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What is the fundamental relationship between grand potential and free energy? In other words, what assumptions need to be made in order to obtain the free energy equation rather than the grand potential?

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The grand potential depend on T, the chemical potentials of the components and on V. The Gibbs energy (previously called Gibbs free energy) depend on T, P and the amount of the components.

There is also the Helmhotz energy which depend on T, V and the amount of the components and the entalpy which depend on S, P and the amount of the components. All derived from the internal energy which depend on S, V and the amount of the components.

S, V and the amount of the components are extensive propertites whih depend on the amount of material. T, P and the chemical potentials are intensive properties, independent of the amount.

The grand potential depend on a single extensive property, V if you replace V by its potential, P then you get zero because you have no material at all.

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