Traffic on the German autobahns reaches speeds of up to 230 km/h. At what temperature (°C) do oxygen molecules have this same average speed?

Verified step by step guidance

1. Step 1: Understand that the problem involves finding the temperature at which oxygen molecules have an average speed of 230 km/h. This requires using the root-mean-square speed formula for gases.
2. Step 2: Recall 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 single molecule.
3. Step 3: Convert the given speed from km/h to m/s for consistency with SI units. Use the conversion factor: 1 km/h = 0.27778 m/s.
4. Step 4: Rearrange the root-mean-square speed formula to solve for temperature \( T \): \( T = \frac{m v_{rms}^2}{3k} \).
5. Step 5: Calculate the mass \( m \) of an oxygen molecule. Use the molar mass of oxygen (O2), which is approximately 32 g/mol, and convert it to kg per molecule using Avogadro's number. Then, substitute all known values into the rearranged formula to find \( T \) in Kelvin, and 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 posits that the average kinetic energy of gas molecules is directly proportional to the temperature of the gas in Kelvin. This theory helps us understand how temperature affects the speed of gas molecules, which is crucial for solving the problem of finding the temperature corresponding to a specific molecular speed.

Average Speed of Gas Molecules

The average speed of gas molecules can be calculated using the equation derived from kinetic theory, which relates the speed of gas particles to temperature. The formula v = sqrt((3RT)/M) shows that the average speed (v) is dependent on the gas constant (R), temperature (T), and molar mass (M) of the gas. This relationship allows us to determine the temperature at which oxygen molecules travel at a given speed.

Conversion Between Temperature Scales

Understanding how to convert between temperature scales is essential for this problem. The Kelvin scale is used in gas laws, and to convert Celsius to Kelvin, one must add 273.15. This conversion is necessary when calculating the temperature that corresponds to the average speed of oxygen molecules, ensuring that the calculations are accurate and consistent with the principles of thermodynamics.