The average temperature at an altitude of 20 km is 220 K. What is the average speed in m/s of an N2 molecule at this altitude?

Verified step by step guidance

1. Identify the formula for the average speed of a gas molecule, which is given by the equation: \( v_{avg} = \sqrt{\frac{8kT}{\pi m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule.
2. Convert the mass of an \( \text{N}_2 \) molecule from grams per mole to kilograms per molecule. Start by finding the molar mass of \( \text{N}_2 \), which is approximately 28.02 g/mol, and then convert it to kg by dividing by 1000.
3. Use Avogadro's number \( 6.022 \times 10^{23} \) to convert the molar mass in kg/mol to kg/molecule by dividing the mass in kg/mol by Avogadro's number.
4. Substitute the values into the formula: \( k = 1.38 \times 10^{-23} \text{ J/K} \), \( T = 220 \text{ K} \), and the calculated mass of an \( \text{N}_2 \) molecule in kg.
5. Calculate the average speed \( v_{avg} \) using the substituted values to find the result in meters per second.

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 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 calculating the average speed of an N2 molecule at a given temperature.

Root Mean Square Speed

The root mean square (RMS) speed is a statistical measure of the speed of particles in a gas. It is calculated using the formula v_rms = sqrt(3RT/M), where R is the ideal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas. This concept is essential for determining the average speed of nitrogen molecules at the specified altitude and temperature.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. While the question focuses on molecular speed, understanding the Ideal Gas Law provides context for how gases behave under different conditions, including temperature and altitude, which can influence molecular motion and speed.