Problem — Tyre on the motorway

Problem

A tyre has pressure $p_1 = 2{,}0$ bar at $T_1 = 15\,^\circ$C. After thirty minutes on the motorway the temperature of the rubber rises to $T_2 = 45\,^\circ$C. Approximating the volume as constant, calculate the new pressure. Why is it recommended to check the pressure “when cold”?

Solution

At constant volume Gay-Lussac’s law holds: $p\propto T$, with $T$ in kelvin. $\frac{p_1}{T_1}=\frac{p_2}{T_2}$ I convert the temperatures to kelvin: $T_1=15+273{,}15=288{,}15\ \text{K}\qquad T_2=45+273{,}15=318{,}15\ \text{K}$ I find $p_2$: $p_2=p_1\,\frac{T_2}{T_1}=2{,}0\cdot\frac{318{,}15}{288{,}15}$ $p_2=2{,}0\cdot 1{,}104=2{,}21\ \text{bar}$ The pressure rises by about 10%. This is why one checks “when cold”: with the tyre cold the temperature is close to the reference nominal value, so the reading corresponds to the correct value; when hot the same tyre would give a higher reading and would throw off the adjustment. $\ev{p_2\approx 2{,}21\ \text{bar}}$

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Topics: Kinetic theory of gases Concepts: Ideal gas law · Pressure · Temperature