The coldest temperature recorded at ground level on Earth was -89.2 °C at the Vostok Station in Antarctica. What is the speed of a nitrogen molecule at this temperature? (LO 10.9) (a) 933 m/s (b) 40.2 m/s (c) 404 m/s (d) 12.8 m/s

Verified step by step guidance

1. Identify the formula for the root-mean-square speed of a gas molecule: \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a nitrogen molecule.
2. Convert the given temperature from Celsius to Kelvin using the formula: \( T(K) = T(°C) + 273.15 \).
3. Calculate the mass of a nitrogen molecule. First, find the molar mass of nitrogen (N2), which is approximately 28.02 g/mol. Convert this to kilograms per molecule by dividing by Avogadro's number (\( 6.022 \times 10^{23} \) molecules/mol) and converting grams to kilograms.
4. Substitute the values for \( k \), \( T \), and \( m \) into the root-mean-square speed formula to find \( v_{rms} \).
5. Compare the calculated \( v_{rms} \) with the given options to determine the correct answer.

Key Concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory explains the behavior of gases in terms of particles in constant motion. It posits that gas molecules move rapidly and randomly, and their speed is influenced by temperature. As temperature decreases, the average kinetic energy of the molecules also decreases, leading to slower molecular speeds.

Root Mean Square Speed

Root Mean Square Speed (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 absolute temperature in Kelvin, and M is the molar mass of the gas. This concept is crucial for determining the average speed of gas molecules at a given temperature.

Temperature Conversion

Temperature conversion is essential for solving problems involving temperature in different units. In this case, the temperature must be converted from Celsius to Kelvin by adding 273.15. This conversion is necessary because gas laws and kinetic energy calculations require absolute temperature measurements to ensure accuracy in the results.