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Studying for a test? Prepare with this lesson on Gases and kinetic molecular theory.

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# Ideal gas equation example 1

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In the last video we hopefully learned the intuition behind the ideal gas equation, that pressure times volume is equal to the number of molecules we have times some constant times the temperature. And that's all nice and it hopefully it makes sense to you how all of these fit together. That pressure should be inverse to volume and that's why you're multiplying both sides by each other. You could take volume and put it on this side of the equation. Or that pressure should be proportional to the number of particles and the temperature. But now let's apply it and actually do some problems. Because just knowing this isn't good enough. So let's say that I have a two liter container, or let's say a two liter balloon, containing hydrogen gas. And that's hydrogen as a diatomic molecule. So each molecule has two hydrogens in it. And let's say I'm measuring it at 30 degrees Celsius. Use different color. 30 degrees Celsius. My brain is really malfunctioning. 30 degrees, not 30 percent, 30 degrees Celsius. And let's say that the pressure on the outside of the balloon, we've measuredat two atmospheres. So my question to you is how many moles of hydrogen do we have? How many moles... So let's apply our ideal gas equation. And since we're dealing with liters and atmospheres, we have to make sure we use the right proportionality constant. But in general, if we keep pressure. So our pressure is given in atmospheres. Let me write down all the units, actually. So we have 2 atmospheres times our volume is 2 liters, is equal to n. n is the number of particles we care about, and we care about it in moles, but let's just write n there for now. Is equal to n times R. I'll do R in a second times. R times T. Now you might be atempted to just put 30 degrees in there. But in all of these problems-- in fact in general, whenever you're doing any of these gas problems or thermodynamics problems, or any time you're doing math with temperature-- you should always convert into Kelvin. And just as a bit of review as to what Kelvin is, it's just a different scale. So for example, the lowest possible temperature that can be achieved in the universe, when you think about it in Celsius, let me draw a little temperature scale here. So if that's the temperature scale. I'll draw two, one for Celsius and one for Kelvin. So the lowest possible temperature that can be achieved in the universe, and when we say the lowest possible temperature that means that the average kinetic energy of the molecules or the atoms are zero. They're just not moving. They're just stationary. So in Celsius, it's minus 273.15 degrees Celsius. So zero might be some place over here. Zero, that's where water freezes. And then 100 degrees, that's where water boils. And you can immediately see, the whole Celsius scale was made based on the freezing point and the boiling point of water. Now, Kelvin. So look at this and you say, if I have something that's 5 degrees and I have another thing that's 10 degrees, when you look at the Celsius scale, you're like, oh, maybe the 10 degree thing it has twice as much energy as the 5 degree thing. It has twice the temperature. But when you look at it from the absolute distance to zero. Let me see if I can draw this. So the 10 degree is all the way over here and the 5 degree is almost as far, that far. So the 10 degrees Celsius is only a slight increment over 5 degrees Celsius, if you were to divide the two. It's not twice as hot. And that's why they came up with the Kelvin scale. Because in the Kelvin scale, absolute zero is defined as 0. Zero Kelvin. So this right here is zero degrees Kelvin. And so zero degrees Kelvin is absolute zero. So what is zero degrees Celsius? And the increments are the same. One degree change in Celsius is one degree change in Kelvin. So at least they keep it, it's just a shift. So this is going to be plus 273 degrees Kelvin. And then 5 degrees would be plus 278 10 degrees would be plus 283 Kelvin. And then you see that 5 and 10 degrees really aren't that different from each other. But in general, if you want to convert from Celsius to Kelvin you just add 273 degrees. So 30 degrees Celsius is what? Well, this 5 and 10 I drew too close to 100. But let's say it's sitting here. It would be 303 degrees Kelvin. So this is equal to 303 degrees Kelvin. All right, so now for our temperature, that's what we were worried about. We wanted to put in the temperature there. So now we can put in our 303 degrees Kelvin. Now we have to figure out what constant to use here. And I've written a couple of down here. Remember, we're dealing with atmospheres and liters. So I wrote down a couple of versions of R right here. Let's see we're dealing with atmospheres and liters. And in the denominator we're always dealing with mole and Kelvin no matter what. So those are always going to be there. So we should use this proportionality constant. R is equal to 0.082 liter atmospheres per mole Kelvin. Let me write that down. So let me rewrite our whole equation actually. So I have 2 atmospheres times 2 liters is equal to n times, I have a bad memory, 0.082 liter atmospheres per mole Kelvin, times 303 degrees Kelvin. So let's see what we can do. Let's see if all of the units work out. So we can always, when you do dimensional analysis, you can treat units like numbers. So if you divide both sides of this equation by atmospheres, the atmospheres cancel out. Divide both sides of this equation by liters, liters cancel out. You have a Kelvin in the numerator, Kelvin in the denominator, that cancels out. And so we have 2 times 2 is equal to n times 0.082 times 303. And then we have just a per mole and a 1 over the mole. So to solve for n, or the number of moles, what we do is we divide both sides of this equation by all of this stuff. So we get 2 times 2 is 4. 4 divided by 0.082 divided by 303. I'm just taking this and putting it on the left-hand side, dividing both sides by it. And when you divide by a per mole, if you put a 1 over a mole here, that's the same thing as multiplying by a mole. So it's good, the units all worked out. We're getting n in terms of moles. And so we just have to get the calculator out and figure out how many moles we're dealing with. So we have 4 divided by 0.082 divided by 303 is equal to 0.16. If we wanted to go more digits, 0.161, but we'll just round. So this is equal to 0.16 moles of H2. I am telling you actually here, the exact number of hydrogen molecules. But if you wanted a number, you'd just multiply this times 6.02 times 10 to the 23 and then you would have a number in kind of the traditional sense. And of course, if you wanted to know what is the mass of the hydrogen you have. You'd say, OK well one mole of H2 has a mass of what? The mass of one hydrogen is one atomic mass unit. The mass of two hydrogen when it's in its molecular form, is two atomic mass units. So a mole of it is going to be 2 grams. So in this case, we have 0.16 moles. So if we wanted to convert that to grams, this in the case of these hydrogen gas molecules would be 0.32 grams. And I just multiplied it by 2 because each mole is 2 grams. Anyway, I hope you found that there's a bunch more of these problems. Because I think the math is pretty straightforward here. The thing that always makes it daunting, I think, is the units and making sure you're using the right units. What is they are using meters cubed instead of liters, or kilopascals instead of atmospheres. So I'll try to do a bunch of examples where we use all the different units and you're able to pick our constants appropriately.