Lifting gas

A lifting gas or lighter than air gas is a gas that has a lower density than normal atmospheric gases and rises above them as a result. It is required for aerostats to create buoyancy, particularly in lighter-than-air aircraft, which include free balloons, moored balloons, and airships. Only certain lighter than air gases are suitable as lifting gases. Dry air has a density of about 1.29 g/L (gram per liter) at standard conditions for temperature and pressure (STP) and an average molecular mass of 28.97 g/mol, and so lighter than air gases have a density lower than this.
Gases theoretically suitable for lifting.
Hot air.
Heated atmospheric air is frequently used in recreational ballooning. According to the Ideal gas law, an amount of gas (and also a mixture of gases such as air) expands as it is heated. As a result, a certain volume of gas has a lower density as the temperature is higher. The average temperature of air in a hot air balloon is about .
Hydrogen.
Hydrogen, being the lightest existing gas (7% the density of air), seems to be the most appropriate gas for lifting. It can be easily produced in large quantities, for example with the water-gas shift reaction, but hydrogen has several disadvantages:
Helium.
Helium is the second lightest gas. For that reason, it is an attractive gas for lifting as well.
A major advantage is that this gas is noncombustible. But the use of helium has some disadvantages, too:
Hydrogen and helium.
Hydrogen (density 0.090 g/L at STP, average molecular mass 2.016 g/mol) and helium (density 0.179 g/L at STP, average molecular mass 4.003 g/mol) are the most commonly used lift gases. Although helium is twice as heavy as (diatomic) hydrogen, they are both so much lighter than air that this difference only results in hydrogen having 8% more buoyancy than helium.
In a practical dirigible design, the difference is significant, making a 50% difference in the fuel-carrying capacity of the dirigible and hence increasing its range significantly. However, hydrogen is extremely flammable and its use as a lifting gas in dirigibles has decreased since the Hindenburg disaster. Helium is safer as a lifting gas because it is inert and does not undergo combustion.
Water vapour.
The gaseous state of water is lighter than air (density 0.804 g/L at STP, average molecular mass 18.015 g/mol) due to water's low molar mass when compared with typical atmospheric gases such as nitrogen gas (N2). It is non-flammable and much cheaper than helium. The concept of using steam for lifting is therefore already 200 years old. The biggest challenge has always been to make a material that can resist it. In 2003, a university team in Berlin, Germany, has successfully made a 150 °C steam lifted balloon. However, such a design is generally impractical due to high boiling point and condensation.
Ammonia.
Ammonia is sometimes used to fill weather balloons. Due to its high boiling point (compared to helium and hydrogen), ammonia could potentially be refrigerated and liquefied aboard an airship to reduce lift and add ballast (and returned to a gas to add lift and reduce ballast). Ammonia gas is relatively heavy (density 0.769 g/L at STP, average molecular mass 17.03 g/mol), poisonous, an irritant, and can damage many metals and plastics.
Methane.
Methane (density 0.716 g/L at STP, average molecular mass 16.04 g/mol), the main component of natural gas, is sometimes used as a lift gas when hydrogen and helium are not available. It has the advantage of not leaking through balloon walls as rapidly as the smaller molecules of hydrogen and helium. Many lighter-than-air balloons are made of aluminized plastic that limits such leakage; hydrogen and helium leak rapidly through latex balloons. However, methane is highly flammable and like hydrogen is not appropriate for use in passenger-carrying airships. It is also relatively dense and a potent greenhouse gas.
Hydrogen fluoride.
Hydrogen fluoride is lighter than air and could theoretically be used as a lifting gas. However, it is extremely corrosive, highly toxic, expensive, is heavier than other lifting gases, and has a high boiling point of 19.5 °C. Its use would therefore be impractical. 
Coal gas.
In the past, coal gas, a mixture of hydrogen, carbon monoxide and other gases, was also used in balloons. It was widely available and cheap; the down side was a higher density (reducing lift) and the high toxicity of the carbon monoxide.
Acetylene.
Acetylene is 10% lighter than air and could be used as a lifting gas. Its extreme flammability and low lifting power make it an unattractive choice.
Hydrogen cyanide.
Hydrogen cyanide, which is 7% lighter than air, is technically capable of being used as a lifting gas at temperatures above its boiling point of 25.6 °C. Its extreme toxicity, low buoyancy, and high boiling point have precluded such a use.
Neon.
Neon is lighter than air (density 0.900 g/L at STP, average atomic mass 20.17 g/mol) and could lift a balloon. Like helium, it is non-flammable. However, it is rare on Earth and expensive, and is among the heavier lifting gases.
