(a) Amonton's law expresses the relationship between pressure and temperature. Use Charles's law and Boyle's law to derive the proportionality relationship between \(P\) and \(T .(\mathbf{b})\) If a car tire is filled to a pressure of \(220.6 \mathrm{kPa}\) measured at \(24^{\circ} \mathrm{C}\), what will be the tire pressure if the tires heat up to \(49^{\circ} \mathrm{C}\) during driving?

Charles's law states the relationship between the temperature and volume of an ideal gas, keeping the pressure constant. The formula for Charles's law is: \[V \propto T\] Boyle's law states the relationship between the pressure and volume of an ideal gas when the temperature is held constant. The formula for Boyle's law is: \[P \propto \frac{1}{V}\]

To derive the proportionality relationship between \(P\) and \(T\), we need to combine both laws. From Charles's law, we can write: \[V = k_{1}T\] where \(k_{1}\) is some constant of proportionality. Similarly, from Boyle's law, we can write: \[P = k_{2}\frac{1}{V}\] where \(k_{2}\) is some constant of proportionality. Now, we can substitute the expression for \(V\) from Charles's law into the expression of Boyle's law: \[P = k_{2}\frac{1}{k_{1}T}\] We can rewrite the equation as: \[P = \frac{k_{2}}{k_{1}} \cdot \frac{1}{T}\] Let \(\frac{k_{2}}{k_{1}}\) be another constant \(k\), so: \[P \propto \frac{1}{T}\] This is the proportionality relationship between \(P\) and \(T\), which is also known as Amonton's law.

Given: Initial tire pressure \(P_{1} = 220.6\,\text{kPa}\) at \(T_{1}=24^{\circ}\mathrm{C}\). We need to find the tire pressure \(P_{2}\) at \(T_{2} = 49^{\circ}\mathrm{C}\). First, convert the Celsius temperatures to Kelvin: \[T_{1} = 24^{\circ}\mathrm{C} + 273.15 = 297.15 \,\text{K}\] \[T_{2} = 49^{\circ}\mathrm{C} + 273.15 = 322.15 \,\text{K}\] Now, we can use the proportionality relationship we derived, which means: \[\frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}}\] Now, substituting the given values and solving for \(P_{2}\): \[P_{2} = \frac{T_{2}}{T_{1}} \cdot P_{1}\] \[P_{2} = \frac{322.15 \,\text{K}}{297.15 \,\text{K}} \cdot 220.6\,\text{kPa}\] Calculating the tire pressure \(P_2\): \[P_{2} = 240.57\, \text{kPa}\] Therefore, the tire pressure when the tires heat up to \(49^{\circ}\mathrm{C}\) during driving will be approximately \(240.57\, \text{kPa}\).