Fred and Art are old friends who get together once in a while, usually for a run on the beach.
The live somewhere on the West Coast.
Fred is a physics teacher, and Art is a programmer.
How much does air weigh?
Fred and Art were at the beach for their usual run. And as usual, it was Art's dog, Shane, a golden Lab, that did most of the running. It was a great day: a gentle breeze blowing, white clouds parked in the blue sky, seagulls drifting overhead, a few small sailboats running up the coastline, every now and again an airliner flying overhead out to sea, and further up the beach, a family flying kites.
Art said, "Great day".
"Yeah, we could be in Florida, getting blown around by Hurricane Daphne."
"Or in Kansas, catching tornadoes."
"So - you ready to move?"
Art took the Frisbee he was carrying and threw it up ahead. It seemed to float on the air. Shane darted after it, raising little clouds of sand after him.
"Why do they like to do that so much?" Fred asked.
"I have no idea. I keep asking him, but all he does is smile and wag his tail. It must be a closely guarded secret of the fraternity."
After a while, Art said, "My son asked me the other day how much air weighs. I told him it was really complicated and I'd have to work on it for a while. So now I'm working on it, Fred. How much does air weigh?"
"Well, I could just tell you - as soon as I look it up, but that wouldn't really help you or Billy. Think about this for a minute: how would you weigh it?"
Art stopped, and held out his right hand, palm up. "Doesn't seem to weigh much at all."
Fred said, "That's because we're standing in an ocean of it. Think about water. It has weight, doesn't it?"
"Sure. Just carry around a couple of gallons of it for a while."
"OK, then, how do you weigh water?"
"You put it on a scale and read it off the dial."
"Exactly. Now how about this: we're under water, swimming around in SCUBA gear. Now, how do you weigh water?"
Art thought a minute. "Somehow I don't think the scale would work. The water would just float off."
"Right," Fred replied. "So what's the real problem down there?"
"It looks like you can't weigh water, under water."
"Because?" Fred asked.
"Well, because you're under water. It would be like ... like ... like trying to weigh air."
Art thought for a while, then said, "OK, you can weigh water, if you get it out of the water, so it seems logical that you could weigh air if you get it out of the air."
Fred stopped for a bit. "I hadn't thought of that. But you're right, that would work. So how do you get air out of the air? Or, more precisely, how do you get the air you want to weigh, out of the rest of the air?"
Now it was Art's turn to stop. After a bit, he said, "You could bottle it, take it up to the Space Station."
"Two problems, Art, maybe three. We'll go back and count later. First, it still costs a bundle to take something up there. And as much as you like running, you're not really in shape to make the trip. And what do you think your scale would read, up there, if you put something on it?"
Art said, "Oh, yeah." He remembered that everything is weightless on the Space Station. "It's coming back to me now. Weight is what things have here on the surface. They weigh less on the moon, more on Jupiter."
"Now you're getting it. By the way, do you know why that's true?" Fred said, looking serious.
"Because gravity sucks".
"That's One," Art said, throwing a handful of sand at Fred.
"No, really," Fred said. "What we call 'weight' is just a counter for a fundamental property that everything has: mass. And mass is a measure of how much 'stuff' there is in anything. The more 'stuff' there is, the more mass there is, and the more it weighs, here on the surface."
"OK, then, what's the 'stuff'?"
"Basically, just atoms. The more atoms there are, the more 'stuff', and the more mass. You remember that old school riddle: which weighs more, a pound of feathers or a pound of lead?"
"No," Art said.
"Barbarian!" Fred scoffed. "OK, then, here it is: which weighs more, a pound of feathers or a pound of lead?"
"They both weigh a pound," said Art, looking puzzled. He was, after all, a programmer, and being quite logical, he figured that a pound's a pound.
"Of course. But you'd be surprised at how many people either have to think about it for a minute, or say that the pound of lead is heavier. So which is bigger?"
"The pound of feathers - a lot bigger."
"Right. Feathers are less dense than lead, so their 'stuff' has a lot more empty space in it, which you can prove by compressing the feathers, and trying to compress the lead. Both have the same mass, it's just arranged differently. So what we're really talking about isn't just 'weight', it's 'weight per unit of volume'. It's more useful to know that a bunch of stuff, like lead, for instance, weighs about 700 pounds per cubic foot, compared to aluminum, at 170 pounds per cubic foot. And that number - weight per unit of volume - is the density of whatever we're measuring."
