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When you fill up a balloon about a quarter way with liquefied butane fuel and let it sit at room temperature it will turn into gas. But why does the gas weigh the same as the liquified butane?

The liquefied butane liquid weighs just about as much as water.

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Assuming you fill the balloon only with liquid butane, the answer is very simple - the gas in the balloon at room temperature IS the liquid in the balloon initially. If you put some amount of liquid butane into the balloon and seal it at t=0, then allow it to warm until the liquid has evaporated, there is essentially no transport of other matter into or out of the balloon during that time. The only thing that has happened is that the initial charge of liquid has absorbed thermal energy from the surroundings and undergone a phase change from liquid to gas.

Because there is no change in the amount of material in the balloon - there are the same number of butane molecules in the gas phase as there were initially in the liquid phase - there can be no change in mass. Liquid and gas phase contain the same number of molecules, at some fixed mass per molecule.

Note that while the mass does not change, the density does.

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    $\begingroup$ The mass of butane will be the same but the weight might be different due to buoyancy. The mass of air displaced by the extra volume of gaseous butane might matter a little. $\endgroup$
    – matt_black
    Sep 1 '16 at 22:34

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