Nitrogen.
Pure nitrogen has the advantage that it is inert and abundantly available, because it is the major component of air. However, because nitrogen is only 3% lighter than air, it is not an obvious choice for a lifting gas.
Vacuum.
Theoretically, an aerostatic vehicle could be made to use a vacuum or partial vacuum. As early as 1670, over a century before the first manned hot-air balloon flight, the Italian monk Francesco Lana de Terzi envisioned a ship with four vacuum spheres.
In a theoretically perfect situation with weightless spheres, a 'vacuum balloon' would have 7% more net lifting force than a hydrogen-filled balloon, and 16% more net lifting force than a helium-filled one. However, because the walls of the balloon must be able to remain rigid without imploding, the balloon is impractical to construct with all known materials. Despite that, sometimes there is discussion on the topic.
Plasma.
Another medium that in theory could be used is a plasma: Ions repelling each other could give a pressure intermediate between vacuum and hydrogen and hence that counteracts the atmospheric pressure. 
The energy and the containment requirements are extremely impractical, so that it may only be interesting for science fiction.
Combinations.
It is also possible to combine some of the above solutions. A well-known example is the Rozière balloon which combines a core of helium with an outer shell of hot air.
Hydrogen versus helium.
Hydrogen and helium are the most commonly used lift gases. Although helium is twice as heavy as (diatomic) hydrogen, they are both significantly lighter than air, making this difference negligible.
The lifting power in air of hydrogen and helium can be calculated using the theory of buoyancy as follows:
Thus helium is almost twice as dense as hydrogen. However, buoyancy depends upon the "difference" of the densities (ρgas) − (ρair) rather than upon their ratios. Thus the difference in buoyancies is about 8%, as seen from the buoyancy equation:
Where FB = Buoyant force (in Newton); g = gravitational acceleration = 9.8066 m/s2 = 9.8066 N/kg; V = volume (in m3).
Therefore, the amount of mass that can be lifted by hydrogen in air at sea level, equal to the density difference between hydrogen and air, is:
and the buoyant force for one m3 of hydrogen in air at sea level is:
Therefore, the amount of mass that can be lifted by helium in air at sea level is:
and the buoyant force for one m3 of helium in air at sea level is:
Thus hydrogen's additional buoyancy compared to helium is:
This calculation is at sea level at 0 °C. For higher altitudes, or higher temperatures, the amount of lift will decrease proportionally to the air density, but the ratio of the lifting capability of hydrogen to that of helium will remain the same. This calculation does not include the mass of the envelope need to hold the lifting gas.
High-altitude ballooning.
At higher altitudes, the air pressure is lower and therefore the pressure inside the balloon is also lower. This means that while the mass of lifting gas and mass of displaced air for a given lift are the same as at lower altitude, the volume of the balloon is much greater at higher altitudes.
A balloon that is designed to lift to extreme heights (stratosphere), must be able to expand enormously in order to displace the required amount of air. That is why such balloons seem almost empty at launch, as can be seen in the photo.
A different approach for high altitude ballooning, especially used for long duration flights is the superpressure balloon. A superpressure balloon maintains a higher pressure inside the balloon than the external (ambient) pressure.
Solids.
In 2002, aerogel held the Guinness World Record for the least dense (lightest) solid. Aerogel is mostly air because its structure is like that of a highly vacuous sponge. The lightness and low density is due primarily to the large proportion of air within the solid and not the silicon construction materials. Taking advantage of this, SEAgel, in the same family as aerogel but made from agar, can be filled with helium gas to create a solid which floats when placed in an open top container filled with a dense gas.
In 2012, discovery of aerographite was announced, breaking the record for the least dense material at only 0.2 mg/cm3 (0.2 kg/m3). These solids do not float in air because the hollow spaces in them become filled with air. No lighter-than-air matrix or shell containing a hard vacuum has ever been constructed.
Submerged balloons.
Because of the enormous density difference between water and gases (water is about 1,000 times more dense than most gases), the lifting power of underwater gases is very strong. The type of gas used is largely inconsequential because the relative differences between gases is negligible in relation to the density of water. However, some gases can liquefy under high pressure, leading to an abrupt loss of buoyancy.
A submerged balloon that rises will expand or even explode because of the strong pressure reduction, unless gas is able to escape continuously during the ascent or the balloon is strong enough to withstand the change in pressure.
Balloons on other celestial bodies.
A balloon can only have buoyancy if there is a medium that has a higher average density than the balloon itself.