Anyone who's studied thermodynamics knows the fundamental relations for the various free energies: \begin{align} \mathrm{d}U &= T\:\mathrm{d}S- p\:\mathrm{d}V\\ \mathrm{d}F &= -S\:\mathrm{d}T- p\:\mathrm{d}V\\ \mathrm{d}H &= T\:\mathrm{d}S+ V\:\mathrm{d}p\\ \mathrm{d}G &= -S\:\mathrm{d}T+ V\:\mathrm{d}p \end{align} We think of $S,V$ as being the "natural variables" for the energy $U$, and similarly for the other free energies. But why? For e.g. an ideal gas, the energy is expressed most easily as a function of $T,V$, where it has the nice form $U=\frac{3}{2} N k T$ (in 3 dimensions, with $N$ particles). Changing variables to $S,V$ renders it much more complicated.

Of course one "obvious" reason to write the free energies as functions of their natural variables is that then the partial derivatives of the free energies are themselves nice functions (namely, the other state variables). But is this the only justification? Or is there some more profound reason to prefer these variables?

marked as duplicate by By Symmetry, Aaron Stevens, John Rennie energy Oct 19 at 15:09

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It is convenient to pick that state variables, where the Arguments of the function can be considered as constant. Then at least one term vanishes. Example: Free Energy $F$ depends on temperature $T$ and volume $p$. If we assume that the temperature is constant, then we have

$F = -SdT-pdV = -pdV$.

This is a much easier expression that can be integrated. Gibbs free enthalpy is very common in e.g. chemical reaction, because the Change in this quantity is only Change in particle numbers (during the chemical reaction) if pressure and temperature is constant (very frequently used assumption).

By Legendre transform of thermodynamic potentials like free Energy, enthalpy, etc. one can define any thermodynamic potential; usually that is used, where the Change in one or more dependent variables can be regarded as small or vanishing.

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