Ykhjuy

It would be nice if there were an atmosphere on the Moon with oxygen; in fact it could be just oxygen at 1/5 the Earth's pressure. People could live there. Now of course given the Moon's lower gravity the atmosphere would escape into space, but it'd take hundreds of thousands of years for that (roughly 1 million years if I recollect well). Enough time for 50 human civilizations to develop one after another.

Now there is oxygen on the Moon, but it is fixed in rocks and probably in some water. What would be the best, easiest way to make an atmosphere out of it?

My guess is that can be done with a powerful enough energy source. Rocks or water can be decomposed by heating.

Using the Sun's light for that could do, if it is concentrated. Nuclear reactors and nuclear bombs could be used too, but that doesn't seem that easy because it costs to send them; making an automated nuclear plant / bomb factory on the Moon may be cheaper, but other problems appear (ores, plant defects etc.)

So my first guess to a solution would be using solar energy, with mirrors that can be built in place or brought from Earth (as thin sheets of plastic). But how much of the Moon's surface would have to be used? I wouldn't cover too much of it with oxygen factories; that would cost a lot too. Let's say we want an atmosphere there in a reasonable time (hundreds of years). What approach to use?

To make a lasting atmosphere on the Moon, we'd need to produce around 10,000 metric tons of gas per day just to keep it stable. Source: http://www.geoffreylandis.com/moonair.html

That is a lot. To give you an idea, in normal conditions, 1m³ of air has a mass of 1.024 kg. 10,000 tons would have a volume of approximatelly 10,000,000m³. That is nearly 50x the discharge per second of the Amazon river.

We can only imagine an industrial setup able to have that output in the realms of sci-fi. By the time we have the technology to set that up on the Moon for real, our solar cells should be more efficient than they are now, and we will have a better understanding of the ice deposits on the Moon. Otherwise, calculating your energy requirements seems pointless to me.

Last but not least, I don't think you want to burn rocks to make an atmosphere. That is wasteful, and the waste by-products may make the air unbreathable. You will also need to rewrite the maps for the lunar surface every few days to make up for all the surface you've been digging and moving around.

Using heat to decompose molecules is a bad idea in a low gravity environment like the Moon.

At high temperature the molecules have an higher average velocity, meaning that an higher fraction of the molecules will have enough velocity to escape the gravity well. This in turn will mean that your estimated life span for the atmosphere will be considerably shorter.

Moreover from a hot mass you would develop both the Oxygen and the material it was bounded to. You will need to separate them quickly, else they will recombine (that's why you don't use water to extinguish extremely hot fires, because by doing so you just put Oxygen and Hydrogen next to a hot body)
And you will also have the additional problem of dissipating the amount of heat, unless you want to have a boiling planet totally unsuited for life.

Once you have an abundant source of energy, go for the electrolytic path: dissolve the material in a suitable solvent, apply current and harvest the developed substances.

On one hand you will get Oxygen for your atmosphere, on the other hand you will get metals which you can use for other purposes on your lunar colony.

Once you have a significant atmosphere with a density of a hundredth of a millibar or so, it will stabilise itself as it becomes much colder with height.
That is as long as you stick to oxygen and nitrogen (don't emit greenhouse gases, and an ozone layer is also bad)! Moon rotates so slowly that no strong winds will occur, I guess.

The important point is that once the pressure has dropped so far (with height) that the mean free path lenght get's into the kilometer range, the temperature must be so low that practically no particle has escape velocity any more.

The problem will be to get beyond the first step, when the gas molecules can still fly directly off the hot lighted moon surface into space, without hitting other molecules.