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I know that some gases are heavier than air and in the case of a gas leak it's best to escape to higher ground, while some gases are lighter than air such as Helium. However unlike Helium I haven't seen a burp burst into flames while I have had the unfortunate pleasure of being present at a party where flatulence was lit on fire.

So I am wondering if there is a difference in how the gas is produced which determines if it's becomes a burp or flatulence.

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The gas expelled as a burp originates from the stomach. It is either gas swallowed together with food or gas that was dissolved in the drink. One prominent example for the latter is sparkling water where $\ce{CO2}$ is dissolved in the water at high concentration as carbonic acid ($\ce{H2CO3}$). Because of the acidic environment in the stomach the equilibrium between carbonic acid and $\ce{CO2}$ gas is shifted. Consequently $\ce{CO2}$ accumulates and needs to be expelled as a burp.

The gas expelled as a fart is mainly produced by microorganisms in the intestine and usually contains a lot of methane which makes it flammable.

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