Balloon Notions

This is for all my "fans" who are impatiently waiting for something long and involved and complicated. :)

Let's start with something known to chemists as the "gram molecular weight" of a substance. It has another and more common name (to chemists): "the mole". It is a quantity of atoms or molecules, and its purpose is quite simple. Consider water: Its molecular weight is about 18 Atomic Mass Units, this being the sum of 16 (the Atomic Weight of one Oxygen atom) and 2 (the Atomic Weight of two hydrogen atoms). Well, "a mole" of water molecules is defined as "that quantity which weighs 18 grams". A mole of oxygen atoms is defined as that quantity which weighs 16 grams. A mole of oxygen molecules (having 2 atoms each) is defined as that quantity which weighs 32 grams. A mole of hydrogen molecules (also having two atoms each) is defined as that quantity which weighs 2 grams. And so on. It happens that that quantity is basically the same for every different type of atom and/or molecule: 602,214,150,000,000,000,000,000 or about 6.02x10E23, and since a fellow named Amadeo Avogadro was first to hypothesize that "equal volumes of gases at the same temperature and pressure contain equal numbers of particles", a critical part of the foundation of Modern Chemistry, it is known as "Avogadro's Number". See link.

From the preceding it can be deduced that if you have a mole of oxygen molecules and a mole of hydrogen molecules, and both are in containers at the same temperature and pressure, then the containers have the same volume. Furthermore, since "the mole" is just a number, we could imagine the phrase, "a mole of air molecules", and not pay a lot of attention to the fact that several different substances are involved -- and still conclude that if it was at the same temperature and pressure as those moles of pure hydrogen and oxygen, then its container would also have the same volume.

Let us investigate that mole of air a little more closely. A quick Google search for "composition of air" reveals that 99.998% of air is just four things: 78.084% nitrogen, 20.947% oxygen, 0.934% argon, and 0.033% carbon dioxide. Next, using the fact that "the mole" is a "conversion factor" that lets us convert Atomic and Molecular Weights into grams, we can multiply those percentages by the weights of the specified substances, and compute the weight of a mole of air:
Molecular weight of nitrogen molecule (two atoms): 28.0134 x 78.084% = 21.874 grams
Molecular weight of oxygen molecule (two atoms): 31.9988 x 20.947% = 6.703 grams
Atomic weight of argon atom (always single atom): 39.948 x 0.934% = 0.373 grams
Molecular weight of carbon dioxide( three atoms): 44.0095 x 0.033% = 0.015 grams
The total number of grams is 28.965, less than a gram more than if air was pure nitrogen (see that 28.0134 above).

The last piece of background information that we need here is the answer to a question, "Just how big is that volume of gas that contains one mole of particles?" Well, that answer depends on the temperature and pressure, and there is more than one "Standard Temperature and Pressure (STP)" out there. See link. Nevertheless, chemists generally work with values that, in the end, let me just say: "22.4 liters is the Gram Molecular Volume, the volume of one mole of gas at STP." And 22.4 liters is a cube that is just less than 11.1 inches (or 28.2 cm) on a side.

Now we are ready to talk about balloons. Let's start with an "ideal balloon", which is a container that has no weight at all, and which has a volume of 22.4 liters at STP. This very conveniently lets us compare Atomic and Molecular weights to each other, to see (1) what makes a good lifting gas in a balloon and (2) how much it can lift. For example, if we had an ideal balloon of pure nitrogen, it would float in air (and even rise slowly), because it has the same volume as a mole of air, but weighs less (as mentioned above). It has the ability to lift 28.965 - 28.013 = 0.952 gram. A real physical balloon skin that weighed more than that less-than-a-gram would sink to the ground, of course. On the other hand, the weight of the skin goes up by the square of the size of the balloon, while the lifting power goes up by the cube of the size of the balloon. So a pure-nitrogen balloon that was twice as big (a cube nearly 22.2 inches on a side) could lift 2x2x2x0.952=7.616 grams, while the amount of material needed to make the skin of this balloon would only be 2x2=4 times as much as for the 22.4-liter balloon. It probably would not be very practical to do, making an actual pure-nitrogen balloon, but it could be done. It just has to be big enough.

