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Meaning of Helmholtz/Gibbs free energy

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This figure indicates the energy required to conjure up the rabbit "out of thin-air". Evidently it takes more than energy, U=E, of the rabbit. Please explain, and also tell me why these energies are called "Free". In other words, why did scientists give them such a name? Which figure is correct or are they both correct ?

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The name ''free energy'' is much to the dislike of most physicists. The reason it is used is, just like in so many other cases, because you can't change traditional names that easily. Using the free energy function you can calculate how much work you can theoretically extract from a system.

You can define free energy as the minimum energy required to create an object ''out of nothing'' inside an environment kept at constant temperature and volume (Helmholtz free energy) or constant temperature and pressure (Gibbs free energy). If the environment is at temperature T then you can save some energy by creating the object at absolute zero. The object will automatically warm up; so you get some of the energy for free. The object also occupies some volume. This means that to create the object you have to perform work to make room for the object. The minimum amount of energy you have to have to create the object is ...

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A detailed explanation is given.