Chemistry: Applications of the Gas Laws

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Let's talk about a couple of applications of the gas laws that you might encounter or that might explain some everyday experience that you've probably had. The first one, which you might encounter in lab, is the practice of collecting a gas over a water to be able to know that your sample is pure. Imagine if you take some magnesium and you react it with hydrochloric acid. We've actually seen that already. It makes magnesium chloride and hydrogen gas. Now how are we going to collect that hydrogen gas to make sure that it's separated from all of the other air that's in the room? And the answer is to use a technique called collecting a gas over water.
Here's our magnesium. Here's some hydrochloric acid. They're reacting. They're making hydrogen gas. It's going through this green tube, and it's bubbling up into this purple container. And the purple container was originally filled entirely with water. And as the hydrogen is produced, it bubbles up to the top and it displaces the water downward. And in here we have our hydrogen gas. And there was nothing else in the bottle, or so it would seem. In fact what is true is that there's also water vapor inside this space. Because the hydrogen bubbled through water, and the presence of the liquid water here, this space has equilibrated with the water. And so there are some H[2]O molecules in this space in addition to the H[2] molecules. Well Dalton's Law of partial pressure tells us that the total pressure is equal to the sum of the partial pressures of the component gases in the container. In this case, there's the partial pressure of so-called dry hydrogen, in other words in the absence of water, plus the partial pressure of water. So the total pressure of the gas inside this space consists of the sum of these two pieces.
If we wanted to calculate how much hydrogen alone was in here, we'd have to use this fact. And I'm going to illustrate that in the following problem. We have magnesium metal and it's reacting with hydrochloric acid and a gas was produced. That gas is hydrogen. It's collected over water. If we can quantify the volume of gas that we've produced, 36.0 milliliters, and we know the temperature at which we collected it--and this should be the temperature of the water--and we know the total pressure inside that space at the top of the bottle, 765.0 Torr, we can solve the problem of how much just hydrogen. So if we're just trying to get at the hydrogen, how many moles of hydrogen are formed? To do that you need another piece of information that you would typically look up in a textbook or in a book that shows chemical data. And that is that the vapor pressure of water, that partial pressure of water that we're going to use in the equation, is only a function of temperature. And in this case it amounts to 17.5 Torr.
Dalton's Law of partial pressure says that we have this sum. We can isolate just the partial pressure of hydrogen by recognizing that it's the total pressure minus the partial pressure of the water, which is 765.0 minus 17.5 Torr, or 747.5 Torr. And the temperature, remember to use the Ideal Gas Law we have to convert things to Kelvins. We started out at 20 degrees Celsius plus 273.15, and using the correct number of significant figures, 293.1 Kelvin.
And we've got to do one more thing that you may have thought of, and that is when we've used the Ideal Gas Law before; we've always had the pressure in atmospheres. And right now, we've got the pressure in Torr. So we've got to convert the pressure from Torr to atmospheres. So we have . Now I just wrote this as a 1, but this is assumed to be an exact number. So we're not limited by significant figures by the conversion factors. You never are. When we work this out, we get 0.9835 atmospheres, which is a little below 1 atmosphere. That makes sense, because this represents slightly less than 1 atmosphere as well.
Let's finish solving this problem now. We're interested in the number of moles of dry hydrogen, and we can use the Ideal Gas Law, . And plugging in . Multiply this all out and divide by the gas constant and the temperature, and it gives 1.47 x 10^-2 mole to the correct number of significant figures. We've got 3 significant figures in the volume over here, so that's going to determine the number of significant figures.
So what have we learned? We've learned how to figure out the number of moles that we collected from our experiment. When we use the technique of collecting gas over water, we have to take into account the vapor pressure.
Let's now look at another problem. And the next problem I want to look at, it's again a gas law problem, is the question of gas density. Density overall is . And we're going to answer two questions that may have occurred to you. Why do helium balloons float? Here's a helium balloon here. And why is it floating? Why does it want to go up to the ceiling? And if we were outside, why would it seemingly go up to heaven? And then the second question, how do hot air balloons work? You've probably seen hot air balloons floating around. How exactly do they get off of the ground and stay up in the air?
We need to talk about a couple of assumptions that we're going to make in order to answer these two questions. And the assumptions we're going to make, and they're not really profound assumptions, but we're going to assume that air is just 100% nitrogen. Not an unreasonable approximation, since it's already really 78%. And then the second thing that we're going to assume is that the pressure on the inside of a balloon is 1 atmosphere. It's probably a little bit over 1 atmosphere, but not enough to make a big difference. So we're just not even going to worry about it.
