# Why do we prefer to write free energies as functions of particular state variables? [duplicate]

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 StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 19 '18 at 15:09

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