Well, "gas" is a pretty vague term but if you're referring specifically to farts, they're not necessarily heavier than air because they don't just consist of methane and hydrogen, but other poo-particles too that are heavier than air.
The realistic answer is: The change in weight is pretty negligible, but you probably still become ever so slightly lighter.
Ignoring stuff like breathing and eating and whatever, as gas builds up from microbes metabolizing shit in your gut, your mass stays the same but you become less dense (and therefore must take up more volume, and be physically bloated by the gas). The boyant force of air trying to get under your volume and push you up is a battle between volume and mass, so since you take up more volume for the same mass, boyancy will technically reduce your weight, but probably not by much.
When you fart, the gas is released and your volume and mass decrease by the same volume and mass of the gas. Because the density of this gas is much lower that your average density, your density increases while farting, removing any extra boyancy created by holding the gas in. Depending on the density of the gas, the weight of mass you lost is more than or less than the boyancy lost
If the gas is lighter than air, the mass weight lost from farting is less than the mass weight of air for an equivalent amount of volume lost. Just like deflating a helium balloon, this results in a net loss of boyancy, making you heavier.
The opposite is true for gas heavier than air. The mass weight lost from farting is more than the mass weight of air for an equivalent volume. Just like dropping sandbags from a hot air balloon, your boyancy increases (but seriously not a lot)
As for the density of farts? They are almost certainly lighter than air because we are usually hotter than our environment (else were uncomfortable) and methane is about half as dense as typical air. Apparently it's complicated but my impression is that farts are usually less dense.
Does this make any sense? Are we really experiencing a boyant force even when our average density is much greater than air? I dont think so, I think boyancy in air is irrelevant for humans. Here's an interesting bit of science that helped me understand why the way we are taught about boyancy isn't always the whole story
I'd imagine your farts would get compressed a bit increasing the air density. This would lead to expelling a larger volume of gas at whatever atmospheric pressure you're at, than what was inside your body under pressure. Maybe this is negligible, by maybe not. Just food for thought.
Agreed, I was trying to work out if that matters and decided to assume that it is negligible. Partly this was because it would change as the fart moves around, and as part of the farting process the air must be compressed to some degree. All of the quantities in this problem are basically negligible though so maybe it is important.
Since we've all participated so far, what not ask the more important question of how much thrust could a standard fart be expected to generate and how best to optimize for thrust?
Great comment, thank you!
You say buoyancy is irrelevant to humans, but would it be measurable?
I don't think so, but it is definitely possible. We have incredibly precise measuring devices, and although it would be complicated by all the other things our bodies do it may be possible to measure.
I used this calculator with density of 1.225 for air and 375ml for the volume of a large fart and it reports a force of about .0045 N. This is about .45 grams, which is about .001 lbs, which is what this company claims their sensitivity is... So since the boyant force excludes the gravitational force on the fart itself the measured force would be much smaller. you would need a scale much more sensitive to measure it. If you farted pure methane the force would be about half. A good measurement should probably have about 10 times the accuracy (.045 grams). Since you can buy a scale for a couple thousand dollars that almost gets there I'm going to guess that this would be measurable, but there may be some hidden gotcha I'm not aware of.
When you fart the expelled gas isn't immediately replaced. So you just become lighter by the mass of the flatulence. (Which will be a rounding error to the mass of your body)
Only a few gasses are lighter than "air", but even those have weight. Helium for example weighs 0.0114 lbs per cubic foot.
Helium only rises because it less weight than airz like how oil floats on water.
When your body contains that matter, your weight is higher. When you expel the matter you lose weight.