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?

Short answer

(a) Average speed in m/s = \(1050 \times 0.44704 \approx 469.4 \,\text{m/s}\) (b) Kinetic energy of a single N2 molecule = \(\frac{1}{2}\times m \times (1050 \times 0.44704)^2\approx 4.71 \times 10^{-21} \mathrm{J}\) (c) Total kinetic energy of 1 mol of N2 molecules = \(KE \times NA \approx 2.83 \times 10^3 \,\text{J}\)

Step-by-step solution

Convert mph to m/s

To convert the given speed to m/s, we use the conversion factor: 1 mph = 0.44704 m/s So, average speed in m/s = \(1050 \times 0.44704\)

Find the kinetic energy of a single N2 molecule

The kinetic energy of a single molecule can be calculated using the formula: \(KE = \frac{1}{2}mv^2\) Where m is the mass of the molecule and v is its average velocity. But first, we need to find the mass of a single N2 molecule. The molecular weight of N2 is approximately 28 g/mol. To find the mass of a single molecule, we will use Avogadro's number, NA = \(6.022\times10^{23} \mathrm{mol}^{-1}\). We also need to convert the molecular mass to kg. Mass of a single N2 molecule in kg: \(m = \frac{28 \times 10^{-3} \, \text{kg/mol}}{6.022\times10^{23}\, \text{mol}^{-1}}\) Now we can find the kinetic energy: \(KE = \frac{1}{2}\times m \times \left(1050 \times 0.44704\right)^2\)

Find the total kinetic energy of 1 mol of N2 molecules

To find the total kinetic energy for 1 mol of N2 molecules, multiply the kinetic energy of a single molecule (found in step 2) by Avogadro's number: Total kinetic energy = \(KE \times NA\) Combining all the steps in the solution, we get: (a) Average speed in m/s = \(1050 \times 0.44704\) (b) Kinetic energy of a single N2 molecule = \( \frac{1}{2}\times m \times (1050 \times 0.44704)^2\) (c) Total kinetic energy of 1 mol of N2 molecules = \(KE \times NA\)