"OK," Art said, "but what's that got to do with the weight of air?"
"Have patience, grasshopper," Fred said inscruitably. "Think about this: which weighs more: a pound of lead or a pound of air?"
"OK, master Fred, I'm ready for that one: both weigh the same. But I'm still not convinced that air has weight, or that you can weigh it."
"Fair enough. It's time to stop theorizing and get down to basics. The first thing we need to do is to show that it does have weight. Then we'll be ready to put some numbers to it. Let's go back to my house. I've got some stuff in the garage that'll do the trick."
"But while we're on our way," Fred continued, "take a look around and see if you can see the effects of air."
"You can't see air moving - you can't see wind," Art said. "All you can see is its effects, like those trees over there, bending in the wind. I suppose that if air were weightless, it couldn't push the trees around."
"Precisely," said Fred. "That alone should convince you that air has weight. But there's a nice experiment I have in mind that'll show, on a small scale, that air has weight. And that's the reason we see the kites and the gulls flying over there, the boats sailing out there, and even the clouds."
Art called to Shane, who came bounding up over a sand dune, with a big piece of driftwood in his mouth.
"Home, boy!" Art said. Shane dropped the driftwood and raced back to the van parked at the edge of the beach.
When they got to Fred's place, they went into the garage, which, like most American garages, had never seen a car, and went over to his workbench, which was just a little bigger than the average workbench.
"Pull a couple of beers out of the fridge, and let's see what we can come up with," Fred said.
"OK," he continued, "first let's convince ourselves that air does really weigh something."
Sitting near one end of the workbench was a balance beam: a horizontal bar with a pivot in the middle, balanced on a vertical post. Hanging from each end of the bar was a round dish, or platform, connected to the bar above by three wires. At the pivot point, in the middle of the bar was a pointer, attached to the bar, sitting against a scale, attached to the post.
"First, we're going to take two identical balloons, and blow them up to as close to the same size as we can." Fred picked one out a drawer in the workbench, and threw a second one to Art. "OK, hit it," he said.
When both balloons were blown up to about the same size, Fred took them and put them on the platforms of the scale. He fastened each balloon to its platform with a paper clip.
"What's that for?" Art asked.
"You'll see." He made a slight adjustment to the balance, and the pointer rested right at the center, on the zero mark. "OK, let's go over to the other side."
At the other side of the workshop, Fred reached into a drawer, pulled out an air psitol, and fired at the balloon on the left. It burst with a loud pop.
Art looked startled. "Do you do that often?" he asked.
"Used to. I liked to do that in class, 'cause it really got their attention, but the principal came in one day and was ready to throw me to the wolves. I switched to a dart, but he didn't like that either. Principals can be real pills, sometimes."
"So what do you use now?"
"You'd be amazed at what I tried. Hatpin? Too sharp. Dangerous. Little pin? Too dangerous. Cigarette lighter? No smoking on campus. Finally I said, 'Well, suppose I just strangle it?'. He wasn't amused. They usually aren't. Finally I rigged up a special balloon with a valve in it.
"But anyway," Fred went on, "what do you see?"
"The side with the full balloon went down, the popped side went up."
"The full balloon is heavier than the popped one."
"What about the balloon parts of the balloons?"
"Well, it doesn't look like any of the rubber disappeared, so the difference in weight has to be because of the air that was in the popped baloon."
"Right," Fred said. "There might be a small piece of rubber carried away by the pellet, but if we did it again, using a pin, the result would be the same."
"So now we know how much air weighs?" Art asked.
"No. At this point, all we know is that it weighs something. But what do we really want, when we ask for the weight?"
"OK, I get it," Art said. "We really want the density."
"Good. And what do we need to get the density?"
"How much there is, and how much it weighs."
"You got it," Fred said. "Can we do that?"
Art thought a minute. "I think we can get the weight by seeing how much we have to add back to the popped side to re-balance the full balloon. And we can get the volume by measuring the full one."
"But wait a minute," Art said. "I'm missing something here. We agreed before that you can't weigh air in air, 'cause that's like weighing water under water. So why does the balloon fulll of air weigh more than the empty balloon?"
