Fun Fact - A Very Long Straw

Numbers in this article are approximate, for readability; I'm not going to quibble about a few percent. They vary anyways, depending on where you live.

It's fun to drink through a straw, but have you tried it in a swimming pool? You don't actually want to drink the chlorinated water, but there is an opportunity for science here. Find a long straw, perhaps one of those floatation tubes, or any other tube that you don't mind putting your mouth on, and suck the water up the tube as though you were drinking through a very long straw. It doesn't have to be very long however. No need to stand on the diving board. You can only suck the water 3 feet (one meter) up the tube, 3 feet above the level of the pool. At that point your lungs are straining, because the weight of the water is pulling down as hard as your diaphragm can pull up. By analogy, your right arm might be able to lift 60 pounds against the pull of the earth, and no more.

It doesn't matter if the straw is as thin as a pencil, or as wide as your mouth, or (theoretically) as wide as a stadium; you can only pull water 3 feet up the tube. If the straw is wide, it might take several breaths to coax the water up the tube. The first breath is easy, as the water rises a few inches. The next breath is not too difficult, but after a while it's hard going, until the weight of the water, per square inch, is the same as the pull of your diaphragm on the air in the tube, per square inch. If the straw is narrow, like a drinking straw, the water jumps up 3 feet in half a breath, but the limit is the same. Try this experiment with different size tubes the next time you're in a pool.

So how strong are your lungs anyways? Can we put a number on it?

Use a straw that is 4 feet tall (you can only suck water up 3 feet anyways), and a little more than an inch wide. The cross sectional area of your straw is one square inch. When the water has climbed 3 feet up the straw, you are holding up, with your breath, the weight of a column of water 3 feet high and 1 square inch in cross section. This is 36 cubic inches of water, weighing about 1.3 pounds. You can suck air in with a force of 1.3 pounds per square inch, or 1.3 PSI.

You have a friend name Bill who always tries to best you at everything. You know the type. He comes over with a 40 foot straw and says, "Watch this!" Course he has to stand on the roof of your two story house. He attaches a vacuum pump to the top of the tube and turns it on. All the air is sucked out of the tube, which is, fortunately, made of glass, so it does not collapse under pressure. (You've probably tried to drink a thick milk shake through a paper or plastic straw and had it flatten on you, right?) His straw is just as wide as yours, 1 square inch in cross section. The water climbs up the tube, but stops at 33 feed. It can climb no higher, even though all the air has been sucked out of the tube. This column of water, 33 feet high and 1 square inch across, weighs 14 pounds. Since it is holding steady, everything must be in balance. The column of air over every other square inch of your swimming pool must also weigh 14 pounds. If it weighed more, it would push the water up higher in the tube. Imagine a column of air, 1 square inch across, like the circle made by your finger and thumb, but the column goes all the way up to the sky. Miles and miles of air, past the clouds, all the way up to space. Air is lighter than water of course, but this column is miles high. It weighs 14 pounds, just like the column of water in Bill's tube. Bill has discovered a handy way to measure ambient air pressure.

Well not so handy really - he has to use a modern vacuum pump, and stand precariously on your roof. Soon Cathy comes along with a better idea. She has a long glass tube, like Bill's, but it is sealed at one end. She lays this tube down in your pool, until it is completely under water. (You've got a really long pool.) She tips the open end up just a bit, but still holds it below the surface, so all the air runs out. After the bubbles stop, there is nothing but water in the tube. Now she points the open end down, and lifts the closed end up out of the water. Soon the tube is standing straight up with its open end still in the water and its closed end at the top. The tube rises 40 feet above the surface of the pool, but the water is only 33 feet high in the tube. Just like bill's tube, the water can only rise 33 feet above the surface of the pool. This balances the air pushing down on the rest of the pool. The last 7 feet of tube have no air and no water - a pure vacuum. "See," she smiles, "I didn't need a vacuum pump to suck all the air out, and I don't have to stand on your roof. I can do it all from your swimming pool." (Cathy is pretty strong, and can swing a 40 foot glass tube about without breaking it.) She has made the same empirical measurement as Bill. Atmospheric pressure at your elevation is 33 feet of water, or 14 PSI.

Gasparo Berti performed this very experiment in 1640, though he didn't understand why the water stopped at 33 feet. He thought air was weightless, as did everyone in his day, even Galileo.

If you happen to live on Venus, which has a much thicker and heavier atmosphere, you would need a tube over half a mile long. Pressure on Venus is 90 atmospheres. But the water in your swimming pool would boil away anyways, so no matter.

At this point, Evan comes along with a tabletop apparatus. His tube is only 3 feet long. Again it is made of glass and sealed at one end. He lays it in a red liquid and tips the open end up until all the air runs out. He then tips the open end down and lifts the closed end up out of the liquid. The tube stands 3 feet high, but the liquid, which is much easier to see than water, rises just 30 inches up the tube. The last 6 inches are empty. "This is called a barometer," he explains, "and it uses mercury, which is much heavier than water. It's the heaviest liquid known. So this column of liquid, 30 inches high, weighs the same as 33 feet of water, or miles and miles of air."

