Ideal gas equation
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Worked example: Calculating partial pressures
- [Instructor] We're told that a 10-liter cylinder contains 7.60 grams of argon, in gas form, and 4.40 grams of molecular nitrogen, once again in gas form, at 25 degrees Celsius. Calculate the partial pressure of each gas and the total pressure in the cylinder. All right, so pause this video, and see if you can work through this on your own before we work through it together. All right, so you might imagine that the ideal gas law is applicable here, and it's applicable whether we're just thinking about the partial pressures of each gas or the total. So the ideal gas law tells us that pressure times volume is equal to the number of moles times the ideal gas constant times temperature. And in this case, we're trying to solve for pressure, whether it's partial pressure or total pressure. So to solve for pressure here, we can just divide both sides by V, and you get pressure is equal to the number of moles times the ideal gas constant times the temperature divided by the volume. And so we can use this to figure out the partial pressure of each of these gases. So we can say that the partial pressure of argon is going to be equal to the number of moles of argon times the ideal gas constant times the temperature, both gases are at the same temperature over here, divided by the volume. And then we can also say that the partial pressure of our molecular nitrogen is equal to the number of moles of our molecular nitrogen times the ideal gas constant times the temperature divided by the volume. So we already know several of these things. We can look up the ideal gas constant with the appropriate units over here. They've given us the temperature, at least in terms of degrees Celsius. We'll have to convert that to kelvin. And they've also given us the volume. So all we really have to do is figure out the number of moles of each of these. And to figure out the number of moles, they give us the mass, we just have to think about molar mass. So let's look up the molar mass of argon, as well as the molar mass of molecular nitrogen. So the molar mass of argon, getting out our periodic table of elements, we look at argon right over here, and it has an average atomic mass of 39.95, which also gives us our molar mass. So a mole of argon will have a mass of 39.95 grams per mole. And then if we want to figure out the same thing for our molecular nitrogen, we look up nitrogen here, we see an average atomic mass of 14.01. So we might be tempted to say that the molar mass of molecular nitrogen is 14.01 grams per mole, but we have to remind ourselves that molecular nitrogen is made up of two nitrogen atoms. So the molar mass is going to be twice this, or 28.02 grams per mole. So this is equal to 28.02 grams per mole. And then we can apply each of these equations. So the partial pressure of argon, let me give myself a little extra space here, partial pressure of argon is going to be equal to the number of moles of argon. Well, that's just going to be, let me do this in another color, so you can see this part of the calculation. That's going to be the grams of argon, so let me write that down, 7.60 grams, times one over the molar mass, so times one over 39.95 moles per gram. And you can see that the units work out. Grams cancel with grams, and this is just going to give you the number of moles of our argon. And then we multiply that times our ideal gas constant. and we have to pick which one to use. In this case, we're dealing with liters, so both of these cases deal with that. And the difference between these is how they deal with pressure. The first is in terms of atmospheres. The second is in terms of torr. So if we want our partial and total pressures in terms of torr, we could use this second one. So let's do that. So in this case, let's use this second ideal gas constant. So that's going to be times 62.36 liter torr per mole kelvin. And then we need to multiply that times the temperature. So 25 degrees Celsius in kelvin, we add 273 to that, so that's 298 kelvin. And all of that is going to be divided by our volume, which is 10.0 liters, 10.0 liters. And we can validate that the units work out. We already talked about these grams canceling out. This mole cancels with this mole. This kelvin cancels with that kelvin. And then this liters cancels with this liters. And we're just left with torr, which is what we care about. We're thinking about a pressure, in this case, a partial pressure. We have 7.60 divided by 39.95 times 62.36 times 298 divided by 10.0 is equal to this business. And now we just have to think about our significant figures here. So we have three here, four here, three here, and three here. So when we're multiplying and dividing, we'll just go to the fewest number of significant figures we have, so it's three. So we'll want to go round to 354 torr. So the partial pressure of argon, 354 torr. And now we can do the same thing for the molecular nitrogen. And let me get myself a little more space here. So the partial pressure of our molecular nitrogen is going to be equal to, I will do this in a different color as well, when I figure out the number of moles, that is going to be the mass of molecular nitrogen, which is 4.40 grams, times one over the molar mass, so that's one over 28.02 grams per mole. And then that is going to be times our ideal gas constant, so we can really just copy the rest of this right over here, times 62.36 liter torr per mole kelvin times 298 kelvin. All of that is going to be over 10.0 liters. And once again, the units work out. Grams cancel with grams. Moles cancel with moles, liters with liters, kelvin with kelvin, and we're just left with torr. And this gets us to 4.40 divided by 28.02 times 62.36 times 298 divided by 10.0 is equal to this. And once again, the lowest significant figures we have here are three, so we'll round this to 292. So this is equal to 292 torr. And so we've figured out the partial pressure of each of these. And if we want to figure out the total pressure, the total pressure, that's just going to be the sum of the partial pressures. So it's going to be the partial pressure of the argon plus the partial pressure of the molecular nitrogen. And so this is going to be, let's see, I think I can do this in my head, 646 torr. And we are done.