13 5.3 The pressure law
The pressure law: At constant volume, the pressure of a fixed mass of gas increases by 1/273 of its pressure at 0 °C for every 1 °C rise in temperature.
Second form (Kelvin scale). From similar triangles: P₁/T₁ = P₂/T₂, i.e.
P / T = constant (fixed mass, constant volume), so P ∝ T
At constant volume, the pressure of a fixed mass of gas is directly proportional to its absolute temperature.
Note that the coefficient of volume expansion at constant pressure αv and the coefficient of pressure increase at constant volume βP are equal for all gases: both = 1/273 K⁻¹.
Example 1. A gas has a pressure of 45 cm Hg at 27 °C (300 K). Its volume is constant. At 127 °C (400 K): P₂ = 45 × 400/300 = 60 cm Hg.
Example 2. A sealed container of air is at 0 °C with pressure 80 cm Hg. It is cooled to −91 °C (182 K): P = 80 × 182/273 = 53.3 cm Hg.
Example 3 (car tyre). A tyre has air at 2.0 atm at 7 °C (280 K). After driving, the air is at 47 °C (320 K). Assuming constant volume, P = 2.0 × 320/280 = 2.29 atm. (That is why tyre pressure should be checked when the tyres are cold.)
Example 4 (find the temperature). The pressure of a gas at constant volume rises from 1 atm at 27 °C to 1.5 atm. T₂ = 300 × 1.5 = 450 K = 177 °C.