A big-league fastball travels at about 45 m/s. At what temperature (°C) do helium atoms have this same average speed?

Verified step by step guidance

1. Identify the formula for the root-mean-square speed of gas molecules: \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( v_{rms} \) is the root-mean-square speed, \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a gas molecule.
2. Rearrange the formula to solve for temperature \( T \): \( T = \frac{m v_{rms}^2}{3k} \).
3. Convert the given speed of the fastball (45 m/s) to the root-mean-square speed \( v_{rms} \) for helium atoms.
4. Determine the mass \( m \) of a helium atom. Use the molar mass of helium (4.00 g/mol) and convert it to kilograms per molecule using Avogadro's number.
5. Substitute the values for \( v_{rms} \), \( m \), and \( k \) into the rearranged formula to calculate the temperature \( T \) in Kelvin, then convert it to Celsius by subtracting 273.15.

Key Concepts

Kinetic Molecular Theory

Kinetic Molecular Theory explains the behavior of gases in terms of particles in constant motion. It states that the average kinetic energy of gas particles is directly proportional to the temperature of the gas in Kelvin. This theory helps relate the speed of gas particles to temperature, allowing us to calculate the temperature at which helium atoms would have a specific average speed.

Average Speed of Gas Particles

The average speed of gas particles can be determined using the equation derived from kinetic energy principles. For an ideal gas, the average speed (v) is related to temperature (T) and molar mass (M) by the equation v = sqrt((3kT)/m), where k is the Boltzmann constant and m is the mass of a single particle. This relationship is crucial for finding the temperature corresponding to a given speed.

Temperature Conversion

Temperature in scientific calculations is typically expressed in Kelvin, which is an absolute temperature scale. To convert from Celsius to Kelvin, one must add 273.15 to the Celsius temperature. Understanding this conversion is essential when calculating the temperature at which helium atoms achieve a specific average speed, as the calculations must be performed in Kelvin.