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?

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

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

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

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%)?

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

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

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

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

$\overline{c}=\sqrt{\overline{c^2}}=\sqrt{1745}=41.8{\text{ ms}}^{-1}$

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?

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