Before we get any further, I really should point out that density is an intensive property. And being an intensive property that means it doesn't depend on how much stuff you've got. So if you've got a sample of gas, and it has a certain density, if you have twice as much gas, it's still going to have the same density, because it will occupy twice the volume. And since it doesn't matter how much stuff we've got, let's assume that we're going to start with 1.00 moles of stuff.
To answer this question, and you recall that , we need to calculate the volume. And if we're going to assume 1 mole of gas, we need to calculate the volume of 1 mole of gas. To be realistic, we're going to calculate this at room temperature, 25 degrees C, and 1 atmosphere. And your first thought might have been, "Oh well the volume of 1 mole of gas is 22.4 liters." And that would be wrong. And the reason why it's wrong is recall 22.4 liters is the volume of a mole of gas only at STP. And the T in STP, standard temperature and pressure, is zero degrees C. And that's not a realistic approach to why this helium balloon is floating at room temperature. So let's adjust the temperature to room temperature and see what that gets us. So we're going to calculate the volume of a mole of gas at 25 degrees C and 1 atmosphere. And to do that , and let's assume that I said it was actually 1.00 atmospheres so that we can actually measure the pressure much more precisely. Okay, work this out and what do you get? 24.4 liters, so the volume of a mole of gas at a slightly higher temperature than STP is slightly higher. That's Charles's Law. That's exactly what we would have expected.
This volume is the volume of a mole of gas. It doesn't even matter what kind of gas we've got. So we can construct the following table. We've got the volume of 1.00 mole of gas, be it helium, 24.4 liters, or nitrogen, 24.4 liters. And we know that the mass of 1.00 mole of helium is 4 grams. And the mass of 1.00 mole of nitrogen is 28.0 grams. Knowing that , we can quickly calculate by dividing 4 into 24.4, that the density of helium is 0.164 grams per mole, compared to the density of nitrogen at the same temperature, 1.15 grams per mole.
How does this explain that helium floats in nitrogen, that helium balloons float in air? It turns out that less dense things float on more dense things. And so ice floats on water because ice is less dense than water. Similarly oil floats on water because oil is less dense than water. So here we have that a helium balloon would float on a nitrogen balloon because it's less dense. The helium balloon is less dense.
Now let's finally turn to what's going on in a hot air balloon. In a hot air balloon, what we need to do is calculate what the gas volume would be if we were at a higher temperature. And I just chose an arbitrary temperature of 600 Kelvin. So once again we're going to calculate the volume of 1 mole of gas. Now it's at 600 Kelvin. That's pretty hot. We have , and once again let's suppose that I wrote 1.00 atmospheres so that we have some reasonable number of significant figures. And let's assume that this is actually three significant figures. So that's going to give us a value for the volume of a mole of gas, any gas, at 600 Kelvin, 49.2 liters assuming it's an ideal gas.
And from that we can calculate the following table very similar to the one we assembled before. Now we're comparing nitrogen at 298, so we're assuming the gas on the inside of the hot air balloon and the outside are both 100% nitrogen. At 298, room temperature, the volume is 24.4 liters. But at 600 Kelvin, it's 49.2 liters. The mass of one mole of nitrogen, since it's the same gas for both, the masses are both the same. And here are the densities. And you can see that because we are dividing by a larger volume, the density at the higher temperature is lower than the density at low temperature. So again, less dense things float in more dense things. So hot nitrogen floats in cold nitrogen. This is why smoke rises. Smoke rises because hot things are less dense than cold things. So smoke rises. And similarly, hot air balloons are going to rise.
An interesting point is that as the day warms up, the sun comes out. This temperature can get higher and higher. As this gets higher and higher, this is going to get lower and lower. As the density of the rest of the air gets lower, your balloon is going to have less lift. So very often you'll see hot air balloons in the early morning when the temperature is relatively low, because they don't have to heat the gas on the inside of their balloon as hot in order to get enough lift to get off of the ground.
Let's summarize. I've shown that you can use gas laws in a couple of different problems, one that you might see in lab, another one that you might see just floating by one day. And it's pretty clear that, from a couple of simple calculations, you can get some explanations that tell you something about how the world around you actually operates.
Gases
The Ideal Gas Law and Kinetic-Molecular Theory of Gases
Applications of the Gas Laws Page [3 of 3]