Physical significance of Maxwell's thermodynamic relations I know the formulations and derivations of Maxwell's thermodynamic property relations but the thing I don't understand is why do they exist in the first place. Is it just a mathematical coincidence or there is some deeper meaning in statistical mechanics. I mean
$$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V.  $$
What can be physically understood out of such relations apart from the fact that they make it easier to calculate certain other properties as they relate them? How could they be equal? Please don't explain mathematically as I already know the multivariable calculus derivation.
I hope I'm clear with the question.
 A: This article by Dan Styer might give you the physics answer you seek:
A thermodynamic derivative means an experiment

American Journal of Physics 67, 1094 (1999); 
Daniel F. Styer 
https://doi.org/10.1119/1.19088
He gives the example of
$$\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_V$$
He says (formatting mine)

Each of the above derivatives is not only a
mathematical expression, but also an invitation to perform an
experiment.

The derivative on the left is measured as follows:

A sample in a container of variable volume (such as a
piston) is placed within a thermostatically controlled bath (so
that the temperature does not change) and is heated in a slow
and carefully monitored way (so that the heat absorbed quasistatically
can be divided by the temperature to find the
entropy change). As the substance is heated at constant temperature,
the volume of the piston must change. Dividing the
heat absorbed by the temperature and the volume change
gives (for small volume changes) the derivative on the left.
This experiment is not impossible, but clearly it is difficult
and expensive.


Consider in turn the experiment on the right.
The sample
is in a ‘‘strong box’’ container of fixed volume and its pressure
and temperature are measured as both of them change.
The change need not be controlled carefully and the heat
absorbed need not be monitored: You can just blast your
sample with a propane torch. Dividing the measured change
in pressure by the measured change in temperature gives (for
small temperature changes) the derivative on the right.


It is far from obvious that the results of these two very
different experiments should always be the same. The fact
that they are shows how thermodynamics can save a lot of
experimental labor!

Mathematically, it seems that the Maxwell Relations
are a result of the equality of area for the same process
on a PV-diagram and a TS-diagram. (Their elements of area are equal.)
A: I'll admit I don't have a magical way to see this relation intuitively either. I'll share my thoughts anyway. And I'd be really interested if anyone really has some intuitive explanation.
All the variables in the Maxwell's relation above are in this
$$
\mathrm{d} F=-S \mathrm{~d} T-p \mathrm{~d} V
$$
If you trust the above equation, and that the Helmholtz free energy $F$ is a state function, then Maxwell's relation follows naturally.
So your distrust of Maxwell's equation can really be traced back to this equation.
If you think about this equation more literally, it's quite Bizarra. Imagine you have a 2D plane with the two axis $T,V$. Each point $(T,V)$ completely specify a system in equilibrium. $F=F(T,V)$ is a differentiable function on this 2D plane, and we can even specify some values $F_1,F_2,...$ and draw contours on the 2D plane.
In this picture, $S$ and $P$ are derivatives at a given point $(T,V)$ in the direction of $T$ and $V$. They tell you how much the energy should change in response to a change in $T,V$. I think that's where the problem lies. It's quite elusive to try to think of entropy as a response of some free energy to the change in temperature, they are simply too vague for one to develop a working intuition.
Perhaps you could see how increasing temperature will increase the average kinetic energy of gas in a chamber such that they will hit the wall harder, and that increasing the volume will increase undertainty of where the particles are. This kind of thinking will not help, for example, when you are studying the temperature of a Ising model, which could be negative, since energy decrease when we increase entropy.
In my opinion, all of this tells you that you don't really understand energy, entropy, or temperature or conservation of energy. They sound familar and innocent, but we've only had experience with them in some limited specific contexts. The first and second law of thermodynamics is really much stronger than what we thought it is. It gave specific definition to our vague intuition about heat, disorder, and conservation of energy.
