(a) Amonton's law expresses the relationship between pressure and temperature. Use Charles's law and Boyle's law to derive the proportionality relationship between P and T.

Video transcript

Hi everyone for this problem, we're told to derive the relationship between pressure and volume of gas at a constant temperature. Using Charles's law and gay lucic's law. Okay, so let's take a look at what both of these laws tells us the first one Charles law, that law tells us that volume and temperature are directly proportional. So that means an increase in volume is equal to an increase in temperature and vice versa. They're directly proportional. And Gay lucic's law tells us that pressure and temperature are directly proportional. So that means an increase in pressure is equal to an increase in temperature and vice versa. And so we need to derive the relationship between pressure and volume of a gas at constant temperature. So for Charles law, we can say that V one over T one Is equal to V two over T two. And at constant temperature, that means that T two over T one Is going to equal one. And so that means V two over V one is going to equal one. And so with that being said, we can do the same thing for gay lucic's law. And gay lucic's law shows us that P one over T one Is equal to P two Over T two. Okay, and when this is at constant temperature, that means that T one Over T two is going to equal one And P one over P two is going to equal one. So that means we're going to essentially ignore our temperatures because they're constant. So not paying attention to our temperatures. When we set P one over T one equal to V two over V one, we get this. So we're going to get P one over P two is equal to V two Over V one. So because our temperature is constant, we're ignoring it. So we're just going to set the two that we're focusing on equal to each other and when we get rid of our fraction, So when we cross multiply and get rid of our fraction, we see that P1 times v one is equal to P two times V two. So we cross multiplied here to get rid of the fraction. And from there we can see the relationship, which is what the question asked derived the relationship between pressure and volume at constant temperature. Using both of these laws, that's the end of this problem. I hope this was helpful.