The CO2 both dissolves into the beer and compresses under pressure so it takes up less space.
Say we took a 5 gallon keg of beer and held it at the appropriate temp and pressure for 2 or 3 weeks to get the carbonation to 3.0 volumes. Then we disconnected the CO2 supply, attached an empty giant balloon to the gas post and let it sit at 32 degrees F long enough for all of the CO2 to come back out of solution, we would have 5 gallons of flat beer in the keg and the balloon would expand to a size of 15 gallons.
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