Ch 18: A Macroscopic Description of Matter

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 18: A Macroscopic Description of Matter

Problem 52

Chapter 18, Problem 52

On a cool morning, when the temperature is 15°C, you measure the pressure in your car tires to be 30 psi. After driving 20 mi on the freeway, the temperature of your tires is 45°C . What pressure will your tire gauge now show?

Verified step by step guidance

Step 1: Recognize that this problem involves the relationship between pressure, temperature, and volume for a gas. Since the volume of the tire and the amount of gas inside remain constant, we can use the ideal gas law in the form of the combined gas law: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \(P\) is pressure and \(T\) is temperature in Kelvin.

Step 2: Convert the given temperatures from Celsius to Kelvin using the formula \( T(K) = T(°C) + 273.15 \). For the initial temperature, \( T_1 = 15 + 273.15 \). For the final temperature, \( T_2 = 45 + 273.15 \).

Step 3: Rearrange the combined gas law to solve for the final pressure \( P_2 \): \( P_2 = P_1 \cdot \frac{T_2}{T_1} \). Here, \( P_1 \) is the initial pressure (30 psi), \( T_1 \) is the initial temperature in Kelvin, and \( T_2 \) is the final temperature in Kelvin.

Step 4: Substitute the known values into the equation. Use \( P_1 = 30 \) psi, \( T_1 = 15 + 273.15 \), and \( T_2 = 45 + 273.15 \). Perform the division \( \frac{T_2}{T_1} \) and multiply by \( P_1 \) to find \( P_2 \).

Step 5: The result \( P_2 \) represents the pressure in the tires after driving. Note that this is the absolute pressure. If the tire gauge measures gauge pressure, you may need to subtract atmospheric pressure (approximately 14.7 psi) to find the gauge pressure.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, and temperature of an ideal gas through the equation PV = nRT. In this context, it helps us understand how the pressure of the air in the tires changes with temperature, assuming the volume remains constant. As temperature increases, the kinetic energy of gas molecules increases, leading to higher pressure.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature when pressure is held constant. Although the volume of the tire does not change significantly, this law illustrates the relationship between temperature and pressure, indicating that as the temperature of the gas increases, so does the pressure, assuming the volume remains constant.

Pressure-Temperature Relationship

The pressure-temperature relationship in gases indicates that an increase in temperature results in an increase in pressure, provided the volume is constant. This principle is crucial for understanding how the tire pressure will change after driving, as the temperature of the air inside the tires rises from 15°C to 45°C, leading to a measurable increase in pressure.

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