A Civilization built on Hot Air

December 2011

Abstract:
All about the remarkable ubiquity of Mr.Boyle's (and M.Charles') gas law.

In the late 1650s and early '60s, Robert Boyle, a well-to-do Englishman and a founder of the Royal Society, then just over thirty years of age, experimentally found a direct relationship between the volume and pressure of gases. According to "Boyle's law," if we double the volume of a given amount of gas, the pressure of the gas is halved; conversely, if we reduce the volume of the gas by a given factor, then the pressure of the gas is increased by the same factor. In short, if you allow the gas more room, the pressure goes down and if you squeeze it into less space the pressure goes up, so that the product of the pressure and volume remains the same before and after the change, or: P₁V₁ = P₂V₂, for the symbolically inclined.

Nearly a century and a half later, in the early 1800s (science was in no hurry in those days), the French chemist Jacques Charles found the very same relationship between the temperature and volume of a gas, as well as between temperature and pressure. When we add Charles' law(s) to Boyle's law, we get what now is called the "Ideal Gas Law". (It's not the law that's ideal, it's the gas. An "ideal gas" is simply an unspecified dry, inert gas.) The Ideal Gas Law gives us reciprocal relationships between temperature, volume, and pressure of a gas: It tells us that if we increase the temperature of a gas (while keeping the volume steady) the pressure will increase. Conversely, if we increase the pressure (again holding the volume steady) the temperature will increase. In fact, whenever we hold any one of these three parameters steady, while varying a second parameter, we'll find the third parameter varying in direct proportion to the second. The upshot of this is that a gas – such as air – can do work, can exert force and move things by expanding in volume, if we add pressure or temperature to it. If we just add pressure we get increased temperature, and – more usefully – if we just add temperature we get increased pressure. You're saying, "big deal, this is high school physics." Well, it is a big deal. And here's the story you probably never heard in high school physics class.

Ancient understanding
Let's go back to our ancestors of the stone age for a moment. Perhaps the most important tool we had was fire. And what was this fire? A self-sustained combustion, or breaking of chemical bonds within the cells of the wood, evolving hot gases which expand according to the gas law into the surroundings, giving us comfort as we gathered around the fire in our hut. (Sure, there was mostly radiant heat, but the hot gases helped a bit.) Granted, we didn't yet make maximum use of the wonder of expanding gas, but as we'll see, we learned.

We eventually learned to cook food in pots. We discovered that if we put a lid on the pot, and even added a rock on top of the lid, our food cooked faster. What were we doing? We were making use of the gas law: The cover meant that pressure could be maintained in the pot (and a heavier cover allowed greater pressure), and that – as we know from M. Charles – results in greater temperature in the pot and faster cooking.

We conquered the sea by inventing the sail. The sail is an interesting application of several physical principles, including Robert Boyle's gas law relating pressure and volume. You may think of it as the wind "pushing" the boat along. And there is some push, but even more pull. The sail, when it's correctly trimmed, builds up a pressure difference between a volume of high pressure air (which tries to expand) on the windward side of the sail and low pressure air on the lee side. The lee side low pressure volume draws the sail toward it in an effort to reduce its volume in accordance with Boyle, trying to move the boat in the direction of the low pressure. The keel resists this, and the resulting movement of the boat is a compromise between the draw of the low air pressure volume (and some help from the windward high pressure) and the resistance of the keel, a question of vector geometry.

Learning to destroy
Some may agree with me that one of the most unfortunate inventions in history has been the gun. Whether large cannon or palm-sized Derringers, their purpose has been to bring death. Which they have achieved, in spades. The gun is built on a single principle: The gas law. The idea is to confine some instrument of destruction, such as a lead ball, in an enclosed space with a single exit, such as a barrel. By quickly heating the gas behind the lead ball we cause the gas to expand and drive the lead ball with rapidity toward its target. And, incidentally, we drive the gun or cannon in the opposite direction, which we call recoil. All in conformity with the gas law. Thinking back on my own high school physics classes, I don't think the full application of this law was made clear to us.

