## 9. Math Puzzle $2 / 66$

At the beginning of the trip, a car tire at $12{ }^{\circ} \mathrm{C}$ temperature has an overpressure of 1.5 at. How great is the overpressure after a longer drive, if the tire has assumed a temperature of $41^{\circ} \mathrm{C}$ due to friction. We assume the tire volume is constant.
Given are $T_{1}=12^{\circ} \mathrm{C}=285 \mathrm{~K} ; T_{2}=41^{\circ} \mathrm{C}=314 \mathrm{~K}$ and $p_{1}=1.5$ at +1 at $=2.5$ at.

Since the volume remains constant, the volume can be neglected in the ideal gas law equation $\frac{p_{1} V_{1}}{T_{1}}=\frac{p_{2} V_{2}}{T_{2}}$, i.e.,

$$
p_{2}=\frac{p_{1} T_{2}}{T_{1}}=2.5 \cdot \frac{314}{285}=2.75 \mathrm{at}
$$

The overpressure is therefore 1.75 at after a longer drive.