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Studying for a test? Prepare with this lesson on Gases and kinetic molecular theory.
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We have a situation here where we have a four meter cubed container. Let's say that's a balloon of some sort. And instead of having just one type of molecule of gas in this container, we have three molecules of gas. We have some oxygen molecules, some hydrogen molecules, and some nitrogen molecules. And what the problem tells us is, we have 2.1 total kilograms of gas. An of that, by mass, 30.48% is oxygen, 2.86% is hydrogen molecules, and 66.67% is nitrogen. So what we need to figure, and it's all at standard temperature at least. It's zero degrees Celsius, which we know is 273 Kelvin. But we need to figure out what is the total pressure in the container, or being exerted on the surface of the container? And then, this is a new concept, we want to figure out the partial pressures of each of these gases. Essentially, how much are each of these gases contributing to the total pressure? And you can imagine, if this is a container and each of these are the three types of gases, some of the pressure is going to be from the blue, maybe oxygen is the blue gas, from the blue gas bumping into the walls. Some of the pressure is going to be from the hydrogen bumping into the walls, maybe that's this yellow gas, and some of the pressure is going to be from the nitrogen bumping into the walls. Let's say that's the brown gas. So the partial pressure due to nitrogen, that's the pressure just due to the brown particles bumping into walls. Let's see if we can figure this out. So the first thing to figuring the total pressure is, we have to figure out the total moles of molecules we have. And the easiest way I can figure out to figure out the total number of moles is to figure out the moles of each of these molecules. So if we have 2.1 kilograms of gas-- let me write this down -- If we want to know moles of nitrogen. Let me do it in the nitrogen color. We know that 66.67% of this, we could say 2.1 kilograms or 2100 grams, we know that that's nitrogen. So let's do it in grams. Because when we talk about molecular mass it's always in grams. It doesn't have to be. But it makes it a lot simpler to convert between atomic mass units and mass in our world. So this is 2/3 of 2100, that's 1400 grams of N2. Now what's the molar mass of this nitrogen molecule? Well we know that the atomic mass of nitrogen is 14. So this molecule has two nitrogens. So its atomic mass is 28. So one of these molecules will have a mass of 28 atomic mass units. Or one mole of N2 would have a mass of 28 grams. So one mole is 28 grams. We have 1400 grams -- or we say grams per mole, if we want to keep our units right. So if we say 1400 total grams so divided by 28 grams per mole we should get the number of moles. So 1400 divided by 28 is equal to 50. That worked out nicely. So we have 50 moles of N2. We could write that right there. All right. Let's do oxygen next. So we do the same process over again. 30% is oxygen. So let's do oxygen down here, O2. So we take 30%. Remember, these percentages I gave you, these are the percentages of the total mass, not the percentage of the moles. So we have to figure out what the moles are. So 30.48% of 2100 grams is equal to about 640. Let's round off. So this is equal to 640 grams. And then what is the mass of one mole of the oxygen gas molecule? The atomic mass of one oxygen atom is 16. You can look it up on the periodic table, although you should probably be pretty familiar with it by now. So the atomic mass of this molecule is 32 atomic mass units. So one mole of O2 is going to be 32 grams. We have 640 grams. So how many moles do we have? 640 divided by 32 is equal to 20. We have 20 moles of oxygen. Let me write that down. We have 20 moles. Now we just have to figure out the hydrogen. 2.86% of all that is hydrogen. So let's see, if we do 2100 grams, remember I want to do everything in grams, so I just want to do a unit conversion there. 2100 grams times 2.86% is equal to about 60 grams. So hydrogen, this 2% of that 2100 grams is 60 grams. And then what's the molar mass of one hydrogen. That's H2. So we know that the hydrogen atom by itself has a mass of 1, doesn't have a neutron in most cases. So the atomic mass of this is 2. Or the molar mass of this is 2 grams. So one mole of H2 is equal to two grams. We have 60 grams. So we clearly have 60 divided by 2, we have 30 moles. So this is interesting, even though hydrogen was a super small fraction of the total mass of the gas that we have inside of the container, we actually have more actual particles, more actual molecules of hydrogen than we do of oxygen. That's because each molecule of hydrogen only has an atomic mass of 2 atomic mass units, while each molecule of oxygen has 32 because there's two oxygen atoms. So already we're seeing we actually have more particles do the hydrogen than do the oxygen. And the particles are what matter, not the mass, when we talk about part pressure and partial pressure. So the first thing we can think about is how many total moles of gas, how many total particles do we have bouncing around? 