Question

A tire is inflated to a gauge pressure of \(28.0\) psi at \(71^{\circ} \mathrm{F}\). Gauge pressure is the pressure above atmospheric pressure, which is \(14.7\) psi. After several hours of driving, the air in the tire has a temperature of \(115^{\circ} \mathrm{F}\). What is the gauge pressure of the air in the tire? What is the actual pressure of the air in the tire? Assume that the tire volume changes are negligible.

Short answer

Answer: The new gauge pressure is 30.62 psi, and the actual pressure of the air in the tire is 45.32 psi.

Step-by-step solution

Convert Temperature to Kelvin

To use the Gay-Lussac's Law, we need to convert the temperatures given in degrees Fahrenheit to Kelvin. The conversion is as follows: \(K = \frac{5}{9} (F - 32) + 273.15\) For \(T_1 = 71^{\circ}\mathrm{F}\): \(T_1K = \frac{5}{9}(71-32) + 273.15 = 294.26\mathrm{K}\) For \(T_2 = 115^{\circ}\mathrm{F}\): \(T_2K = \frac{5}{9}(115-32) + 273.15 = 318.71\mathrm{K}\)

Use Gay-Lussac's Law to find the new gauge pressure

Using the given initial gauge pressure, \(P_1 = 28.0\,\text{psi}\), and the temperatures in Kelvin, we can find the new gauge pressure, \(P_2\), using Gay-Lussac's Law: \(\frac{P_1}{T_1K}=\frac{P_2}{T_2K}\) Rearranging for \(P_2\): \(P_2 = \frac{P_1 \cdot T_2K}{T_1K} = \frac{28.0\,\text{psi} \cdot 318.71\,\text{K}}{294.26\,\text{K}} = 30.62\,\text{psi}\) So, the new gauge pressure is \(30.62\,\text{psi}\).

Convert gauge pressure to actual pressure

To find the actual pressure of the air in the tire, we need to add the atmospheric pressure, \(P_{atm}=14.7\,\text{psi}\), to the gauge pressure: \(P_\text{actual}=P_\text{gauge}+P_\text{atm}=30.62\,\text{psi} +14.7\,\text{psi}=45.32\,\text{psi}\) The actual pressure of the air in the tire is \(45.32\,\text{psi}\). The new gauge pressure is \(30.62\,\text{psi}\), and the actual pressure of the air in the tire is \(45.32\,\text{psi}\).

Key concepts

Gauge Pressure

Gauge pressure is a term you'll often encounter in physics, especially when dealing with fluids and gases. It refers to the pressure of a system above the atmospheric pressure. Essentially, it's the difference between the absolute pressure within a container (like a tire or a boiler) and the pressure of the air outside. This distinction is crucial because the pressure we experience everyday, atmospheric pressure, acts on everything.

In the exercise with the tire, gauge pressure is what a tire pressure gauge measures. The value doesn't consider the omnipresent atmospheric pressure. That's why, when calculating the actual pressure inside the tire, the atmospheric pressure of about 14.7 psi (pounds per square inch) at sea level is added to the gauge pressure to get the full picture of forces acting on the tire walls.

Understanding gauge pressure is essential for various applications - from inflating tires to industrial processes. It helps ensure safety and efficiency in systems where pressure is a critical factor.

Temperature Conversion

Temperature conversion is an indispensable tool in science for communicating findings and performing calculations when temperatures are expressed in different scales. There are three primary temperature scales used globally: Fahrenheit (°F), Celsius (°C), and Kelvin (K). Fahrenheit is often used for everyday temperature readings in the United States, while Celsius is more common internationally. Kelvin, on the other hand, is the temperature scale of choice for scientific endeavors.

In our exercise's context, converting Fahrenheit to Kelvin is necessary to apply Gay-Lussac's Law correctly, since the law requires absolute temperature for accurate calculations. The Kelvin scale starts at absolute zero, the theoretical point where particles have minimal thermal motion. Understanding how to convert between these scales assures you can apply scientific laws accurately and bridge the communication gap between different systems of measurement.

As demonstrated in the solution, creating a reliable method to convert temperatures is key to performing accurate physics calculations. Whether it's monitoring weather patterns, designing HVAC systems, or working out the expansion of gases, mastering temperature conversion unlocks the ability to navigate between different measurement systems.

Kelvin Temperature Scale

The Kelvin temperature scale is fundamental to the physical sciences because it is based on absolute zero, the coldest possible temperature where molecular energy is at a minimum. Unlike Celsius and Fahrenheit, the Kelvin scale does not use degrees. Instead, it uses Kelvin (K) as the unit of measurement. One of the most important characteristics of the Kelvin scale for scientists and engineers is the direct proportionality to energy.

When working with gas laws, the Kelvin scale provides a way to compare temperatures on an absolute scale – that's why temperatures for our tire pressure problem are converted to Kelvin. This scale ensures that calculations take into account the absolute energy present in the system. Using Kelvin allows for a clearer understanding of thermal dynamics and aids in accurate predictions of gas behavior under varying temperature conditions.

To students and professionals alike, getting familiar with the Kelvin scale and how it relates to other temperature measurement scales is pivotal for a thorough grasp of thermodynamics, physics, and chemistry.