On a cool morning, when the temperature is 15°C, you measure the ...

Question

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?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. So before a race, a bike mechanic inflated a road bike tire to a gauge pressure of 110 P si the air temperature inside the tire was 15 degrees C. At the end of the race, the air temperature inside the tire increased to 35 degrees Celsius. Calculate the tires gauge pressure at the end of the race. So our end goal is to calculate the tires gauge pressure at the end of the race. Awesome. So we're given some multiple choice answers and they're all in the same units at P si pressure per square inch. So let's read them off to see what our final answer might be is. 47.5 B is 105 C is 133 and D is 257. Awesome. So first off, in order to solve this problem, you must assume that the volume of the tire and the quantity of air, which is the number of moles inside the tire will remain constant throughout the entire race. So assuming that we can recall and use the Ideal Gas Law equation, which states that the pressure that the pressure multiplied by the volume is equal to the number of moles multiplied by the universal gas constant multiplied by the temperature. So that's our ideal gas law. Now, we can apply the ideal gas law to examine the pressure of the gas before and after the race. So let us write the relationship between pressure and temperature as a ratio. So using ideal gas, so we can write the following that the initial pressure divided by the initial temperature is equal to the final pressure divided by the final temperature. Awesome. And then that is equal to the number of moles multiplied by the universal gas constant divided by volume and that is equal, all of that is equal to some constant value. Awesome. So using this relationship, we'll call it equation one or relationship one, we can write relationship two which states that the final pressure is equal to the initial pressure multiplied by the final temperature divided by the initial temperature. Awesome. So before we can solve for the final pressure of the tire, we must convert the pressure value of the gauge to absolute pressure because the gauge pressure is above atmospheric pressure. So to convert the gauge pressure to absolute pressure, all we have to do is add the atmospheric pressure value to it. So let's make a quick little note that atmospheric pressure in P SI is 14.7. So that equals atmospheric pressure. Awesome. OK. So let's do that. So the initial pressure equals 110 P SI plus the atmospheric pressure which we, we determined it to be 14.7 P SI. So when you add those two values together, you get 124.7 P si now we can sell it to our known variables to solve for P F. But before that, for P F, the final pressure. But before we could do that, we need to quickly convert, we'll do it up here. We need to quickly convert our final temperature an initial temperature from degrees Celsius to degrees Kelvin. So to do that, all we have to do is just take our degree Celsius value and add 273. So to do that for the final temperature, which was we determined the final temperature, which is the temperature inside the tire was 35. So 35 plus 2, 73 is 308 Kelvin. And then the initial temperature of the tire inside of the tire was 15 degrees C so 15 plus 273 equals Kelvin. So now we can solve for P F using equation two. So let's plug in our known variables. So the initial pressure was 124.7 P si multiplied by the final temperature which was Kelvin divided by 288 Kelvin. So when you plug that into a calculator, you should get 133 P si because the Calvin temperatures cancel out. Awesome. So that means that our final answer must be C 133 P si Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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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?

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