"Good question," Fred said. "When you put air in a balloon, it gets compressed a little. So there's more air in the same space. Incidentally, you could do this experiment with basketballs, and you'd get a more conclusive result, because basketballs hold quite a lot of it."
"Yeah," Art said. "Everybody knows why there's air: to blow up volleyballs."
"Right. And you don't even need two of them. I don't know if my little air pistol would pop a basketball. They're pretty strong. What you do there is, empty one out as much as you can, weigh it, then pump it up as much as you can, and weigh it again. It'll weigh more. Trust me - I'm a teacher."
"OK, Mr. Wise Guy, I trust you. But how 'bout them numbers?"
"Let's do the easy one first," Fred said. "Let's see how much weight it takes to re-balance the balloons."
He got some small paper clips out of a box, and put one of them on the platform with the popped balloon. Nothing happened. He added another, and another, until the scale moved a little and brought he pointer just to the left of center (still not quite zero). One more finally took the pointer past zero to the other side.
"OK, then," Fred said, "it looks like the air lost from the balloon is between 6 and 7 paperclips."
"I don't seem to remember the International Standard Paperclip as a unit of weight," Art said. "What have we learned?"
"Don't get excited - I know exactly how much each clip weighs. I weighed a block of 100 of them; it came to 48.7 grams, so each clip weighs .487 grams. They're as identical as peas in a pod, or maybe clips in a box.
"We figured between 6 and 7 clips, so that's 2.92 to 3.41 grams."
Art, who has been scribbling in a notebook, said, "I get 3.4055 grams."
"Just like a programmer," Fred said. "Lots of digits of precision, but no accuracy. All I know for sure is 3 digits in the weight of the clips, so all we can use is three digits in the result."
"But we're getting off track," Fred continued. "Suppose it's right in the middle - 3.16 grams," he said, punching keys on a little calculator. "What else do we need to know?"
Art thought a minute, then said, "Well, we know how big the balloon was, but we don't know how much air was in it, because it got compressed."
"Right," Fred said. "Now we're at what's technically known as 'the tricky part'. Any ideas?"
"Nope. Insufficient data."
"So we're stuck, then?"
"Looks like it," Art said, starting to get impatient.
"Don't give up just yet," Fred said. "Have another beer and focus the awesome power of your mind on this little problem."
"Well, I don't know about the awesome power, but I will have just one more, to clear away the cobwebs." After a couple of minutes he said, "I give up. Unless we know how much air is really in that balloon, we don't have anything to work with."
"You mustn't give up. It's just what they want you to do. What if we could get the air out of the balloon, into a place where it could be measured?"
"OK, that sounds reasonable. But how do we get it out, and how do we know it's not just as compressed in whatever we put it in?"
"First things first. Here's how you get it out." Fred got another balloon from the drawer, and a short glass tube from another drawer. He fastened the end of the balloon around one end of the glass tube with rubber bands. "Now we're going to blow up this one to the same size as the others. When you're done, put your finger over the end of the tube. While you're doing that, I'll get the next part ready."
Fred got a clear plastic tub, set it on the workbench, and filled it with water. Then he got some laboratory clamps, and a large graduated measuring glass (called, oddly enough, a "graduate"). He held the graduate under the water in the tub, so that it filled, then pulled up the closed end, leaving the open end just about an inch under the surface. Water stayed in the graduate, which was about a foot long. He held the graduate in place with the clamps.
"OK, Art, take a look at this. The graduate is full of water, even though it stands about a foot out of the water. Know why?"
"I think the laser-like focus of your mind is starting to fade," Fred said. "Inside the cylinder, there's no air, just water. Ouside, on the surface of the water in the tub, is the weight of a column of air that's about 30 miles high. It works out to almost 15 pounds per square inch. And that's what's pushing the water up into the graduate.
"Now what we're going to do is take the balloon you've got, hold the end of the glass tube under the graduate, and let the air out. It'll bubble up into the graduate, push some of the water out, and when the balloon's empty, we'll just read the volume of air from the scale on the side."
Art put the tube in the water and let go with his finger. The air bubbled up into the graduate, displacing some of the water.
"OK," he said, "it looks like there's just about 1600 cubic centimeters of air in there. Where's the cubic inch scale?"
