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Chapter 10: Problem 27

\((\mathbf{a})\) 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 32.0 \(\mathrm{lb} / \mathrm{in.}^{2}\) (psi) measured at \(75^{\circ} \mathrm{F},\) what will be the tire pressure if the tires heat up to \(120^{\circ} \mathrm{F}\) during driving?

Short Answer

Expert verified
The proportionality relationship between pressure and temperature can be derived from Charles's law and Boyle's law as \(P \propto \frac{1}{T}\). When a car tire is filled to a pressure of 32.0 psi measured at \(75^{\circ} \mathrm{F}\), and the tires heat up to \(120^{\circ} \mathrm{F}\) during driving, the tire pressure will be \(34.66 \frac{\text{lb}}{\text{in}^2}\).

Step by step solution

01

Understand Charles's Law

Charles's law states that the volume of a given amount of gas is proportional to its temperature at a constant pressure. Mathematically, we can write it as: \[V \propto T\] where V is the volume of the gas and T is the temperature (in Kelvin).
02

Understand Boyle's Law

Boyle's law states that the volume of a given amount of gas is inversely proportional to its pressure at a constant temperature. Mathematically, we can write it as: \[V \propto \frac{1}{P}\] where V is the volume of the gas and P is the pressure.
03

Combine Charles's Law and Boyle's Law

From Charles's law and Boyle's law, we can write: \[V \propto T \quad \text{and} \quad V \propto \frac{1}{P}\] By using the fact that the proportionality constant must be the same, we get the relationship: \[T \propto \frac{1}{P}\] Now, rearrange it to get the proportionality relationship between pressure and temperature: \[P \propto \frac{1}{T}\]
04

Apply the proportionality relationship to find tire pressure

Given the initial pressure \(P_1 = 32.0 \frac{\text{lb}}{\text{in}^2}\) and initial temperature \(T_1= 75^{\circ}\mathrm{F}\), we need to find the pressure \(P_2\) at a higher temperature \(T_2 = 120^{\circ}\mathrm{F}\). First, let's convert the temperatures from Fahrenheit to Kelvin: \[T_1(K) = \frac{5}{9}(75 - 32) + 273.15 = 297.038 K\] \[T_2(K) = \frac{5}{9}(120 - 32) + 273.15 = 322.038 K\] Now, using the proportionality relationship we derived earlier, we have: \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\] Rearrange and solve for \(P_2\): \[P_2 = P_1 \frac{T_2}{T_1} = 32.0 \frac{\text{lb}}{\text{in}^2} \cdot \frac{322.038 \text{K}}{297.038 \text{K}}\]
05

Calculate the final tire pressure

Now, we can calculate the final tire pressure \(P_2\): \[P_2 = 32.0 \frac{\text{lb}}{\text{in}^2} \cdot \frac{322.038 \text{K}}{297.038 \text{K}} = 34.66 \frac{\text{lb}}{\text{in}^2}\] Therefore, when the tires heat up to \(120^{\circ}\mathrm{F}\), the tire pressure will be \(34.66 \frac{\text{lb}}{\text{in}^2}\).

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