Question

\((\mathbf{a})\) 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

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

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).

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.

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: \[P \propto \frac{1}{T}\]

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 \(P_2\): \[P_2 = P_1 \frac{T_2}{T_1} = 32.0 \frac{\text{lb}}{\text{in}^2} \cdot \frac{322.038 \text{K}}{297.038 \text{K}}\]

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 \(34.66 \frac{\text{lb}}{\text{in}^2}\).

Key concepts

Amonton's Law

Amonton's Law, also known as the pressure-temperature law, describes how the pressure of a gas changes with temperature at a constant volume. This law states that pressure is directly proportional to the temperature, provided the volume remains unchanged. Mathematically, this can be expressed as:

• \( P \propto T \)

Where \( P \) represents pressure and \( T \) represents temperature.
This relationship implies that if the temperature of a gas increases, the pressure also increases, assuming the volume is constant. This law helps us understand how heating and cooling affect the pressure in closed systems like car tires.
In practical applications, Amonton's Law is crucial for systems where volume stays unchanged, making it useful for predicting changes in pressure with temperature adjustments.

Charles's Law

Charles's Law focuses on the relationship between volume and temperature, asserting that the volume of a gas is directly proportional to its temperature when pressure is constant. This relationship is expressed as:

• \( V \propto T \)

Here, \( V \) is the volume, and \( T \) is the temperature in Kelvin.
Charles's Law means if you heat a gas, its volume expands if the pressure is kept constant. Conversely, cooling the gas will cause the volume to decrease.
It is particularly significant in scenarios where volume changes are measured at a constant pressure, such as in balloons expanding in warm environments or contracting in cooler settings.

Boyle's Law

Boyle's Law highlights the relationship between pressure and volume at a constant temperature. It states that the pressure of a gas is inversely proportional to its volume, shown as:

• \( P \propto \frac{1}{V} \)

Where \( P \) is the pressure, and \( V \) is the volume.
This means if the volume of a gas decreases, its pressure increases, assuming the temperature doesn't change. Conversely, an increase in volume leads to a reduction in pressure.
Boyle's Law is essential for understanding how pressurizing and depressurizing affect gas systems, such as in breathing, where lung volume changes leading to pressure variations control the inhalation and exhalation process.

Pressure-Temperature Relationship

The pressure-temperature relationship describes how pressure changes with temperature in a closed container when volume is constant. This derives directly from Amonton's Law and is critical in understanding various everyday phenomena.

• Increases in temperature lead to increased pressure if the volume is kept constant.
• Decreases in temperature result in lower pressure under the same conditions.

This relationship is crucial for practical applications like assessing tire pressures, as heating them leads to higher pressure levels. Using this concept, you can predict and calculate changes in pressure with temperature changes, useful for maintaining safe and efficient functioning of closed-contained gas systems.

Ideal Gas Law

The Ideal Gas Law is a combined equation that encompasses the relationships described by Boyle's, Charles's, and Amonton's laws. It is formulated as:

• \( PV = nRT \)

Here, \( P \) is pressure, \( V \) is volume, \( n \) is the amount of gas in moles, \( R \) is the ideal gas constant, and \( T \) is temperature.
This law allows one to calculate unknown properties of a gas if other conditions are provided. It serves as a fundamental principle for predicting gas behavior under varying conditions.The Ideal Gas Law assumes the behavior of an "ideal" gas, which may not be perfectly accurate for real gases under extreme conditions, but it provides a reliable approximation for most practical scenarios.