Evangelista Torricelli performed this experiment in 1644, and surmised, correctly, that air had weight, and the atmosphere exerts pressure on us all, pressure that exactly balances the 30 inches of mercury in his glass tube. (Mercury is not actually red, but a dye is added to make it easier to see.)

Other 17th century scientists soon learned that the pressure is less at the top of a mountain, since there is less air above you pushing down. Even at a fixed location, the air pressure varies just a bit from day to day. Low pressure often indicates bad weather on the horizon, while high pressure portends a nice sunny day. Thus the barometer became an important instrument for short term weather forecasts. It's better than anything they had before, which was nothing.

Ok, let's return to your swimming pool. You can suck water 3 feet up the tube, but how far down can you blow? Maybe your lungs are stronger blowing out than sucking in. Sorry, they aren't. You can push air down the straw about 3 feet below the level of the water, and that's it. There's no blowing bubbles through a 6 foot straw. Ain't gonna happen.

Perhaps, in one of your heroic day dreams, you find someone who is trapped below water. His leg is caught in something and he can't come up for air. Quick thinking, you come to the rescue with a hose. That works, as long as his head is only a foot under water. If he is pinned several feet below the surface, the hose is useless. His diaphragm isn't strong enough to pull air down to that depth. You better think of something else.

There isn't much variation from person to person. You and I can blow at a pressure of 1.3 PSI, while a professional trumpet player blows at 1.9 PSI. There is one gentleman however, Brian jackson, who seems to be endowed with a special gift - he is able to blow at around 170 PSI, besting the rest of us by a factor of 100! He holds several Guinness world records (yes, there is a record for just about everything). He could easily inflate a car or bicycle tire with his mouth, and he has literally blown hot water bottles apart on television, e.g. America's Got Talent season 6. Medically, it is unexplainable. If a man was 100 times as strong as us, he could lift ten thousand pounds. But of course he can't, because ligaments would tear and bones would break. Somehow Brian achieves pressures that should, seemingly, blow his chest apart. Yet there he stands unharmed, with shattered water bottles all around him.

Atmospheric Half Life

As you gain altitude, atmospheric pressure drops exponentially. Thus our atmosphere, or any atmosphere, has a half life of sorts. On earth, pressure drops by half every 18,000 feet, or 5500 meters. So at 36,000 feet, where airplanes fly, the air is a fourth as thick, a fourth as dense, as it is at sea level. At 54,000 feet the pressure is one eighth ATM, one eighth the pressure at sea level. Even this is too thick for supersonic flight, as the plane would overheat, so the Concorde flew at 60,000 feet. 60/18 = 3.33, 23.33 = 10, thus a tenth the pressure at sea level.

Demonstrating exponential air pressure is easy and fun; let's derive it. I will abuse the word density to mean molecules per unit volume, having nothing to do with mass. Assume an ideal gas, compressible, where the density at altitude a is the weight of the air above a, is proportional to the integral above a, of density times molecular weight times gravity. p(a) = c × ∫(t=a,∞) p(t). Written as an ordinary differential equation, p′ = -cp. thus p = b×exp(-ca). On earth, and if you measure pressure in atmospheres, and altitude at feet above sea level, then you get p = ½a/18000.

What about other planets, with other gases in the atmosphere? Halflife is the most interesting parameter, since some planets, like Jupiter, don't even have a surface to stand on. Follow the path through the constant c in the exponential. If gravity is multiplied by k, (and I'm assuming constant gravity which is not realistic if you go too far up or down), then halflife is divided by k. If molecular weight is multiplied by k then halflife is divided by k. In the upper atmosphere of Jupiter, gravity is 2.5 times earth's, so divide 18,000 by 2.5. But the air is all hydrogen, weight 2, whereas our air has weight 28, so multiply halflife by 14. The result is 100,000 feet, or 19 miles. When Galileo's 750 pound atmospheric probe, half of it heat shield, plunged into Jupiter's atmosphere, it experienced 228 g's of deceleration, fine for a probe, not so great for an astronaut. It dropped its heat shield, almost all of it ablated, then deployed its parachute, and started taking readings. It passed through 97 miles of Jovian atmosphere in 58 minutes. The pressure around it doubled every 19 miles, until it was crushed like an egg. It stopped functioning at 23 atm pressure and 300 degrees f, probably due to heat rather than pressure. Farther down, the hydrogen gets squashed down to a liquid, largely incompressible, and thereafter pressure is linear with depth, rather than exponential. And then less than linear as gravity decreases, as you travel thousands of miles inward toward the center of the planet, but that's another story.

Mars has a third of our gravity, and CO2 11/7 as heavy as N2, thus a half life of 34,000 feet. But it starts at 1% atm pressure on the surface, so not a lot of air there.

Venus starts at 90 atm, a thick corrosive atmosphere, with a half life of 11,500 feet. If you lived on Venus, and there was an Empire State building, and you rode the elevator up to the top, say 1,150 feet, pressure would change by 7.5%, which doesn't sound like much, but it's 6.7 atm, which would blow your ears out long before you got to the top floor. Even a 5 story building is impractical. I suppose the entire building could be held at earth pressure inside, that works, but it has to have mighty strong walls.

Further Reading

Barometric pressure and the barometer
Gasparo Berti
Various pressure levels
Brian jackson