It is now obvious why a great lifting gas like hydrogen is desired: It lets the balloon be smaller. How much lift are we talking about? Our small ideal balloon can lift 28.965 - 2.016 (the molecular weight of hydrogen) = 26.949 grams! Well, here is a list:

substance; weight; lifting power; comment
helium; 4.003; 24.962gm; very small atom tends to escape confinement; somewhat expensive
monatomic hydrogen; 1.008; 27.957gm; as small as helium, flammable, and explosive all by itself (2H-->H2)
ethylene (C2H4); 28.053; 0.912gm; flammable
carbon monoxide; 28.010; 0.955gm; rather toxic
diborane (B2H6); 27.67; 1.295gm; toxic, flammable, and reacts with water (vapor)
hydrogen cyanide (HCN): 27.025; 1.94gm; very toxic
acetylene (C2H2); 26.037; 2.928gm; flammable
neon; 20.18; 8.785gm; probably too expensive to be used in balloon-scale quantities
hydrogen fluoride; 20.006; 8.959gm; very toxic and chemically reactive (will eat holes in gas bag)
hydrogen oxide (water); 18.015; 10.95gm; is a liquid at STP, not a gas
hydrogen nitride (ammonia); 17.031; 11.934gm; rather toxic
hydrogen carbide (methane); 16.042; 12.923gm; flammable
carbon dihydride (CH2); 14.027; 14.938gm; ???, certainly flammable, may self-react to make ethylene
boron hydride (BH3); 13.835; 15.13gm; does not appear to exist; individual molecules of boron hydride seem to be as elusive as individual molecules of silicon dioxide; the conditions for making either tend to cause them to polymerize into more-complicated molecules, such as is diborane. Well, diborane is actually a hydrogen-bonded pair of boron hydride (borane) molecules, and we all know (or should know) that water is a liquid at STP because of hydrogen bonds. Nevertheless, even if we could make this stuff without the hydrogen bonds, it still probably wouldn't be satisfactory. It would still be flammable.

Well, that's about it; I've pretty much racked my brain and chemical knowledge for gaseous substances of low weight. The first things to be too heavy to be a balloon gas are nitric oxide (NO, 30.006) and formaldehyde (H2CO, 30.026) and ethane (C2H6, 30.069). Carbon dihydride I never heard about before, with respect to it being an ordinary molecule; all the reactions I find on the Web for it seem to show it as being a temporary intermediary in the middle of other things going on, that consume it as fast as it is made. Theoretically it could be stable like carbon monoxide (carbon is sometimes happy with just two chemical bonds), but if it was reasonably well known, I should have heard about it before. I'm guessing it's another super-monomer, like silicon dioxide and boron hydride.

If you ever wondered why airship travel died with the Hindenburg, just looking at that list of flammable and toxic gases, and their LACK of lifting efficiency, should make the explanation perfectly obvious. (But did you know that 2/3 of the people on-board the Hindenburg survived? Look it up!)

You might notice that I've not mentioned hot-air balloons so far. That's because, by definition, they do not function at STP, Standard Temperature and Pressure. But since the STP gases have basically been exhausted, let's do that now. We start with the Gas Law: (Pressure)(Volume)/(Temperature) is a constant. We don't need to worry about the value of that constant here, and can arbitrarily set it equal to 1. Then we can change the other three variables any way we like, as long as the result of the expression is still equal to 1. For example, we can double the pressure and double the volume and quadruple the temperature, and that is a valid manipulation: If (P)(V)/(T) = 1, then (2P)(2V)/(4T) = 1. Next, we also need to pay attention to Standard Temperature, which is 0 degrees Celsius or 273 degrees Kelvin; all changes in temperature as far as the Gas Law is concerned, must be associated with the Absolute Temperature, the number of degrees above Absolute Zero. So I wll use the Kelvin scale here (but just subtract 273 if you want to know the Celsius temperature. Converting that to Farenheit is left as an exercise for the reader).

All right, the first thing to notice about a hot-air balloon is that the bottom of the balloon is completly open. This means that the pressure of the gas inside the balloon never really gets to be much greater than ordinary atmospheric pressure (just like a sealed-lifting-gas balloon), because all the overpressure can leak out very very easily. As a working starting assumption therefore, we will concern ourselves here only with temperature and volume changes. However!! A hot-air balloon also has a rather fixed total volume; once it gets inflated, it doesn't expand any further! So, if the pressure and the volume both stay the same, then exactly how does heating the gas lift the balloon???

The answer is simple, if a trifle sneaky. For the moment I will ignore the fact that the hot-air-burner puts large amounts of hot water vapor and hot carbon dioxide into the balloon; I will even pretend that we are heating the balloon with electricity, so I can work with the known composition of ordinary air. Now let's start with a doubling of the temperature of the air in the fully inflated balloon, to 546K. Since the volume is fixed, the pressure tries to double --but instead it leaks out of the open bottom end of the balloon. Note that it isn't JUST pressure that leaks out; it is also air that was originally inside the balloon that leaks out! We can be quite sure that fully HALF the original air will escape the balloon if the gas temperature is doubled. So if we started with an ideal balloon that had 22.4 liters of volume and a weight of 28.965 grams, and now half of that air --and its wieght-- has been pushed out by doubling the temperature, then the remaining weight of the gas inside the balloon is 14.48 grams, and our ideal balloon will also have a lifting power of 14.48 grams.

