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 L and a gauge pressure of 36.0 psi at a temperature of 12.0 °C. On a hot day, the tire warms to 65.0 °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.)

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Accepted Answer

Hey there, welcome back. Alright. So here, when the temperature is 18°C, the gas has a volume of 43 liters and the pressure of tours if the volume changes, so we're changing the volume here. So we have two volumes, 26 liters and the temperature changes to 86 degrees Celsius. What is the pressure of the if the amount of gas stays the same? So if the end, if most of gasses the same, what is the pressure of that gas? So we're gonna use tours um or we we want the answering tours using three significant figures. Alright, So here we are obviously going to be using ideal gas law. Right? And we have all of these variables happening PV equals NRT. Now we are told that and here number of moles of gas is the same. So it's constant. So we're gonna go ahead and just say we can ignore that are here is also constant because it's always constant. Right? It doesn't change the gas constant and temperature here is actually changing. So what we have here is PV equals T. Right? So, these are the things that are changing now notice that we have to to temperatures, we have two volumes. Right? Um, volume one, volume 22 pressures one is known. One is unknown that that's what we're solving for. So we can actually go ahead and say that we have P one V one and T one. We're going to go ahead and move this tea over to this side and of course we're going to go ahead and divide those sides by T. So it's going to be PV over tea and it's going to equal to P two Times V two over T two. Right? So the left side is going to be our first set of um variables and then the right side is the second one. And we're going to be solving for the second pressure. So this is what we want to solve for. Right? So let's go ahead and actually rearrange this equation Um to sell for peace. So realize that this is that we're dealing with combined gas law here. Right, Alright, so P two. So we're gonna go ahead and multiply both sides by T2 and then divide both sides by V two. And we're going to have P one times v one times T two Divided by T one and V two. Right again, we're just solving for P to everything else. We have, we just don't have it in the correct units unfortunately. So pressure here has to be in atmospheres, the volume has to be in leaders and temperature has to be in Calvin right, for the pressures. We actually Have that in tours. So let's go ahead and take the 765 tours and convert that into um atmospheres. So the conversion factor is one, atmosphere Is going to have 760 tours. So this is almost one atmosphere is just a little bit over one atmosphere. So it's going to be 1. atmospheres. Alright, so we have that for the volume so volume here. No, I'm sorry. So volumes are in the correct units there and leaders so we have that at least but the temperatures are not in the right units. So The first temperature that we have is 18. Okay, so we have 18 degrees Celsius. We're going to go ahead and convert that into Calvin by adding to 73.15 to that. That will give us 2 91.15 Calvin And then the second temperatures 86°C plus to 73.15 and that gives us 3 59.15 Calvin. Alright, so we have This is T one. This is T two. This right here is P one, P one and then we have the volume. So let's go ahead and just plug in the numbers. All right. So P one is 1. atmospheres. Volume one is 43 l and then the temperature too um is 3 59. Right? Yes, 3 59 countless. Right, so divide all of that by the first temperature which was 18 or to 91. Calvin and then volume here is going to be volume two. Um let's see 26. So the first one was 43. The second one is 26 liters. Alright, so let's go ahead and multiply and divide everything. So notice that leaders here will cancel out and the calvins cancel out. And we're going to have our answer in atmospheres, we do want the answer and tours, but let's go ahead and just calculate this first. Alright, so once you multiply and divide everything correctly here, we are going to get to point um Yeah, atmospheres. Okay, now they want the answer and tour. So let's just go ahead and convert that back into tour. So one atmosphere, Remember from before tours? There we go. And that will give us one point. They want three significant figures to 1.5, 6 And let's put that in scientific notation times 10 to the third and that will be tours. Oops. Alright, so that is going to be our final answer for this question. Now, even though we are supposed to be using atmospheres when we are dealing with ideal gas law equation, if you notice, because this is um this is just the ratio here, we actually could have just gotten away with using um the pressure one in um tours instead of converting into atmospheres and that would have given us the answer back in tours because all the other units here canceled out since we had, you know, to two versions of each to two volumes and two temperatures um and two pressures, they actually cancel out. So it didn't really matter here, but it's always just safe to use the units that you're supposed to. So that's why we took the tours converted them into atmospheres before using the ideal gas or the combined gas law. But then at the end, we just have to convert that back. Alright folks, that is it. Thank you so much for watching and we'll see you in the next video.

Verified Solution

Key Concepts

Gauge Pressure vs. Absolute Pressure

Ideal Gas Law

Charles's Law