Worked example: Vapor pressure and the ideal gas law

this exercise is from chapter 12 of the COTS treacle and Townsend chemistry and chemical reactivity book and I'm doing it with their permission so they tell us you place two litres of water in an open container in your dormitory room the room has a volume of 4.25 times 10 to the 4th litres you seal the room and wait for the water to evaporate will all of the water evaporate at 25 degrees Celsius and then they tell us at 25 degrees Celsius the density of water is 0.997 grams per milliliter and it's vapor pressure it's vapor pressure is twenty-three point 8 millimeters of mercury and this is actually the key clue to tell you how to solve this problem and just as a bit of review let's just think about what vapor pressure is let's think about what vapor pressure is let's say it's some temperature and in this case we're dealing at 25 degrees Celsius I have a bunch of water and let me do that in a water color I have a bunch of water molecules sitting here in a container sitting here in a container at 25 degrees Celsius they're all bouncing around in every which way and in every which way and every now and then one of them is going to have enough kinetic energy to kind of escape the hydrogen bonds and all the things that keep liquid water and slick would stayed in the little scape it'll go off in that direction and then another one will and this will just keep happening the water will naturally vaporize in a room but at some point enough of these molecules have vaporized enough of these molecules have vaporized over here that they're also bumping back into the water and maybe some of them can be captured back into the liquid state now the point the pressure at which this happens is the vapor pressure as you can imagine is more and more of these as more and more of these water molecules vaporize and go into the gaseous state more and more will also create pressure downward pressure more and more will also be colliding with the surface of the water and the pressure at which the liquid and the vapor State are in equilibrium is the vapor pressure and they telling us right now it is twenty three point eight millimeters of mercury now what we need to do is figure out to figure out this problem and say okay how many if we could figure out how many molecules need to evaporate how many molecules of water need to evaporate to give us this vapor pressure we can then use the density of water to figure out how many liters of water that is so how do we figure out how many how many molecules let me write this down how many how much how many molecules how many molecules of water need to evaporate to evaporate to give us the vapor pressure to give us the vapor pressure to give us the vapor pressure of 23.8 millimeters of mercury so what I guess law or formula I don't never like to just memorize formulas but we've given this formula in the past and it's probably one of the top most useful formulas in chemistry or really I all of science what formula or law deals with pressure they give us they give us the volume of the room because that's where the pressure will be inside of so we have pressure the equilibrium vapor pressure we have a volume of a room right over here we know the temperature of the room right over there and we are trying to figure out the number of molecules that need to evaporate for us to get that pressure in that volume at that temperature so what what deals with pressure volume number of molecules let's say in moles so our lowercase n number of molecules and temperature well we've seen this many many times it's the ideal gas law pressure times volume is equal to the number of moles of our ideal gas in this case we're going to we're going to use water as our ideal gas or vapor as our ideal x times the universal gas constant times temperature and this should never seem like some bizarre formula to you because it really really makes sense if your pressure goes up then that means that either the number of molecules have gone up and we're assuming volume is constant that means either the number of molecules have gone up which makes sense more things bouncing on to the side of the container or your temperature has gone up the same number of things but they're bumping with higher kinetic energy or if you're but if your pressure stays the same and your volume goes up then that also means that your number of molecules went up or your temperature went up because you now have a bigger container in order to exert the same pressure you need more either more molecules or more kinetic energy for the molecules you have and you could keep playing around with this but I just want to make it clear this isn't some mysterious formula the first time I was exposed to this I kind of did view it as some type of mysterious formula but it's just relating pressure volume number of molecules and temperature and then this is just the ideal gas or sorry the universal gas constant so let's just get everything into the right units here and then what we're trying to solve for we want to figure out the number of molecules of water so we want to solve we want to solve for n and if we know the number of moles of water we can figure out the number of grams of water and then given the density of water we can figure out the number of milliliters of water we are dealing with so let's just rewrite the ideal gas law by dividing both sides by the universal gas constant and temperature so that you get n is equal to pressure times volume over the universal gas constant times temperature now the hardest thing about this is just making sure you have your unit's right and you're using their right ideal gas constant for the right units and we'll do that right here so what I want to do because the ideal the ideal gas cottle gas constant I have is in terms of atmospheres we need to figure out this vapor pressure this equilibrium pressure between vapor and liquid we need to write this down in terms of atmospheres so let me write this down so the vapor pressure the vapor sure the vapor pressure is equal to 23 point 8 millimeters millimeters of mercury and you can look it up at a table if you don't have this if you just don't have this in your brain one atmosphere one atmosphere is equivalent to 760 760 millimeters of mercury so if we wanted to write the vapor pressure as atmospheres let me get my calculator out the calculator out put it right over there so it's going to be twenty three point eight twenty three point eight times one over 760 or just divided by 760 and we have three significant digits so it looks like 0.03 one three so this is equal to 0.03 one three atmospheres that is our vapor pressure so let's just