Balloon not floating? Did your party store give you “fake” helium? Don’t be fooled by fullness – Dalton’s Law of Partial Pressures tells us that there could be any combination of gases in there and, as long as the total # of gas molecules is the same, they’ll have the same pressure. But if we weigh we have a way to check whether we need to sue!

The pressure comes from the gas molecules banging into one another and the walls of the balloon as they try to escape and while you can’t feel the individual collisions, you can poke it and see how firm it feels &, regardless of what gas or gases are in there, your balloon would be just as “full”

But that doesn’t mean that the identity of the gas molecule doesn’t matter for other things! So we often want to know what the proportions of the different gases are (mole fraction) and how much of the total pressure comes from each gas (partial pressure).

The ideal gas law tells us that If you fill a balloon to fullness with your breath & fill an identical balloon to the same fullness with helium, each balloon will have the same total number of gas molecules. This is because gas molecules are traveling so fast and are so far apart from one another in space that if we make some assumptions (like that they don’t interact with one another other than by ricochetting directly off) we can treat them as point particles.

I like to think of the ideal gas law as filling a book with words that want to “jump off the page” & the spacing between the words is so great that “supercalifragilisticexpialidocious” and “cat” behave the same. Much more about it here:  http://bit.ly/2Px1Mvq

More formally, it says that, in a gas (be it pure or a mixture) there’s this relationship where the pressure (P) multiplied by the volume (V) is equal to the # of gas molecules (n) times a constant called the “ideal gas constant” (R = – 0.0821 L × atm × mol-1 × K-1) times the temperature (T).

IDEAL GAS LAW: PV = nRT

And, if it’s pressure we’re interested in, we can rearrange to rewrite it as P = (nRT)/V

If the molecules get to ignore one another, why can’t we? Dalton says we can (at least when it comes to pressure). Since the ideal gas law says that, when it comes to pressure, it’s the # of particles, not their identity, that matters, Dalton tells us we can add together the pressure that would be generated by each gas separately and that would tell us the total pressure. And we can go the other way too – if we know what proportion of a mixture is a certain gas we can calculate what the pressure would be if we removed all of the other gases in there – and we call that the partial pressure.

Law of Partial Pressures: Ptotal = P1 + P2 + P3 …..

Each of those P’s is the partial pressure coming from one of the components and each can be calculated with P = (nRT)/V. And the only thing different between them will be the n (# of molecules of gas). So there’s a “shortcut” if we know what proportion of the gas mixture is that gas, and we call that value the “mole fraction” We can multiply the mole fraction of the component we’re interested in by the total pressure to get the partial pressure.

For example, if we had a gas mixture that was 1/2 A & 1/2 B, 1/2 of the gas molecules would be A and half would be B (each would have a mole fraction of 1/2) – so if we had 1 mol total we’d have 0.5 of each, if we had 2 mol total, we’d have 1 mol each, etc.

And 1/2 of the total pressure would come from A & half from B – each would have a partial pressure of 1/2 the total pressure (e.g. if the total pressure was 1 atm, the partial pressure of each would be 0.5 atm)

If we were to actually remove B, the A molecules could move out more, so there’d be fewer collisions and lower pressure. How much lower? The new pressure would be half what it is.

And the proportions don’t have to be equal – for example, we could have a mixture of 1/4 A & 3/4 B – A would have a mole fraction of 1/4, a partial pressure of 1/4 the total pressure – and B would have a mole fraction of 3/4 & partial pressure of 3/4 the total pressure.

And it doesn’t matter how many different gasses there are – If our mixture was 1/3 A, 1/3 B, & 1/3 C, each would have a mole fraction of 1/3 and the partial pressure of each would be 1/3 of the total pressure.

Where’s that “mole” thing come from? It’s not a freckly thing or a cuddly thing, it’s the chemist’s “dozen” – but instead of 12 it’s 6 x 10^23. So if you ordered a mole of helium, you’d get 6 x 10^23 He’s. And if you ordered a mole of oxygen gas (which likes to hang out as the diatomic molecule O2) you’d get 6 x 10^23 O2s. And the ideal gas law tells us that both would occupy the same amount of space (if temp & pressure are constant) and generate the same amount of pressure (if temp & volume are constant).

