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Thermodynamics part 5: Molar ideal gas law problem

Sal uses the molar version of the ideal gas law to solve for the number of moles in a gas. He also shows how to convert this answer into number of molecules using Avogadro's number. Created by Sal Khan.

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Video transcript

I told you that the two most important things you should know in thermodynamics that will get you most of your way through most exams is that the pressure times the volume is equal to a constant, and that the pressure times the volume divided by the temperatures is equal to a constant. They all change such that the initial pressure times the volume divided by the initial temperature is equal to the final pressure times volume divided by the final temperature. Assuming that you're not changing the energy of the system, and we'll do more on that later. The other thing you should remember is that pressure times volume is equal to n, where n is the number of moles. Mole is a number like dozen, but mole is a huge number-- 6 times 10 to the 23 times R. R was the universal gas constant-- that's 8.31 joules per mole Kelvin times the temperature. Remember, just to be safe, always convert to Kelvin first. Let's see if we can do a problem that I can make up on the fly of this situation. Let's say I have a balloon, and the volume of the balloon is 1 meter cubed, so this is a big balloon. That's fairly large, if you imagine a cubic meter. The volume is a cubic meter, and let's say the pressure is equal to 5 pascals, and that's newtons per meter squared. And let's say we're at a reasonably warm temperature, so temperature is equal to 20 degrees Celsius, and let's say that balloon is filled helium. My question to you is, how many molecules of helium do I have in the balloon? Let's just substitute into the equation. We have pressure, which is 5-- and I'll actually write the units, I never do it, but you should, and you should always do it on an exam-- 5 newtons per meter squared times the volume. 1 meter cubed is equal to my number of moles, n, times the universal gas constant, 8.31 joules per mole Kelvin, times temperature. Remember, and I can't repeat this enough, always convert the temperature to Kelvin-- so whatever our Celsius temperature is, add 273. Add 273 to that, and you get 293 Kelvin. So I get 5 times 1, and meters square, meters cubed cancels out and just becomes a meter. Newton meter is joules. 5 joules is equal to n moles times 8.31 joules per mole Kelvin. This Kelvin and this Kelvin cancel out, so 8.31 times 293 is equal to 2,434.83 joules per mole. To get to the number of moles, we just divide both sides of this equation by that. And the units should work out, so you get 5. So n is equal to 5 joules times 1 over 2,434.83. Since we're dividing by this, this flips; moles per joule. This joule cancels out with this joule, so we just have to divide 5 by this, and we'll get the number of moles. Let's take the inverse of what I had there times 5, and so I get 0.002 moles. So this equals 0.0021 moles. That might seem like a small number to you, but let's figure out how many molecules that is. Let me make some space free, so I can write Avogadro's number down. Did I even say what Avogadro's number is? Avogadro's number is the number of molecules per mole-- it's that number. So, number Avogadro is equal to 6.022 times 10 to the 23 molecules per mole. The top is molecules, and the bottom is moles-- I know you can't read that. I have 0.0021 moles, so how many molecules do I have? I just multiply that 0.0021 times how many moles per molecule-- because this is moles-- times Avogadro's number, which is molecules per mole. That's molecules, this is moles-- maybe I should write the whole thing-- so then the moles cancel out, and Avogadro's number is 6.022 times 10 to the 23-- let's just remember that-- and let's just multiply that times 0.0021. It equals 0.0126 times 10 to the 23 molecules. This is 0.0126. That's the same thing as 1.26 times 0.01, and then of course times 10 to the 23. What's 0.01? That's 10 to the negative 2-- 10 to the negative 1 is 0.1, so this 10 to the negative 2. Then we get 1.26-- 10 to the negative 2, times 10 to the negative 3, and we add the exponents times 10 to the 21. It's roughly 1.26, and then another 19 zeroes-- or roughly 1 followed by 21 zeroes-- is how many molecules of, in this case, helium we had in the balloon. It's not too difficult. The hard part is really just remembering Avogadro's number, remembering the universal gas constant is 8.31 joules per mole Kelvin, remembering to always convert your temperature to Kelvin, and then just making sure all your units match up. Sometimes that might be tricky: they might give volume in liters, and you have to-- especially in this case-- convert it to meters cubed before you do it. They might give pressure, atmospheres, or bars, and you should know the conversion, and convert it to pascals or newtons per meter squared. Other than that, it's just substituting and just doing the hairy math and the scientific notation. Hopefully, that was vaguely clarifying. See you in the next video.