An automobile tire has a maximum rating of 38.0 psi (gauge...

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.

Key Concepts

Ideal Gas Law

The ideal gas law (PV = nRT) is a fundamental equation in thermodynamics that relates the pressure (P), volume (V), temperature (T), and amount (n) of a gas through a constant (R). It allows us to predict how a gas will behave when one or more of these variables are changed, making it essential for analyzing processes involving gas expansion or compression.

Combined Gas Law

The combined gas law integrates Boyle's Law, Charles's Law, and Gay-Lussac's Law into one relationship (P1V1/T1 = P2V2/T2) that describes how the pressure, volume, and temperature of a fixed amount of gas change in relation to one another. This law is crucial for solving problems where more than one variable changes, such as in the scenario of a tire heating up and expanding.

Gauge Pressure vs Absolute Pressure

Gauge pressure is the pressure measured relative to the ambient atmospheric pressure, whereas absolute pressure is the total pressure including atmospheric pressure. Converting between these two is important in practical applications, as devices like tires have ratings based on either one, and the correct interpretation of these readings is key for ensuring safety limits are not exceeded.

Temperature Conversion

Temperature conversion, especially between Celsius and Kelvin, is vital when using the ideal or combined gas law because these equations require temperature to be in an absolute scale (Kelvin). This conversion ensures accurate calculations when analyzing changes in a gas's properties due to temperature variations.

Volume Expansion Effects

When a gas is heated, its volume can increase if it is not held constant, which in turn can affect the pressure depending on whether the container allows for expansion. Recognizing and accounting for volume expansion is important when determining if changes in temperature could lead to pressure exceeding a system’s design limits.

Transcript

00:01 All right, so in this question, we are determining the pressure of a tire after it's been driven for a while to see if it expands past its maximum gauge pressure.

00:12 So two things to start.

00:14 First off, this is obviously going to be using the ideal gas law, i'm sorry, the combined gas law, which can be derived using the ideal gas law, but states this.

00:28 Sorry, it's supposed to be a one.

00:32 All right, so the next thing is gauge pressure, which the book explains.

00:35 To us is the difference between the total pressure and the atmospheric pressure.

00:39 So let's see, pressure gauge is equal to total pressure minus the pressure from the atmosphere.

00:50 So i'm actually going to be using total pressure in this problem and then switching back to gauge pressure after.

00:56 So how i'm going to do that, i can just obviously using algebra, just say that the total pressure is equal to the gauge pressure minus the atmospheric pressure.

01:10 I'm sorry, plus the atmospheric pressure, and then converting back and forth the same way.

01:16 All right, so if i'm finding volume, i'm looking for pressure two, i'm looking for the second pressure.

01:21 So using some rearrangements, p2 is obviously equal to p1 v1, t2 over t1 v2.

01:33 And that just comes from algebraic rearrangement.

01:37 So now i can follow these kind of, just kind of plug in values here.

01:43 With the first pressure, which it gives us is going to be 36 psi, and that's the gauge pressure...

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