Speaking of guns, exactly the same principle – the gas law – applies in bombs. There are two kinds of mechanical bombs. In one, the expanding gas caused by the initial explosion (heating of the gas) drives ingredients in the bomb – such as sharp steel fragments – into the surroundings to cause death and destruction. This is preferred for killing whoever is in the vicinity. In the other type, used for demolishing structures, the bomb depends on an expanding pressure wave (employing the gas law directly) which in turn causes expansion of the gas within the concrete, clay, or other building material of the structure, facilitating its collapse. Even a nuclear bomb is just a more powerful way to bring about expansion of gas in accordance with the gas law.

What we use explosions for is of course up to us. They can be beneficial, as in dynamiting for a mine or a construction site, or imploding a structure at the end of its useful life, or it can kill hundreds of thousands of human beings at the push of a button. In either case, it's the gas law that does the damage.

Learning to move things
It's hard to think of an invention that has had a greater impact on civilization than James Watt's steam engine, patented in 1769. His engine, actually an improvement on earlier, less efficient steam engines, led directly to the Industrial Revolution. Within a decade, trains, factories, and steam ships were making use of Watt's steam engine. I hardly need to say that the steam engine is an application of a single principle: The gas law. The steam engine simply heats air (by boiling water), and makes use of the increased pressure of the gas to drive the engine's pistons. For fifty years, this principle drove the Industrial Revolution forward, bringing with it both advances and miseries.

And now we're getting to the modern age. Here we meet an invention every bit as important to industry as Watt's steam engine had been: the internal combustion engine. If the steam engine permitted the Industrial Revolution, internal combustion brought about the modern age. Yet no single inventor is credited with the internal combustion engine. This complex invention was made practical little by little, after dozens of patented improvements by mainly British, Dutch, Italian, French, and German inventors over the span of a century, from the late 1700s to the late 1800s. The end product of this development, a 2- or 4-stroke engine with internal cylinder compression and ignition, precisely what we still use in our cars, was given its final form by the German engineers Otto, Daimler, and Benz in the 1880s. The internal combustion engine runs by one principle: The gas law. In short, in each cylinder (akin to cannon barrels) fuel is introduced to cause a rapid burning, resulting in expanding gas. The force of the gas expansion drives the engine's pistons which in turn rotate a driveshaft and the wheels. The principle is the same as in the steam engine and in the gun, and in prehistoric pots. Incidentally, the carburetor in the engine that mixes fuel and air depends directly on a consequence of the gas law, the Bernoulli effect. So when we mash the gas pedal (which stands for "gas" as in air, not as in gasoline) in a carbureted car, we are directly using the gas law to regulate air pressure, and thus fuel, in the engine.

Taking to the air
Humanity first took to the air in the early 1800's. In Europe, hot air balloons were all the rage. Get the point? It was a direct test of the gas law. When you heat the gas in the balloon, the pressure increases and the balloon expands. The expanded gas is now lighter and therefore rises through the heavier air like a cork rises through water. (Technically, the balloon and the cork don't so much rise on their own as they get pushed up by denser, heavier air (or water in the case of the cork) flowing down under the balloon in response to gravity.) As we see, the gas law took us into the heavens.

The Wright brothers in 1904 showed a highly sophisticated grasp of the gas law, as they needed to merge three different understandings of the law in order to succeed in powered flight. First, the internal combustion engine, which - though already invented - they needed to lighten for the flight. Second, the concept of the wing: They were the first to understand that the principle of the wing is the same as that of the sail. (They understood to begin with the revolutionary concept of "relative wind," that is, that moving a wing through stationary air would have the same relative effect as a blowing wind moving past a stationary sail.) By designing a wing that was "fatter" on the top than on the bottom, thus giving the wind a longer path over the top than across the bottom of the wing, they calculated that, by the gas law, (and help from Bernoulli's law) the air mass would become less dense on top because it has to move farther and faster, filling more volume, and therefore the pressure on the top of the wing would be less than on the bottom, and the draw of this relative low pressure on top of the wing – which we know as "lift" – could perhaps overcome the weight of the aircraft. It "only" remained to calculate how this pressure differential related to the relative speed of the wind over the wing, in other words, how much speed would be required for the wing to generate enough lift to exceed the weight of the aircraft. That would be the takeoff speed, or stall speed. (I'll remind the reader that, just as in the case of the sail, it's not a matter of air pressure on the bottom of the wing pushing the aircraft up, it's a matter of reducing the air pressure on the top of the wing, which yields a pressure differential. This grasp of the gas law by two brilliant bicycle mechanics is rarely appreciated.)