20 moles of oxygen, 30 moles of hydrogen, 50 moles of nitrogen gas. Add them up. We have 100 moles of gas. So if we want to figure out the total pressure first, we can just apply this 100 moles. Let me erase this. I want to keep the problem statement there the whole time. There you go. And I can erase some stuff that you're not seeing off the screen. And now I'm ready. So we have 100 moles. So we just do our PV is equal to nRT. We're trying to solve for P. P times 4 meters cubed is equal to n. n is the number of moles. We have 100 moles. Is equal to 100 moles times R. I'll put a blank there for R, because we have to figure out which R we want to use. Times temperature, remember we have to do it in Kelvin. So 0 degrees Celsius is 273 Kelvin. And then which R do we use? I always like to write my R's down here. So we're dealing with meters cubed, we're not dealing with liters, so let's use this one. 8.3145 meters cubed pascals per mole Kelvin. The units there, I think I should keep, these are in meters cubed pascals divided by moles Kelvin. And then our temperature was 273 Kelvin. Now let's do a little dimensional analysis to make sure that we're doing things right. These meters cancel out with those meters. We divide both sides of the equation by meters. These moles cancel with these moles. The moles of the numerator, the moles of the denominator. Kelvin in the numerator, Kelvin in the denominator. And all we're left with is pascals. Which is good because that is a unit of pressure. So if we divide both sides of this equation by 4. I'll just divide the 100 by 4. 25 times 8.3145 times 273. And the only unit that we were left with was pascals. Which is nice, because that's a unit of pressure. So, let's do the math, 25 times 8.3145 times 273 is equal to 56,746 pascals. And that might seem like a crazy number. But the pascal is actually a very small amount of pressure. It actually turns out that 101,325 pascals is equal to one atmosphere. So if we want to figure out how many atmospheres this is, we could just divide that. Let me look it up on this table. Yes, 101,325. So if we wanted to, we could write this in kilopascals. That's 56.746 kilopascals. Or if we wanted it in atmospheres we just take 56,746 divided by 101,325. It equals 0.56 atmospheres. So that's the total pressure being exerted from all of the gases. I deleted that picture. So this is the total pressure. So our question is, what's the partial pressure? We could use either of these numbers, they're just in different units. What's the partial pressure of just the oxygen by itself? Well, you look at the moles, because we don't care about the actual mass. Because we're assuming that they're ideal gases. We want to look at the number of particles. Because remember, we said pressure times volume is proportional to the number of particles times temperature. And they're all at the same temperature. So the number of particles is what matters. So oxygen represents 20% of the particles. 20/100. So, the partial pressure of oxygen, let me write that as pressure due to oxygen, due to O2. It's going to be 20% of the total pressure. 20% times, let me write 56.746 kilopascals. I just took this pressure measurement. If I wanted atmospheres, times 0.56 atmospheres. So the partial pressure of oxygen -- well, I already have the 0.56 written there. So times 0.2 is equal to 0.112 atmospheres. It's just 20% of this. How did I get 20%? We have a total of 100 mole molecules in our balloon. 20 moles of them are oxygen. So 20% are oxygen. So 20% of the pressure is going to be due to oxygen. So it's this many atmospheres. If I took 20% times the 56,000. 0.2 times 56, you get roughly 11.2 kilopascals. I'm just multiplying 20% by any of these numbers. And the numbers are going to change depending on the units. So you do the same process. What is the partial pressure due to nitrogen? Well even though 2/3 of the mass is nitrogen, only 50% of the actual particles are nitrogen. So 50% of the pressures is due to the nitrogen particles. Remember, you have to convert everything to moles. Because we only care about the number of particles. So if you want to know the partial pressure due to the nitrogen molecules, it's 50% of these. So it's 28,373 pascals. That's roughly. Or if you took half of this, approximately 28.4 kilopascals. Or approximately 0.28 atmospheres. And then finally, if you want to figure out the partial pressure due to the hydrogen atoms. The partial pressure due to the hydrogen atoms. Hydrogen, even though it's a very small part of the mass, it actually represents 30% percent of the molecules. And it's the molecules that are bumping into things. We don't care so much about the mass. So 30% of the molecules. And remember, when we talk about kinetic energy, if something with a small mass has the same kinetic energy, it's actually moving faster. So when we talk about temperature, that's average kinetic energy. So maybe in this we could imagine that the hydrogen might be moving faster than the nitrogen or the oxygen. But we don't have to think about that too much right now. But the partial pressure due to hydrogen is just 30% of any of these numbers. Pick one. Let's do it in atmospheres. 0.3 times 0.56 is equal to 0.168 atmospheres. And so the total pressure should be equal to the pressure of each of the partial pressures of each of the gases. Plus the partial pressure of oxygen, plus the partial pressure of hydrogen. And so this one, we figured out was 0.28 atmospheres. The oxygen was 0.112 atmospheres. And this was 0.168. And if you add these together, you will see indeed, that they add to 0.56 atmospheres. Which was the total pressure of our system. Anyway, this is a very long and hairy problem. But the key takeaway is that every molecule in the system contributes to the total pressure in proportion to the number of moles it has as a percentage of the total number of moles in the system. I hope that didn't confuse you too much.