# What are the state functions telling me or how are they related to total energy?

I am quite new to thermodynamics and statistical mechanics so this might be an easy question:

In thermodynamics you get a bunch of thermodynamics potentials, so as for example enthalpy, internal energy, gibbs energy, helmholtz energy and so on. Now my idea was, that you use them if the natural variables related to this quantity are constant. But I do have a question of interpretation: It is always true that if you have some process that the difference of the thermodynamic potentials, whose natural variables are left unchanged is equal to the difference in total energy?

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"Total energy" is in fact, the internal energy $U$ (unless you give a more precise definition).

We may consider also enthalpy $H = U+pV$, the Helmholtz free energy $F= U - TS$, the Gibbs free energy $G = U + pV − TS$.

For instance, take Helmholtz free energy, we have (assuming the chemical potentials being zero) :

$dU = -pdV+TdS$, and $dF = -pdV - SdT$

The natural variables associated to $F$ are $V$ and $T$, so if they are unchanged in some process, that means that $F$ is constant in this process ($\Delta F =0$). But in this same process, you have $dU = TdS$, so unless the process is isentropic, you will have a variation in $U$ ($\Delta U \neq0$),so you have, in the general case: $$\Delta F \neq \Delta U$$

A analog demonstration can be done for the enthalpy and the Gibbs free energy.

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