A-level Physics (Advancing Physics)/Kinetic Theory/Worked Solutions

'''1. Five molecules are moving at speeds of 1,5,6,8, and 36ms−1. What is their mean square speed?'''

$$\bar{c^2} = \frac{1^2 + 5^2 + 6^2 + 8^2 + 36^2}{5} = 284.4\mbox{ m}^2\mbox{s}^{-2}$$

2. What is the mass of one molecule of N2 (atomic mass 14, 1u = 1.66 x 10−27kg)?

$$2 \times 14 \times 1.66 \times 10^{-27} = 4.648 \times 10^{-26}\mbox{ kg}$$

3. Atmospheric pressure is 101,325 Pa. If one mole of Nitrogen takes up 2.3 m3 at about 10 °C, what is the mean square speed of the molecules in the air outside, assuming that the atmosphere is 100% nitrogen (in reality, it is only 78%)?

$$pV = \frac{1}{3}Nm\bar{c^2}$$

$$101235 \times 2.3 = \frac{1}{3} \times 6.02 \times 10^{23} \times 6.648 \times 10^{-22}\bar{c^2}$$

$$\bar{c^2} = \frac{3 \times 101235 \times 2.3}{6.02 \times 10^{23} \times 6.648 \times 10^{-22}} = 1745\mbox{ m}^2\mbox{s}^{-2}$$

4. What is the average speed of a nitrogen molecule under the above conditions?

$$\bar c = \sqrt{\bar{c^2}} = \sqrt{1745} = 41.8\mbox{ ms}^{-1}$$

'''5. The particles in question 1 are duplicated 3000 times. If they have a completely unrealistic mass of 1g, what is their pressure when they are crammed into a cube with side length 0.5m?'''

$$p = \frac{Nm\bar{c^2}}{3V} = \frac{5 \times 3000 \times 10^{-3} \times 284.4}{3 \times 0.5^3} = 11376\mbox{ Pa}$$