(c) Calculate the most probable speed of an ozone molecule in the stratosphere, where the temperature is 270 K.

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Identify the formula for the most probable speed of a gas molecule, which is given by the equation: \(v_{mp} = \sqrt{\frac{2kT}{m}}\), where \(v_{mp}\) is the most probable speed, \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of one molecule of the gas.

Convert the molar mass of ozone (O\(_3\)) from grams per mole to kilograms per molecule. The molar mass of ozone is approximately 48 g/mol. To find the mass of one molecule, use the relation: \(m = \frac{M}{N_A}\), where \(M\) is the molar mass in kg/mol and \(N_A\) is Avogadro's number.

Substitute the given temperature (270 K) and the calculated mass of an ozone molecule into the formula for the most probable speed.

Use the value of the Boltzmann constant \(k = 1.38 \times 10^{-23}\) J/K in the equation.

Simplify the expression to find the most probable speed of the ozone molecule in the stratosphere.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Molecular Theory

Kinetic Molecular Theory explains the behavior of gases in terms of particles in constant motion. It posits that the temperature of a gas is directly related to the average kinetic energy of its molecules. This theory helps in understanding how temperature influences the speed of gas molecules, which is crucial for calculating the most probable speed of an ozone molecule.

Most Probable Speed

The most probable speed of gas molecules is the speed at which the largest number of molecules are moving at a given temperature. It can be calculated using the formula v_mp = sqrt(2kT/m), where v_mp is the most probable speed, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of the gas molecule. This concept is essential for determining the speed of ozone molecules in the stratosphere.

Ideal Gas Law

The Ideal Gas Law (PV = nRT) relates the pressure, volume, temperature, and number of moles of a gas. While the question focuses on speed, understanding the Ideal Gas Law provides context for how temperature and molecular mass influence gas behavior. It is foundational for comprehending the conditions under which ozone molecules exist in the stratosphere.