deal with this right here so the number of molecules of water then that are going to be in the air and the gaseous state in the vapor State or is going to be equal to our vapor pressure that's our equilibrium pressure above if more while more water molecules evaporate after that point then we're going to have a higher pressure which will actually make them favor more of them going into the liquid state so we'll go kind of past the equilibrium which is not likely or another way to think about more water molecules are not going to evaporate at a faster rate than they are going to condense beyond that pressure anyway the pressure here is zero point zero three one three atmospheres the volume here they told us right over here is four point so that's the volume four point two five four point two five times ten to the fourth times 10 to the fourth liters and then we want to divide that by you want to make sure that your universal gas constant has the right units I just looked mine up on Wikipedia zero point zero eight see everything has three significant digits so I'll just let me just a lot more significant digits and we'll just round at the end zero point zero eight to zero five seven five seven and the unit's here are liters atmospheres liters atmospheres per mole per mole per mole Kelvin and this makes sense this leader will cancel out with that leader that atmospheres cancel out with that atmospheres I'm about to multiply it by temperature right here in Kelvin will cancel out there and then I don't have a 1 over moles in the denominator a 1 over moles in the denominator will just be a moles because you're going to invert it again so that gives us our answer in moles and so finally our temperature and you got to remember you got to do it in Kelvin so 25 degrees Celsius let me write it over here 25 degrees Celsius is equal to you just add 273 to it so this is equal to 200 and 298 298 Kelvin so times 298 Kelvin and now we just have to calculate this so let's do that so we have so let me clear this out so we have use my my keyboard so point 0 3 1 3 atmospheres times 4 4.25 4.25 times 10 to the fourth that's just that II just means times 10 to the fourth that's the way that it works on this calculator and then divided by point zero eight two zero five seven divided by so I could let you actually let me make it just to make it clear let me show you that I'm dividing by this whole thing so let me insert let me insert some parentheses right here some parentheses and so in the denominator we also are multiplying by 298 and let me close the parentheses let me close the parentheses and then we get 50 4.4 we only have three significant digits so this is equal to this is equal to 54.4 moles and we could see this leaders cancels out with that leaders Kelvin cancels out with Kelvin atmospheres with atmospheres you have a one over mole in the denominator so then you just des agree to one over one over moles is just going to be moles now this is going to be fifty four point four moles of water vapor in the room to have our vapor pressure if our if more evaporates then more will condense we will be beyond our equilibrium so we won't ever have more than this amount evaporate in that room so let's figure out how much liquid water that actually is so fifty four let me do it over here so fifty four fifty four point four moles let me write it down moles of h2o that's going to be in this vapor form it's going to evaporate but let's figure out how many grams that is so what is the what is the molar mass of water well it's roughly eighteen I actually figure it out exactly it's actually eighteen point zero one if you if you actually use the exact numbers on on the periodic table at least the one that I used so we could say we could say that there's eighteen point zero one grams of h2o for every one mole of h2o and obviously you can just look up the atomic weights of hydrogen which is a little bit over one and the atomic weight of oxygen which is a little bit below 16 so you have two of these so two plus sixteen gives you pretty close to eighteen so this this right here will tell you the grams of water that can evaporate to get us to that equilibrium pressure so let's get the calculator out so we have the fifty-four point four fifty four point four times eighteen point zero one is equal to nine hundred and seventy well we only have three significant digits so nine hundred if you round this point seven it becomes nine hundred and eighty so this is 980 grams of h2o needs to evaporate for us to get to our equilibrium pressure to our vapor so let's figure out how many milliliters of water this is so they tell us the debt the density of water right here 0.997 limit is in darker darker color 0.997 grams per milliliter or another way you could view this is for every one milliliter for every one milliliter you have 0.997 grams of water of water at 25 degrees Celsius so for every milliliter this is grams per milliliter we want milliliters per gram because we want this and this to cancel out so we're essentially just going to divide 980 by 0.997 so what is that get the calculator out so we have 980 not cover up our work divided by 0.997 is equal to 980 we'll just around this 983 so this is equal to this is equal to 983 this and this cancelled out or that and that cancelled out so 980 3 milliliters of h2o so we've figured out using the ideal gas law that 900 at 25 degrees celsius which was 298 kelvin that nine hundred and eighty-three milliliters of h2o will evaporate to get us to our equilibrium vapor pressure if any nothing more will evaporate because beyond that if we have more higher pressure than that then you'll actually have you'll also have more vapor going to the liquid state because you'll have more stuff more stuff bouncing here so this is the this is if this much if this much volume of water evaporates we'll have the state where just as much as evaporating as just as much as condensing so you will never have a high you will never get to a higher pressure than that at that temperature so going back to the question we figured out that 980 3 milliliters of water will evaporate the question was is that we place two liters of water in an open container so we just figured out that only 980 3 milliliters of that so that's a little bit less than a liter so this is a little bit less so this is a little bit less than 1,000 milliliters and this is one liter so a little bit less a little bit less than half of this will evaporate for us to get to our vapor pressure so to answer our question will all of the water evaporate at 25 degrees Celsius no if the water swimming the room is sealed well no all of it will not only a little bit less than half of it will