Final answer:
The process that requires water to gain 2260 joules of energy per gram is evaporation.
Step-by-step explanation:
The process that requires water to gain 2260 joules of energy per gram is the process of evaporation. The heat of vaporization of water is about 2250 J per gram, which means that every gram of water that goes from liquid to gas (vapor) requires an energy input of approximately 2250 J. This process involves the absorption of heat energy from the surroundings, causing the water molecules to gain energy and change phase from liquid to gas.