So the gas law provided both the engine and the lift for the Wright Flyer. But that was not all. An engine does not move an airplane. A propeller does. And a propeller, whether on the front of an airplane or on the top of a helicopter (a "rotor"), is nothing but a rotating wing. By moving very fast it develops a great deal of "lift," and when attached to the front of an airplane it develops "lift" in the forward direction, thus pulling the aircraft. (By the way, a ship's propeller – or screw – works similarly. Water is compressible because of the gas dissolved in it, and the screw expands the gas and therefore reduces the gas pressure ahead of it, causing greater pressure behind. To this extent it makes use of the gas law. But we should add that the ship's screw also makes use of the leveraging principle of the "inclined plane," the same principle as when we drive a screw into a piece of wood, since water, like the wood, is only partly compressible.) And there we have the truth about the propeller aircraft. The gas law powers it, lifts it, and moves it. (And lands it safely on compressible pneumatic tires, we might add.)

Going faster
But now we have jet aircraft. It will be no surprise to learn that this is the most direct use we have made to date of the gas law for transportation purposes. Nothing could be simpler: The jet engine operates just like the cannon. But here it's the recoil we're interested in. We take in a gas (air) through intakes, heat this in a chamber closed on one end like a gun barrel, the gas expands in volume according to the gas law, and rushes out the open back end. In accordance with Newton's laws of motion, the aircraft is accelerated forward with a force that is a product of the mass and acceleration of the gas being expelled.

And rockets use the same principle. The main difference is that rockets carry their own "drive mass," or gas to be heated and expelled out the back. Rockets going into air-less space cannot simply gulp in ambient air and heat that as the drive mass, as jets do. (This requirement adds hugely to the bulk of rockets intended to escape the Earth's gravity, though once they're up to speed in space, largely outside the Earth's gravitational effect, the engine can be turned off and they'll maintain their speed.)

And everything else ...
So it seems that expanding gas is the principle that drives our cars, airplanes, ships, and rockets, cooks our food, and kills our enemies. But don't we have other power sources? What about electricity?

Good question. We have one and only way to make electricity: By rotating conducting wires in a magnetic field. But how do we bring about the rotation? Whether our electric power station is fired by coal, oil, or nuclear power, the answer is the same: We create heat, and use it to boil water that gives us steam, whose power of expansion is then used, in accordance with the gas law, to turn a turbine. Our power stations are all, in effect, nothing but steam engines operating by the gas law to produce electricity.

But don't we also get power from hydroelectric dam power plants? (Good question, even if I had to ask it myself.) True, there we don't make steam, and we don't seem to need the gas law. But hold on; the question takes us to the natural operation of the Earth. The fact is that the water behind the dam is there because solar heating of surface waters and vegetation evolves water vapor, a gas which escapes from the liquid phase and then rises in accordance with the gas law. And let's wrap up by mentioning a few other natural forces that occur directly as the result of the law of expanding gases: Storm systems, including hurricanes and tornadoes and in fact the entire weather system. And of course, fires. And all the movement of the Earth's crust, including volcanic eruptions, seafloor spreading and continental drift, mountain building, earthquakes and tsunamis – all caused directly by heating and pressure build-up of gases from the Earth's mantle. Quite a lineup.

(Here's a paragraph in parentheses because this stuff won't be on the test. Or because the following are not really our inventions, but ... the Creator's, let's say:  Did you know that Boyle's law makes our breathing possible, regulates our body's acidity balance and drives the critical gas exchange that happens in all the cells of our body? On a more gross (as in "large") plan, the trillions of bacteria in our gut use Boyle's law – possibly without even knowing it – to evolve gas whose pressure helps to move our waste through our intestine to its final destination. The accompanying f*rt should serve as a reminder of that, and I want you to appreciate that it also serves to make your day a whole lot more comfortable.)

So that's our survey of expanding gas and Robert Boyle's law. All we use it for is to make electricity, cook our food, heat our homes, run our cars, ships, airplanes and every other piece of machinery, and try to wipe each other out with guns, bombs, and rockets. God uses it to keep us alive, make weather, and move the continents.

That's a lot from a puff of air!