10^{23} molecules/mol).

Key Concepts

Average Speed of Molecules

The average speed of gas molecules is influenced by their temperature and mass. According to the kinetic molecular theory, as temperature increases, the average kinetic energy of the molecules also increases, leading to higher speeds. For gases, this speed can be calculated using the root-mean-square speed formula, which incorporates the temperature and molar mass of the gas.

Kinetic Molecular Theory

Kinetic molecular theory explains the behavior of gases in terms of particles in constant motion. It posits that gas molecules are in continuous, random motion and that their speed is related to temperature. This theory helps in understanding how temperature affects the kinetic energy and speed of gas molecules compared to macroscopic objects like cars.

Comparison of Speeds

To compare the average speed of a Ferrari and a gaseous UF6 molecule, one must consider the units and context of each speed. The Ferrari's speed is given in miles per hour, while the speed of the UF6 molecule must be calculated in terms of molecular speed, typically in meters per second. This requires converting the temperature of UF6 into kinetic energy to determine its average speed, allowing for a direct comparison.