Problem 91 - Chemistry: The Central Science

Question

At \(20^{\circ} \mathrm{C}\) (approximately room temperature) the average velocity of \(\mathrm{N}_{2}\) molecules in air is \(1050 \mathrm{mph}\). (a) What is the average speed in \(\mathrm{m} / \mathrm{s}\) ? (b) What is the kinetic energy (in J) of an \(\mathrm{N}_{2}\) molecule moving at this speed? (c) What is the total kinetic energy of \(1 \mathrm{~mol}\) of \(\mathrm{N}_{2}\) molecules moving at this speed?

Step-by-Step Solution

Answer

(a) The average speed in meters per second is \(1050 \times \frac{1609.34}{3600} \approx 469.62 \, \mathrm{m/s}\). (b) The kinetic energy of a single N2 molecule is approximately \(KE \approx 6.389 \times 10^{-21} \, \mathrm{J}\). (c) The total kinetic energy of one mole of N2 molecules moving at this speed is approximately \(8.452 \times 10^{3} \, \mathrm{J}\).

Step 1: Convert velocity from mph to m/s

To convert the velocity given in mph (miles per hour) to m/s (meters per second), you can use the following conversion factor: 1 mile = 1609.34 meters and 1 hour = 3600 seconds. The average velocity of an N2 molecule is 1050 mph. Let's now convert it to m/s: Average velocity (m/s) = 1050 mph * (1609.34 m / 1 mile) * (1 hour / 3600 s) Calculating this will give us the average speed in m/s.

Step 2: Calculate kinetic energy

Now, let's find the kinetic energy of a single N2 molecule moving at this speed. The formula for the kinetic energy is: KE = \(\frac{1}{2}\) * m * v^2 where KE is the kinetic energy, m is the mass of the molecule, and v is the average velocity. We are given the average velocity of N2 molecules in m/s from part (a) and the mass of one N2 molecule can be calculated as: Mass of one N2 molecule = (28 g/mol) * (1 mol / 6.022 * 10^23 molecules) * (1 kg / 1000 g) Now, substitute the values into the equation for the kinetic energy and calculate the result.

Step 3: Calculate the total kinetic energy of one mole

For part (c), we need to find the total kinetic energy of one mole of N2 molecules moving at the average velocity given. We know there is exactly 6.022 × 10^23 molecules per mole. So, the total kinetic energy for 1 mole of N2 molecules can be found by multiplying the kinetic energy of each single N2 molecule (found in part (b)) by the number of molecules in one mole: Total Kinetic Energy = Kinetic Energy of 1 molecule * number of molecules in 1 mole Now, just plug in the values and calculate the result.