The mean square speed \langle \(c^{2}\) \rangle is the average of the squares of the speeds of
Thermal Physics - OCR A-Level Physics
$$pV = \frac{1}{3}Nm\langle c^2 \rangle$$
$$\frac{1}{2}m\langle c^2 \rangle = \frac{3}{2}kT$$
- The mean square speedThe average of the squared speeds of all molecules: ⟨c²⟩ = (c₁² + c₂² + ... + cₙ²)/N. \langle \(c^{2}\) \rangle is the average of the squares of the speeds of all molecules.
- The root-mean-square speedThe square root of the mean square speed: c_rms = √⟨c²⟩. It represents a typical molecular speed. $c_{rms} =$ $\sqrt{\langle c^2 \rangle}.$
- The factor of $\frac{1}{3}$ arises because molecules move randomly in 3 dimensions, and on average $\frac{1}{3}$ of the total KE is associated with each dimension.
- Doubling the absolute temperature doubles the mean KE and increases $c_{rms}$by a factor $of \sqrt{2}.$
- At the same temperature, lighter molecules have higher rms speeds (since $\frac{1}{2}$m\langle \(c^{2}\) \rangle is the same for all gases at temperature T).
Examiner Tips and Tricks
When deriving pressureForce per unit area. Measured in pascals (Pa), where 1 Pa = 1 N m⁻². from kinetic theory, you must state: the 1/3 factor comes from averaging over three perpendicular directions of random molecular motion.