Steampunk airships typically use either helium or hydrogen as a source of lift (depending on the world). However, the source of the gas is often not fully explained, or even not mentioned at all. How could a steampunk civilization with limited (although rapidly expanding) understanding of chemistry acquire the large amounts of helium and hydrogen that are required to lift airships?
It turns out it's not that hard to make hydrogen gas for an airship, and you can do it yourself.
- Take some dilute sulphuric acid.
- Put small pieces of metal in the acid - iron, zinc, and aluminum should all work.
- Capture the resulting hydrogen gas, and then put it in your airship.
This method of hydrogen production was first specifically used for airships in the 18th century, in a time of technology even less advanced than your steampunk civilization - and at the time, hydrogen was not yet known to exist! It seems quite plausible that they could produce the gas in large quantities.
My guess is that you can't really have a steampunk civilization without having roughly 19th century understanding of chemistry and with that, you're probably home free.
As evidence I would like to quote Wikipedia:
The first gas balloon made its flight in August 1783. Designed by professor Jacques Charles and Les Frères Robert, it carried no passengers or cargo. On 1 December 1783 their second hydrogen-filled balloon made a manned flight piloted by Jacques Charles and Nicolas-Louis Robert, 10 days after the first manned flight in a Montgolfier hot air balloon.
Extrapolating from this I'd say that it means the lighter than air properties (and the manufacturing/extraction of) hydrogen was known as early as the 18th century. Which coincides nicely with James Watts continuous rotary motion steam engine in time, suggesting that the components of a steampunk style civilization align rather nicely in time.
For a history of Hydrogen extraction (And further evidence in the case of lighter-than-air-flight), we can turn to Ebbe Almqvist's "History of Industrial Gases" which states.
During the 18th century many worked on ideas surrounding the lighter-than-air principle, but suitable means were not available until in 1766 Henry Cavendish succeeded in producing hydrogen gas in pure form (the known as inflammable air), and discovered that it was 14 times lighter than air. The dream of the century, that of "air sailing," could now become a reality.
In order to produce the gas for the Charles/Robert balloon trip, Almqvist tells us, 500 kilograms of iron and 250 kilograms of sulfuric acid was used. Suggesting, once more, that the industrial production of Hydrogen was well within the limits of late 18th-century chemical science.
A more effective and ultimately cheaper way of producing hydrogen is the electrolysis method where you use electricity to split water into hydrogen and oxygen. This is also well within the capacity of most steampunk type civilizations (although it takes us into the 19th century.
In 1800, Alessandro Volta presented his so-called voltaic pile, the forerunner of the electric battery. A few weeks later, William Nicholson and Anthony Carlisle constructed a voltaic battery and manufactured considerable quantities of oxygen and hydrogen. The electrolysis method remained expensive until the Belgian Zénobe Gramme invented the first steam-driven dynamo in 1873. From 1890 on, when large hydroelectric power stations were built, the method was used on a large scale wherever cheap hydroelectric power was found.
Helium was first found in large quantity in gas from oil wells in 1902 and large-scale industrial production for airships started during WW1. This is late, but arguably still within in the steampunk timeframe. Oil and natural gas wells were operating from the mid 19th century and are suitably steampunk technology.
Acid and metal reaction was already mentioned, but usually, a cheaper way was used. Lavoisier Meusnier iron-steam process was invented in 1784, it generated hydrogen by passing water vapor over a bed of red-hot iron at 600 °C. So only some iron rods and fuel were consumed - no expensive acid required.
Union Army Balloon Corps mobile hydrogen generators used acid-metal reaction, so I guess iron-steam generators were too bulky for mobile use.
Since I don't see a hard science tag....
Surely we can do better than mundane hydrogen or helium?
Even the most ill-educated brute knows that the sky consists of two layers, Aer - the dim, lower part of the sky, and the Aether, the brighter upper part.
Using observation we thus know that the brighter, cleaner air is - the higher it rises.
So our goal is to cause an airship to rise. Logically this means we must trap cleaner air within the gas bag - so how do we clean the air?
Obviously, as educated folks, we know that the purest element is Quintessence - and the studies done by the royal alchemist's guild show that we can create Quintessence and infuse it into the air - transmuting mundane Aer into Aether! The most cost-effective method is, of course, mixing together the transcendent elements Sulphur and Mercury.
So we combine them, and using some sort of heating element - I recommend a glass lens focusing sunlight during the day or heated Luminiferous Aether at night to further enhance the heavenly properties of the admixture - burning the impurities out of the Aer and creating Aether - which will lift the airship! Allow for vents at the top of the bag to release the Aether and you can control your descent as well.
Mining helium gas
If you don't mind a little handwaving, you've got two potential sources of helium that I think would both be great. Most commercial helium is extracted from natural gas deposits that have a high (1-10%) fraction of helium. We build refineries to separate the helium and natural gas. I'm not sure if the typical steampunk world would allow for the mining of hydrocarbons, but oil mining started in the mid 1800's, so that sounds about the right time period. If you want to make it easier on your people then they could find a deposit of nearly pure helium, so it's just a matter of getting it out of the ground. Such deposits have never been found on earth, but that doesn't mean that it is impossible or obviously violates the laws of physics.
Another source of helium is from nuclear reactions. Helium-3 has a number of uses in modern industry and technology, and virtually all Helium-3 used for these purposes is produced by the decay of tritium. Of course tritium itself isn't something you find just lying around (if you did, you could just use the hydrogen as your lifting gas), but it still gives some hints for helium production. Alpha radiation is actually just energetic helium, so any radioactive substance that decays through alpha radiation effectively generates helium gas (which includes Uranium 235). You would need some handwaving, but if you want to get really dangerous then your steampunkers might discover that this strange, cakey, yellow substance produces lightweight gas all on its own. Fortunately for them, it is near impossible to generate a critical nuclear reaction using natural uranium (which is primarily composed of the more stable U-238 nucleus). To get sufficient production of He through alpha decay you may need a less stable element anyway, which is where more handwaving comes in (because less stable elements aren't normally naturally occurring). So this avenue may require more handwaving then you want, or it may be worth a lot of handwaving to mine helium gas from natural nuclear power (and who doesn't want to blow up a steampunk city with an accidental nuclear explosion, right??).
Depending on your level of "technology" you can always have your people produce hydrogen via the electrolysis of water. You don't need to understand the chemistry for your people to try to do it, and it is so simple that just about everyone has the materials to do it at home. Doing it on an industrial scale and keeping the hydrogen gas separate are a bit trickier of course, but depending on your steampunk world generating hydrogen via electrolysis should be quite doable. If you want to keep the use of electricity simple then you can always come up with some sort of "lightning farm" that uses natural lightning in a stormy environment to split water for hydrogen. That's probably not possible in practice, but this is fiction after all and might be fun for a story.