Question

An automobile tire has a maximum rating of 38.0 psi (gauge pressure). The tire is inflated (while cold) to a volume of 11.8 \(\mathrm{L}\) and a gauge pressure of 36.0 psi at a temperature of \(12.0^{\circ} \mathrm{C} .\) When the car is drivenon a hot day, the tire warms to \(65.0^{\circ} \mathrm{C}\) and its volume expands to 12.2 L. Does the pressure in the tire exceed its maximum rating? (Note: The gauge pressure is the difference between the total pressure and atmospheric pressure. In this case, assume that atmospheric pressure is 14.7 psi.

Short answer

The pressure in the tire exceeds its maximum rating if the final gauge pressure is greater than 38.0 psi.

Step-by-step solution

Convert Celsius to Kelvin

To use the ideal gas law, temperatures must be in Kelvin. Convert the initial and final temperatures from Celsius to Kelvin by adding 273.15 to each Celsius temperature. Initial temperature: \(12.0^\circ\mathrm{C} + 273.15 = 285.15\mathrm{K}\).Final temperature: \(65.0^\circ\mathrm{C} + 273.15 = 338.15\mathrm{K}\).

Calculate the Initial Absolute Pressure

Convert the initial gauge pressure to absolute pressure by adding atmospheric pressure to it. Initial absolute pressure: \(36.0\mathrm{psi} + 14.7\mathrm{psi} = 50.7\mathrm{psi}\).

Apply the Ideal Gas Law

Use the ideal gas law in the form \(P_1V_1/T_1 = P_2V_2/T_2\) where \(P\) is pressure, \(V\) is volume, \(T\) is temperature, and the subscripts \(1\) and \(2\) refer to the initial and final states, respectively. We can solve for the final pressure \(P_2\) using the known values.

Solve for Final Absolute Pressure

Rearrange the ideal gas law to solve for \(P_2\): \(P_2 = P_1V_1T_2/T_1V_2\). Plug in the initial and final temperatures, volumes, and the initial absolute pressure to find the final absolute pressure.

Convert Final Absolute Pressure to Gauge Pressure

Subtract atmospheric pressure from the final absolute pressure to find the final gauge pressure. If this pressure exceeds the maximum rating of \(38.0\mathrm{psi}\), then the rating is exceeded.

Determine if the Maximum Rating Is Exceeded

Compare the calculated final gauge pressure with the maximum rating to determine if the tire's pressure exceeds the safe limit.

Key concepts

Gauge Pressure

Gauge pressure is a measure often used in daily life, particularly when filling tires or checking the pressure of various systems. It's the pressure measured relative to the ambient atmospheric pressure. Imagine atmospheric pressure as a baseline or the 'zero' point; gauge pressure is what we measure above this baseline. When you check the tire pressure and read '36.0 psi', that's gauge pressure. This means that inside the tire, the pressure is 36.0 psi higher than the atmospheric pressure outside.

In formal settings, the gauge pressure might not give a complete picture since it ignores the actual atmospheric pressure. This can be crucial in scientific calculations where precision is necessary. So, if you need to convert gauge pressure to absolute pressure, simply add the value of atmospheric pressure, which is commonly accepted as 14.7 psi at sea level, to the gauge pressure reading.

Absolute Pressure

Absolute pressure takes the gauge pressure concept further by adding the atmospheric pressure to it. It represents the total pressure exerted by a gas, including the atmospheric pressure. To clarify, if you measure a gauge pressure of 36.0 psi inside a tire and the atmospheric pressure is 14.7 psi, the absolute pressure inside the tire is 50.7 psi.

Absolute pressure is essential in calculations that involve the ideal gas law. Since it accounts for the full extent of exerted pressure, including the environmental contribution, it provides a more precise understanding of a system's status. For scientific and engineering problems, knowing the absolute pressure allows for accurate and consistent comparisons of pressure measurements.

Temperature Conversion

Temperature conversion is necessary because different situations require different temperature scales. When working with the ideal gas law, temperature needs to be expressed in Kelvin, because it is the scale used in scientific equations to ensure accuracy and avoid negative numbers.

To convert Celsius to Kelvin, one must add 273.15 to the Celsius temperature. This conversion is pivotal since it places the temperature on an absolute scale where 0 K (known as absolute zero) is theoretically the lowest possible temperature where there's no thermal energy left in a substance. In the given exercise, converting temperatures from Celsius to Kelvin ensures that we use the correct units to apply the ideal gas law accurately.

Charles's Law

Charles's Law is a fundamental principle in the study of gases which states that the volume of a gas is directly proportional to its temperature when the pressure and amount of gas are held constant. This means that as the temperature of a gas increases, its volume also increases, assuming the amount of gas and the pressure do not change.

In the context of the exercise, when the car tire heats up from the original temperature of 12.0°C to 65.0°C, Charles's Law helps us understand that the volume of the tire will expand as long as the amount of air in the tire and the surrounding pressure remain unchanged. This expansion is reflected in the change from an 11.8 L volume at the cooler temperature to a 12.2 L volume at the higher temperature. This relationship is critical to solving real-world problems involving changes in gas temperature, such as predicting how much a tire will expand on a hot day.