But they aren’t interchangeable in all ways….

Say you get a new job and and you want to win your bosses’ approval. So you bring in a box of bagels. If you go to the bakery usually, unless you want some super fancy bagels, you pay the same price for a dozen bagels of any kind and they fill up the same amount of space in the box. So you order 6 sesame seed bagels and 6 plain bagels. You have a “dozen fraction” of 0.5 for each.

But you bring them to work and, turns out your boss is allergic to sesame seeds – so even though the bakery charged you the same as if you’d ordered any other combo, your boss is now charging you with attempted murder!

Just like not all bagels are the same, not all gas molecules are they same – while they might share similar properties that allow us to treat them as “identical,” they also have unique properties that make them useful for different situations.

Because the gas molecules in an ideal gas are so far apart and oblivious to one another, the actual size of the particles doesn’t matter when calculating things like pressure and (overall) volume.

But the molecules do have masses – and the masses of some gases are heavier than others. How much an individual gas molecule weighs depends on how many atoms the molecule contains and how many smaller parts (subatomic particles) those atoms have. (note: I’m going to use mass and weight interchangeably here, but weight takes into account gravity which mass does not)

Earth’s atmosphere by volume (and mole fraction in parentheses because of that direct relationship we’ve been talking about):

~78% (0.78) N2

~21%  (0.21) O2

~0.9% (0.009) Ar

There are also trace amounts of CO2, H2O, etc.

If you’re at 1 atm partial pressures are -> 0.78 atm, 0.21 atm, 0.009 atm, respectively

What if we look at exhaled air? Mole fractions in exhaled air: N2: 0.741; O2: 0.151; H2O: 0.037; CO2: 0.063; Ar: 0.010

For simplicity’s sake, let’s pretend that the air we’re filling our balloon with is 80% O2 & 20% N2. And our helium balloon is 100% He.

Each oxygen atom has 8 protons (always) and 8 neutrons (on average) and each nitrogen atom has 7 protons (always) and an average of 7 neutrons. And both of them travel in pairs (as diatomic molecules (O2 & N2)

But helium travels alone and it’s really light. It only has 2 protons and 2 neutrons (electrons are so light we don’t count them).

So 1 mole of helium and 1 mole of air will take up the same amount of space (same volume) and have the same pressure (balloon feels just as tight) but the helium is much lighter, so it’ll rise above the heavier air.

Less do some calculations to check on that potential helium fraud situation…  O has a molar mass of ~16 g/mol and it hangs out as pairs (O2). So a mole of oxygen gas has a mass of ~32 g. Similarly, N weighs ~14 g/mol, so a mole of nitrogen gas (N2) weighs ~28g. But for every mole of atmospheric gas, only 20% of it’s O2 & only 80% of it’s N2. So there’d be 0.2mol O2 & 0.8mol N2.

And if we multiply them by their molar masses…

32 g/mol * 0.2 mol = is 6.4 g

28 g/mol * 0.8 mol = 22.4 g

So 1 mol of atmospheric air weighs ~ 6.4 + 22.4 = 28.8 g

And at standard pressure & temperature, it takes up 22.4L. So if we had a 1 L balloon, it’d have 1/22.4 = ~0.04 mol. So 28.8g/mol * 0.04 mol = ~1.3 g.

Now let’s look at helium. It has a molar mass of ~4g/mol. So our 1L balloon would weigh ~ 0.04*4 = ~0.16 g. So it’ll float up & displace the heavier air.

But don’t get too worried if the balloon isn’t exactly 0.16g (taking into account the skin weight of course). “Balloon gas” isn’t usually pure helium – the really pure stuff’s saved for more important things where purity really matters. Instead, balloon gas has traces of O2, nitrogen & other atmospheric gases. You need at least 92% for the balloon to float well & balloon gas is typically 92-98% pure

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