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?

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