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SYFY WIRE Bad Astronomy

How Do You Weigh a Helium Balloon?

By Phil Plait

My friend Jenny Lawson—aka The Bloggess—is weird.

This isn’t casting any aspersions! She would be the first one to admit it. In fact, she does, constantly, on her blog. If you read her book, Let's Pretend This Never Happened, you'll find this assessment is ironclad.

For further evidence: Just the other day, she was in a conversation with her husband, Victor.* She asked him, “What weighs more, five pounds of helium, or five pounds of cheese?” She wrote up the exchange on her blog, and tweeted a link it. I laughed when I saw it, and immediately replied:

She replied:

To which I made the obvious rejoinder:

It only occurred to me recently to wonder about filling the Swiss cheese holes with helium.

But in reality, I do know the answer. This question is an old variation on the riddle, “What weighs more, a pound of lead or a pound of feathers?” The answer is neither: They both weigh a pound. But I remember hearing this riddle when I was but a wee lad, and being momentarily baffled. We humans sometimes confuse weight and density; lead is very dense, but a pound’s a pound the world around. A pound of feathers would take up a lot more space, but it would still weigh a pound.

But in the case of Jenny’s question, we get even more confused. After all, helium floats! If you had a balloon full of helium, and tried to weigh it on a scale, it would float away. If you could somehow tie it to the scale—and you had a scale with the decidedly odd characteristic that it could measure numbers less than zero—it would say the balloon has negative weight!

But that’s not really the case. Here’s the answer: A balloon filled with five pounds of helium would weigh exactly the same as a five pound block of cheese.

How can this be? Ah, let me explain.

What we think of as weight is really a force—the acceleration due to gravity acting on our mass. Mass is a property of matter that is independent of weight. I have a mass of about 80 kilograms, and I would have this same mass on the Earth, the Moon, or the surface of a neutron star. It’s basically telling you how much matter makes up me.

On Earth, with its one Earth gravity, I feel that mass as a weight. The Earth pulls down on me, and if I step on a scale it says I have a weight of about 175 pounds. If I were on the Moon, with 1/6th the Earth’s gravity, I’d weigh just shy of 30 pounds. My mass is still the same (80 kilos), but there’s less gravity pulling on me. I weigh less!

All matter has mass. Even helium. So let’s say I have some amount of helium—borrowing from Jenny, call it 2.3 kilos. On Earth, that would weigh five pounds. Earth’s gravity still pulls on that helium, and given that mass, the force works out to be five pounds.

But the difference between helium and cheese is that helium is very low density, lower than the air around it. That makes it buoyant. Buoyancy is a force too, like gravity, but it works in the opposite direction: It pushes things up. It only makes objects float if they are less dense than their surroundings, like a boat on water, or a helium balloon in air.

Buoyancy works due to displacement. If I take a chunk of iron a centimeter on a side (the size of a six-sided die) and drop it in water, it sinks. The amount of water it displaces (pushes aside due to its volume) weighs less than the iron does, so down the iron goes.

But if I spread the iron out, make it bowl-shaped, it can actually displace a lot more water. If it pushes out the amount of water equivalent to its own weight, then it’ll float! The water it pushes out wants to flow back under it, and that is a force pushing up on the iron. When the water displaced weighs the same as the iron, the two forces balance. The iron floats. That’s how big steel ships float; they’re spread out, and they displace their own weight in water.

In the case of a balloon, the helium inside the balloon weighs less than the same volume of air the balloon displaces. This air outside pushes on the balloon, and up it goes!

So a helium balloon has two opposing forces acting on it: gravity pulling it down, and buoyancy pushing it up. Buoyancy wins, so you can’t really weigh a balloon, even though it does have weight.

Weird, eh? Think of it this way: If you could put a balloon with five pounds of helium in it in a vacuum chamber, then there’d be no buoyancy force pushing it up. Only gravity acts on it, so if it sat on a scale it would register as five pounds!

Tadaa! See? A five pound balloon weighs the same as a five pound piece of cheese. It would just be less tasty.

Still, I started thinking … how would you weigh a helium balloon? How could you know it weighs five pounds if you can’t weigh it on a scale?

I can think of a few ways. (Note: For all these methods, I will ignore the weight of the balloon material itself, which is hopefully small compared with five pounds.)

For one, you could look up the density of helium in a balloon, then multiply it by the volume you’d need so that it weighs five pounds. In air, helium has a density of 0.2 kilograms/cubic meter, so you’d need about 11 cubic meters of helium to weigh five pounds, which would be a balloon about three meters across. That’s a lot of helium. In reality it would be a bit smaller, since the pressure inside a balloon is higher than the air around it, but eh. Close enough.

I thought of another way, which is to mount a scale upside down on the ceiling and let the balloon rest against it. But then I realize that won’t work; the force of buoyancy pushing it up is actually greater than its weight, that’s why it keeps floating up when you let it go outside (which sometimes leads to another inexorable force of nature, the force of a child crying).

But then I realized you could let a balloon go outside. It’ll float up, getting higher and higher, where the air is thinner (less dense) until eventually the weight of the air displaced by the balloon is equal to the weight of the helium in the balloon. At that point it will stop rising. You can then determine the altitude of the balloon (perhaps using a few spotters with known locations, and a bit of trig as they measure the balloon’s angle in the sky). Then look up the known density of air at that height, and the volume of the balloon (oops: measure that before you let it go) to get the weight of the air displaced. That’s equal to the weight of the helium! Well, more or less; the balloon will expand as it rises, but in principle you could find the helium weight this way. It’ll also be very high up, probably a few miles, but in principle this would work.

Still, that’s pretty complicated. A much easier and practical way is to buy a big tank of helium. Set it on a scale and weigh it. Now grab a balloon (a big one) and start filling it. When the scale says the tank has lost five pounds, you’re done! Note: My wife thought of this one, and then scolded me for making this too complicated. I suspect my own marriage is much like Jenny’s.

There are other ways, too, but they get a bit impractical (like cooling the helium until it’s a liquid, no longer buoyant in air, and then putting it on a scale; The problem with this is helium liquefies at 4 Kelvins, or -269° Celsius, which is -452° Fahrenheit. That’s a tad pricey and difficult of a procedure, as well as incredibly dangerous to do at home).

But the point is, it’s entirely possible to have a five pound balloon of helium. Mind you, it’ll lift a lot more than five pounds; a balloon that size can lift about 25 pounds. You could tie the cheese to it and it would still float away! That’s a waste of a balloon, and cheese. I don’t recommend this either.

So there you go, Jenny. I told you I knew the answer to this, because science. In fact, I even tweeted it:

*I sometimes wonder if she makes these conversations up for the sake of humor, but then—having met them both—I rid myself of that outlandish notion.

Update, Aug. 18, 2014 at 14:30 UTC: I originally said buoyancy only works on objects less dense than their surroundings, which was somewhat sloppy of me. It works on any object surrounded by a less dense medium, too; it's just that in that case the object won't float. I tightened up the language here to clear that up.

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