"Come on, Art," Fred scoffed, "We're all metric here. What's the size of the engine in your car?"
"It's a 2-liter .... OK, OK, we're metric. Now, how do we know it's not compressed, like it was in the balloon?"
"Thought you'd never ask." Fred adjusted the graduate until the level of the water inside was the same as it was outside.
"You see the water level inside matches the level outside. That tells us that the pressure inside the graduate is the same as it is outside. If it were compressed inside, it'd push the level down, and if it weren't, the level would go up. So we're pretty sure that the pressure is just about exactly the same inside as out. Now let's see if we can read the volume a little more carefully."
Art got his close-up glasses on, squinted at the markings, and announced: "1853 cc".
"OK," Fred said, "now I think we've got everything we need to know. Volume: 1853 cc. Weight: 6.5 paperclips, which is 3.16 grams. Divide weight by volume to get density, which is ..." He punched the numbers into his calculator, and showed Art the result: 0.0017053.
"What can we say about the result?" Fred asked.
"I think we can take 3 digits, which gets us 0.0017 grams per cc."
"OK, that's good. It turns out that we measure air in grams per liter, since it's so light, and that makes the numbers a little more convenient. To get liters from cc, we divide by 1000, so we end up with the density of air at 1.7 grams per liter." [see Note]
"Doesn't seem like much," Art said. "A liter is one of those big soft drink bottles. How much did you say those paper clips weigh?"
"0.487 grams - about half a gram. Here - catch." He threw one to Art.
"Hardly seems to weigh anything. There's not much to air, is there?"
"True, but what it lacks in density, it makes up for in volume. For example, you remember what the air pressure is here at sea level?"
"Yes, about 14.7 pounds per square inch."
"That's because we're sitting at the bottom of an ocean of air. The atmosphere goes up to about 200,000 feet, or or about 40 or 50 miles. There's even detectable amounts of the atmosphere up to around 75 miles up.
"Let's try something else," Fred continued. Let's work out the weight of air in this laboratory."
"Looks like a garage to me," Fred replied.
"Barbarian," Fred scoffed again. You may have a garage over at your place, but this is a laboratory." He pronounced it "lab-OR-a-tory". "It's about 30 feet long, 16 feet wide, 8 feet high. So the volume is 3840 cubic feet. Convert to liters, we get 108700 liters, times 1.7 grams per liter, gives 184790 grams, or 184.7 kilograms, times 2.2 pounds per kilo gives about 407 pounds of air in this room."
"That's what makes hurricanes so deadly. You've got tons and tons of air, moving at 50 or 60 miles per hour, and when it hits somethings, something's going to get blown over."
Note: That's not exactly correct. If you go to a handbook of chemistry and physics, you'll find that it says that air, at standard temperature and pressure, weighs 1.28 grams per liter. Standard temperature is 0° C (a pretty chilly 32° F; the freeing point of water), and standard pressure is one atmosphere - what we usually find at sea level.
It's important to use Standard Temperature and Pressure (STP) when talking about gasses, because the volume of a gas depends a lot on its temperature and pressure. It's one of the things that make hot-air balloons work.
So how did our diligent investigators come up with 1.7 g/l?
In any experiment, there are a number of factors that lead to the wrong result.
In our case, they could have made errors in:
Let's assume, for now, that Fred and Art were very careful about their procedures and their measurements. And that, like good experimenters, they did the experiment all over again a second time, and came up with just about the same result.
Can you think of a reason why their result is so different than the actual one? The difference between 1.7 and 1.28 is 32.8% (their result is 32.8% too large).
With the equipment they used, they should have been able to get within 5% of the correct value - somewhere between 1.22 and 1.34 g/l.
It turns out that there is a very good reason, and a very fundamental one, that explains their result. If this were a true scientific experiment, the reason would be considered a serious error.
For our purposes, though, it isn't really an error, because we were interested more in how to find the weight of air - and we did use a pretty good method.
Think about it for a while. Try to come up with a reason for the discrepancy. Play the role of another teacher checking up on their result. Look at every step. Remember that we said that their measurements were accurate, and that they repeated the experiment, and the measurements, and came up with the same result. So the error isn't in the measurements, it's somewhere else.