# Thermodynamic potentials and Legendre transformations

## Background

In thermodynamics one can perform a Legendre transformation to switch between different chemical potentials. For concreteness consider the following:

We know that the natural variables of the internal energy $U$ is the entropy $S$ and volume $V$. Furthermore, the thermodynamic identity can be written as $$dU = TdS - p dV.$$ This means that we can compute the pressure $p$ by keeping $S$ constant such that $$p = - \left(\frac{\partial U}{\partial V}\right)_S.$$ Similarly by keeping $V$ constant we can calculate the temperature $$T = \left(\frac{\partial U}{\partial S}\right)_V.$$

In principle we could now determine the pressure by measuring the internal energy as a function of the volume and just differentiate. However, can we actually measure the internal energy as a function of volume in a laboratory?

Furthermore, the expression for temperature is rather useless in an experimental setting because we can not control the entropy $S$. The solution to this is usually to perform a Legendre transformation to obtain the free energy

$$F(T,V) = U(S,V) -TS.$$

By now differentiating the free energy we can find expressions for the volume and temperature in terms of $F$. We like this expression because $T$ and $V$ are very easy to measure in a laboratory, however once again I'm uncertain as if we can actually determine the free energy as a function of volume and temperature.

## Question(s)

With the background out of the way I want to ask three questions:

1. Can we actually measure the various thermodynamical potentials as a function of their natural variables in a laboratory?
2. If not, do we simply have to derive theoretical expressions for the potentials and use it to compute other observables such as the heat capacity which can be measured directly? (Much like the partition function encountered in statistical physics)
3. If we can not measure the potentials and all of the thermodynamical potentials contains the same information about the system, why should we bother Legendre transforming between them?
• As for question 1, I would say that, yes. we could measure a thermodynamic potential (to some experimental accuracy) in the lab over some temperature and pressure range but it would involve a lot of work since it would require taking data as a function of temperature and compression over a 2D grid of points. One could use the partial differential equations above to determine how much the free energy changes from between adjacent data points on the 2D grid. Don't think that it's worth the effort unless there is some specific need which is probably why it is rarely, if ever, done.
– user93237
May 3, 2018 at 22:07

1. We can only measure changes in $$U$$, $$H$$, $$G$$, $$F$$ and $$S$$. We cannot measure the absolute enthalpy of water, for example. However, by agreeing on a reference state we can report "absolute" values for all of the above properties.