Problem 29 - College Physics

Question

A steel-belted radial automobile tire is inflated to a gauge pressure of \(30.0 \mathrm{lb} / \mathrm{in}^{2}\) when the temperature is \(61^{\circ} \mathrm{F}\). Later in the day, the temperature rises to \(100^{\circ} \mathrm{F}\) Assuming the volume of the tire remains constant, what is the tire's pressure at the elevated temperature? [Hint: Remember that the ideal gas law uses absolute pressure.]

Step-by-Step Solution

Answer

The tire's gauge pressure at 100°F is approximately 33.37 lb/in².

Step 1: Convert Temperatures to Kelvin

First, convert the given temperatures from Fahrenheit to Celsius, and then to Kelvin.The formula to convert from Fahrenheit to Celsius is: \[C = \frac{5}{9}(F - 32).\]For the initial temperature:\[C_1 = \frac{5}{9}(61 - 32) = 16.1^{\circ}C.\]Convert to Kelvin:\[T_1 = 16.1 + 273.15 = 289.25 \, K.\]For the final temperature:\[C_2 = \frac{5}{9}(100 - 32) = 37.8^{\circ}C.\]Convert to Kelvin:\[T_2 = 37.8 + 273.15 = 310.95 \, K.\]

Step 2: Convert Gauge Pressure to Absolute Pressure

Gauge pressure is measured above atmospheric pressure, so we need to add atmospheric pressure to convert it to absolute pressure. Atmospheric pressure is approximately 14.7 lb/in².\[P_1 (\text{absolute}) = 30.0 \text{ lb/in}^2 + 14.7 \text{ lb/in}^2 = 44.7 \text{ lb/in}^2.\]

Step 3: Use Ideal Gas Law to Find Final Pressure

Assuming the volume of the tire remains constant, apply the ideal gas law relationship for constant volume conditions:\[\frac{P_1}{T_1} = \frac{P_2}{T_2},\]where \(P_2\) is the final absolute pressure. Rearrange to solve for \(P_2\):\[P_2 = P_1 \times \frac{T_2}{T_1}.\]Plug in the values:\[P_2 = 44.7 \text{ lb/in}^2 \times \frac{310.95 \, K}{289.25 \, K} \approx 48.07 \text{ lb/in}^2.\]

Step 4: Convert Final Pressure Back to Gauge Pressure

To convert the absolute pressure back to gauge pressure, subtract atmospheric pressure:\[P_2 (\text{gauge}) = P_2 - 14.7 \text{ lb/in}^2 \approx 48.07 \text{ lb/in}^2 - 14.7 \text{ lb/in}^2 = 33.37 \text{ lb/in}^2.\]

Key Concepts

Temperature Conversion

When working with gas laws, it's crucial to use the correct temperature scale, which is Kelvin. This is because Kelvin is an absolute scale and ensures positive values for all physical real temperatures.
To convert temperatures, start by switching from Fahrenheit to Celsius. The formula is: \[C = \frac{5}{9}(F - 32)\]
Fahrenheit measures heat relative to water's freezing point, but Celsius modernizes that to more universal values.
Next, convert the Celsius temperature to Kelvin by adding 273.15. This step is important to align with the Ideal Gas Law, which requires Kelvin to accommodate thermal dynamics fully.

Absolute Pressure

Absolute pressure is the total pressure exerted on a system, accounting for atmospheric pressure. Unlike gauge pressure, which measures pressure relative to the atmosphere, absolute pressure always includes atmospheric contributions.
In practical terms, absolute pressure is critical in calculations involving gases because it provides a full picture of the forces at play. To convert gauge pressure to absolute pressure, simply add the atmospheric pressure (commonly approximated as 14.7 lb/in² at sea level) to the gauge pressure value.
For example:

• If the gauge pressure is 30 lb/in², the absolute pressure would be 30 + 14.7 = 44.7 lb/in².

Understanding absolute pressure helps ensure more accurate and reliable results when applying the Ideal Gas Law.

Gauge Pressure

Gauge pressure is the pressure measurement relative to atmospheric pressure. It effectively tells you how much pressure is in excess of the ambient pressure surrounding the object.
When you inflate a tire, the pressure you read on your gauge doesn't account for the universal atmospheric pressure pressing on everything at Earth's surface. It is critical to recognize this distinction, especially in dynamics involving variable temperature settings like in tires, as seen with weather-related changes.
In some contexts, like car tires, gauge pressure provides convenience. Still, when performing precise scientific calculations, this needs converting to absolute pressure by adding atmospheric effect.

Gas Pressure Calculations

Gas pressure calculations often rely on the Ideal Gas Law, which correlates pressure, volume, and temperature of a gas. For cases like a fixed-volume automobile tire, the changing outside conditions necessitate adjustments in pressure if temperature shifts.
The formula for constant volume provided by the Ideal Gas Law is simplified to:\[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]
This equation helps forecast the resultant pressure after a temperature change, considering the initial and final pressures \(P_1\) and \(P_2\), and temperatures \(T_1\) and \(T_2\), all in Kelvin.
Remember, once the calculation gives the new absolute pressure, you must convert it back to gauge pressure by subtracting the atmospheric pressure. This conversion helps in real-world applications like checking tire pressure more practically.