Question

Calculate the word done in joules when 1.0 mile of water vaporizes at 1.0 atm and 100 degrees Celsius. Assume that the volume of liquid water is negligible compared with that of steam at 100 degrees celcius, and ideal gas behavior.
Note the correct spelling of celsius.
Did you mean 1 mole instead of 1 mile? I will assume you did.

Use PV = nRT and solve for V in L. The problem says to ignore volume of liquid water so you call that zero.
Then work = -p(Vfinal-Vinitial)
The unit is L*atm. If you want to convert to J just multiply by 101.325.
-3022.8

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