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.