# Free Energy

### Free Energy

You cannot get something for nothing, and the term "free energy" does not mean energy that has no cost.  Rather it refers to energy that is available to do something useful.  The use of the word "free" is in the sense of "liberated."  Free energy is energy that can be liberated to do something useful.

We know that the second law places limitations on how much energy can be used to do useful work.  In most situations, some of the energy must be dissipated as heat that cannot be used to do something useful.  The portion of the energy that can do something useful is the free energy.

At constant volume, the free energy is the Helmholtz Free Energy, given the variable A.

### The Helmholtz Energy

The Helmholtz energy is equal to the internal energy, U, minus the product of the absolute temperature, T and the entropy S.

A = U - TS

The internal energy is defined in my post on the first law of thermodynamics.  T is the temperature in Kelvin, and S is the entropy defined in this series.

As an example, suppose that temperature is kept constant.  In such cases:

ΔA = ΔU - TΔS

The delta symbol, Δ, indicates the change in the quantity. By convention:

ΔA = A(final) - A(initial)

### The Gibbs Energy

The Gibbs energy is analogous to the the Helmholtz energy except that it is the free energy when pressure is kept constant.  The Gibbs energy is equal to the enthalpy, H, minus the product of the absolute temperature, T and the entropy, S.

G = H - TS

The enthalpy is defined in a post on enthalpy as:

H = U + PV

So the Gibbs energy can also be written:

G = A +PV

A constant temperature:

ΔG = ΔH - TΔS

Application

At constant pressure, a system is in equilibrium if

ΔG = 0

On the other hand, if delta G is positive:

ΔG > 0

Energy must be provided to drive the the change.

If a change releases Gibbs free energy, i.e., delta G is negative.

ΔG < 0

The change is spontaneous.  Oxidation of hydrocarbons to form carbon dioxide and water is a spontaneous change.  It provides energy that can be used to drive other changes.  In thermodynamics the term spontaneous is used a little differently than in plain language.

For example diamond changes spontaneously into graphite because ΔG < 0.

### References

1. Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
2. McQuarrie, Donal d A., Statistical Thermodynamics,  University Science Books, Mill Valley, CA, 1973
3. Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
4. Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.

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