Well, obviously we can do better than that, if we heat the gas even more! Just for fun, suppose we turn the problem around, and figure out how hot the air needs to be, for it to have the same lifting power as hydrogen? Simple: We want 2.016 grams of gas in our ideal balloon, which is a reduction of 28.965 / 2.016 or 14.37 times. That means the initial 273K temperature must be increased by 14.37 times, to 3923K, pretty hot!!! (For reference: temperature at surface of Sun: 5880K) The gas bag would have to be made of carbon-fiber fabric, or something even more heat-resistant, to stand up to that!

On the other hand! Time for mix-n-match! Suppose we started with a lighter-than-air gas, like water vapor, and heated that inside a sealed balloon?? If the initial weight of our ideal balloon is 18.015 because of water vapor, and we wanted to heat it to hydrogen-equivalence, how hot must it be? 18.015 / 2.016 is a reduction of 8.936 times, so 273K x 8.936 is 2440K, still plenty hot, and much more reasonable than 3923K. Actually, that's still too hot; water starts to dissociate into hydrogen and oxygen at temperatures higher than 1500K or so. Well, suppose we were satisfied with helium-equivalence? 18.015 / 4.003 is a reduction of 4.5 so 4.5 x 273K is 1229K, a possibly quite practical value. And of course I chose water because it is cheap and nontoxic and nonflammable! (And you also now know why I included it on the earlier list. :)

The net effect of the preceding is that for EVERY 22.4 liters of gas-volume of your hot-steam airship, you put in 4.003 grams of water, and then you heat all of it to 1229K, to have the same lifiting effect as that much helium. Or put in slightly less and heat it to almost 1500K, for an effect intermediary between helium and hydrogen. Of course you will also need a gas bag able to survive that hot steam (its lots more chemically reactive than mere water; and carbon fiber won't survive). My suggestion here is to seek a variation of glass fibers, and weave your balloon bag from it. Since glass is already an oxidized substance (silicon dioxide), that's why it won't chemically react with the oxygen in the hot steam -- but glass also cannot withstand that much heat. Other oxides (aluminum oxide, for example) can, though. Modern industries have a lot of experience handling aluminum oxide, but making fibers is only a recent development, so I don't know if they have started weaving fabrics from it yet.

The preceding counts as an Idea, a suitable reason for posting this on the HalfBakery, but I'm not done yet! I'd like to mention a science fiction story I read a while back, it was written in the 1930s by John W. Campbell, Jr, and is considered to be a classic: "The Black Star Passes". The book actually contains three related tales (the third lends its title), and the first story , "Piracy Preferred", is about an air pirate who uses a highly penetrating gas to put the occupants of a giant 3000-passenger plane to sleep, so he can board in mid-air and do robbery. The author poses the problem of how do you capture a sample of a gas that penetrates every known solid substance? He decides that a bottle containing pure vacuum is the answer. The gas will penetrate it, but to get OUT again it has to fight air pressure, and will probably stay in the bottle.

Well, now!!! That was science fiction, but hydrogen and helium are genuine and fairly penetrating gases. Is there a way we can use that author's trick? Let us imagine a double-walled balloon. The two walls are connected together at many places, with short lengths of some sort of attached fibers. The separation between the walls is no more than a couple of millimeters. We may now somewhat-pressurize the region in-between the two walls, and the overall bag-within-a-slightly-larger-bag will be maintained, thanks to all the connections. Also, of course, we fill the inner bag with hydrogen or helium at ordinary air pressure. I will suggest using neon gas as the pressurizing agent between the two walls, because it is a fairly small single atom, and so is naturally suited for bouncing off the small places in the inner balloon wall, through which hydrogen or helium might try to escape. The net effect here is that the hydrogen or helium at one pressure must overcome a higher pressure to escape confinement, similar to the vacuum-bottle idea of that old story. I'll focus on using hydrogen now. One reason we don't like hydrogen as an airship gas is because it leaks out and constitutes an explosion hazard. However, with a gas-pressure wall holding in the hydrogen, the rate of leakage should be very much slowed down and so the airship should be much safer. Also, the THREE walls (two of fabric and one of neon) more thoroughly isolate the contained hydrogen from the oxygen in the outer air, and so that also is a reason why such an airship should be safer.

Any takers?