At what temperature does the average speed of an oxygen molecule equal that of an airplane moving at 580 mph?

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Convert the speed of the airplane from miles per hour (mph) to meters per second (m/s) using the conversion factor: 1 mph = 0.44704 m/s.

Use the formula for the average speed of a gas molecule, \( v = \sqrt{\frac{3kT}{m}} \), where \( v \) is the speed, \( k \) is the Boltzmann constant (\(1.38 \times 10^{-23} \, J/K\)), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a gas molecule in kilograms.

Calculate the mass of an oxygen molecule (O2). Since the molar mass of oxygen is approximately 16 g/mol, the molar mass of O2 is 32 g/mol. Convert this mass to kilograms per molecule by using Avogadro's number (\(6.022 \times 10^{23} \, molecules/mol\)).

Substitute the converted speed of the airplane into the formula for \( v \) and solve for \( T \). Rearrange the formula to isolate \( T \) on one side: \( T = \frac{m \cdot v^2}{3k} \).

Plug in the values for \( m \), \( v \) (converted speed of the airplane), and \( k \) into the rearranged formula to find the temperature \( T \) in Kelvin.

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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 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 determining the temperature at which an oxygen molecule's speed matches that of an airplane.

Average Speed of Gas Molecules

The average speed of gas molecules can be calculated using the equation v = sqrt((3RT)/M), where v is the average speed, R is the ideal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas. For oxygen (O2), the molar mass is approximately 32 g/mol. This relationship allows us to find the temperature at which the average speed of oxygen molecules equals a specified speed, such as that of an airplane.

Conversion of Units

To solve the problem, it is essential to convert the speed of the airplane from miles per hour (mph) to meters per second (m/s) for consistency with the SI units used in the kinetic molecular equations. The conversion factor is 1 mph = 0.44704 m/s. Understanding how to convert units is crucial for accurately applying the formulas and ensuring that all measurements are compatible.

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