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The free energy, $F$ of a thermodynamic system at a given temperature $T$, is defined as, \begin{equation} e^{-\beta F} = \mathcal{Z} = \sum_{\{configuration\}} e^{-\beta E(configuration) } \end{equation} where $\beta = 1/k_BT$ and $E$ is the energy of a certain configuration.

Now my question is:

Why the free energy is called 'free'? I mean it is free in which sense?

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Free in the sense of without obligation, therefore available. Not in the sense of without cost. – Nic Jul 31 '13 at 11:35
up vote 4 down vote accepted

Free Energy refers to the energy in a system that is free to do work i.e. the internal energy minus any energy that is unavailable to perform work. Internal Energy accounts for the total energy of the system.

In 1882, the German physicist and physiologist Hermann von Helmholtz coined the
phrase ‘free energy’ for the expression E − TS, in which the change in F (or G) 
determines the amount of energy ‘free’ for work under the given conditions.
*From wikipedia*

It's normally called the Gibbs energy more recently, though at my Uni it's often been refered to as the 'Gibbs Free Energy'. I believe the same is true of the Helmhotz Energy.

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Free energy is (Internal energy - energy which cannot be used to perform work) .
Free energy means that this energy is available in the form of useful work

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An other way to see the Helmholtz free energy : $F = U - TS$ (where $U = \langle E \rangle$ is the internal energy) is to write (with $\beta = \frac{1}{kT}$):

$$\beta F = \beta U - \frac{S}{k}$$

This could be read :

non-uniform information = Total information - uniform information

If you have only uniform information (entropy), no change can happens, and so, if we consider the energetic point of view, no work can be done.

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