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The piston of a cylinder containing a quantity of ideal gas is advanced so that the volume of the gas is decreased by one-half. A pressure gauge on the cylinder sho...

Question

The piston of a cylinder containing a quantity of ideal gas is advanced so that the volume of the gas is decreased by one-half. A pressure gauge on the cylinder shows the pressure of the gas to have increased threefold in the process. By what factor does the temperature of the gas change?

The piston of a cylinder containing a quantity of ideal gas is advanced so that the volume of the gas is decreased by one-half. A pressure gauge on the cylinder shows the pressure of the gas to have increased threefold in the process. By what factor does the temperature of the gas change?



Answers

A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume?

Hi, guys. I'm doing problem 13 in chapter 18. It says that the volume of Amon Atomic ideal gas triples and I so thermal expansion. By what factor does this pressure change? So whenever we're dealing with ideal gases, we can use the ideal gas law PV equals N R t. Or, if you prefer P vehicles and Katie doesn't matter, Um, course, in the second case, you have big end. So you're talking about number of particles of stuff moles, But here we'll use P vehicle on Artie and we're told that an ice with thermal expansion is going on, okay, and then the volume is tripling. Okay, So what we noticed about this equation is that if we're talking about one gas and doing an isotope thermal expansion, the entire right side is going to remain the same. And so this this equation is always true for an ideal gas. So if it's true before, has to be true. After and so, if you're going to change one of these variables by a factor, some other variables must change by a factor in order to keep this equation true. Okay, so if we undergo a nicer thermal expansion the entire right side remains the same. Okay, so this really means that therefore, that p p one V one equals p to V two. Okay. And so this is a separate equation that sometimes people memorize. But you don't really need to memorize it. Because if you just have the the ideal gas law memorized already, then it's very simple to derive this just by noticing in this first equation that in nicely thermal expansion, the entire right side is the same. So then the stuff that's changing must be on the left side. Okay, so then it says the volume is tripling, so V is giving up by a factor of three. So what must happen to pee in order to make this the same? Well, P as you should immediately notice P is going to go down by a factor of three or P p going 2 to 1/3. But it wasn't originally eso another way to see this. If if you didn't immediately see it from above, you can say V two equals three V one. Okay, this is what it means when the volumes tripling right? This just means the volumes tripling. And then put that into the second equation, right? And so you'll get something like, um, so you you'll get something like P one V one equals three p to V one V of fee one. Right. And then, since you have the one on each side, you can cancel out the V ones dividing through and you'll see that P one equals three p two or pee, too. P two equals 1/3 p one, which is the same thing as we said before that the pressure is changing is going down by a factor of three or being scaled by a factor of 1/3.

So here we have The new cylinder has brought the final equals three times. We initial on the new pressure Final pressure is half the initiate the pressure from here. We're going to use the one we want over t one equals our initial over the Final Three final over the final. So from here we can find that Do you finally, questi initial times be finally over Be initial times we finally over fee any feel in this year in temperature waas 25 C or to 73 plus 25 Gavin pressure final overpressure initially is half times Why didn't find other initiatives three times so bad gives that then bring it to be 447 Jazmine which is 4 47 minus 273 degree centigrade. It was 1 74 C.

Okay, so in this question, we have a gas and its initial temperature is 23 by 27 degrees T initial 27 degrees Celsius. And, you know, with these problems, I make it a habit to just convert it to Teoh. Calvin. Um, I think it's a good habit because you we'll get the wrong answer if you use the ideal gas law in general. Um, took out with the Celsius degree. I might be wrong. There might be an edge case that I'm not thinking of, but it's definitely a safe bet, Okay? And, um, the pressure is constant and we want the volume to double. So let's write down the ideal gas law and then solve for T so PV equals NRT so we can do PV divided by an R to get t. And then we want V What's going on with B? Um, we want this to double. We're told that this stays the same and then presumably you're not secretly adding and molecules. And of course, our is a constant, so v doubles. That means the temperature has to double, so there's no squared or square roots or anything like that. So there simply linearly related and again use longest you're using, um, Calvin and not Celsius or any other temperature measurement. So 600 Kelvin would be it, because again, you're multiply the volume by two, so you have to multiply that the temperature by two.

Yeah, so in this problem, let's suppose the initial pressure is given is the initial pressure and we even is expose initial volume. And let's say the final pressure is equal to you too. And final volume is V two. We are given that if we do is equated to times of even then we have to find me too in terms of preventing so as the processes as determined as a thermal process. And as we know that necessary process, even we one is equal to B two b two. The Saudi remembered here. So if you want to find me too, we live in this question. We get p two is equal to even. We went divided by me too. So there was a shoot. We do mhm. Even we wondered by me two is equal to two times Given even and everyone can slot and we are left with p two is equal to one or two. Stephen. That means Benedict Volume increases two times, then pressure because is to one half, which is the answer of this question. Thank you for what you do


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