Use the van der Waals equation and the ideal gas equation to calculate the pressure exerted by 1.000 mol of Cl2 in a volume of 5.000 L at a temperature of 273.0 K. Explain why the two values are different.

Video transcript

Hey everyone, let's check out this problem. Using the ideal Vander wal's equation. And then the ideal gas equation were being asked to calculate the pressure of a 3.5 to 4 most sample of into gas and were given the volume and the temperature and once we do that, we need to see if the ideal gas equation overestimates or underestimates the pressure and if so by how much. So we're going to use two equations to solve this problem. The first is the Vander wal's equation and that equation is P plus and squared times a over V squared times V minus N B. Is equal to N R T. And we want to isolate P here because we want to calculate the pressure. So let's go ahead and divide both sides of our equation by v minus N B. And when we do that we're left with P plus n squared times a over V squared is equal to n R T Over V -9B. So remember we're trying to isolate pressure. So let's go ahead and subtract and squared times a over V squared from both sides. And when we do that we get pressure is equal to N R T over v minus and be minus and squared times a over V squared. So now our pressure is isolated and we can go ahead and plug in what we know based off what's given in the problem. So let's go ahead and do that. We get pressure is equal to and is our number of moles and they tell us that we have 3.5 to 4 moles. So 3.5 to 4 moles of N two R.R. r gas constant 0.08206. Leader atmosphere over more Calvin And our temperature is 415°C. We need to convert this to Kelvin so we'll do that by adding 273.152 r 415°C. and one size Calvin. Now this is all over V -9b. Our volume We're told at 0. l -N is our moles 3.5-4 malls times B Which is zero 0391 leaders over more. So guys, this is just the left side. So we need to do the right side. The n squared times a over V squared. So we get minus R N squared is Our most three 5 to malls squared times A Which is 1.39 leaders squared times A T. M over moles squared. And this is all over V squared. 0.8500 L Squared. Alright, I know that was a lot. So now we've plugged in everything and we can go ahead and make sure our units cancel, are moles cancel, our leaders cancel, our kelvin's cancel. And we're left with ATMs on the left side and same thing on the right are most squared cancel and our leaders squared cancel. So our pressure is equal to 0.5 A. T. M. And that's using our Vander wal's are ideal Vander wal equation. Alright, so now we need to do the same thing but using the ideal gas equation and this will be a little simpler, ideal gas equation is pressure times volume equals N. R. T. And we want to isolate our pressure and when we do, we get our pressure is equal to N. R. T over V. So let's go ahead and plug in what we know. We have 3.5- malls Times r gas constant 0.08206 leaders atmosphere over mall Calvin. And we have our temperature of 688 0. Calvin. And this is all over our volume of 0.8500 leaders. So we'll make sure our units cancel, are moles cancel, our leaders cancel and our kelvin's cancel. And we're left with A. T. M. Which is for pressure, what we're looking for. So once we do this we get a pressure of .18 cm. So we have two pressures here. And the question asks us Will the ideal gas equation overestimate or underestimate the pressure. So our pressure using the Vander Wal's is 255.5 and our pressure using the ideal gas is less than that. So here we see that it underestimates. So we'll write that here underestimates And it asked by how much. So we'll take the difference between the two numbers 255.5 -234.1 So it underestimates by 0. and that is our final answer, and that is the end of this problem. I hope this was helpful.