# Gibbs free energy

In thermodynamics the Gibbs free energy is a state function of any system defined as

G = H − T·S

where

G is the Gibbs free energy, measured in joules
H is the enthalpy, measured in joules
T is the temperature, measured in kelvins
S is the entropy, measured in joules per kelvin

Each unit in the equation above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy.

The Gibbs free energy is one of the most important thermodynamic functions for the characterization of a system. It determines outcomes such as the voltage of an electrochemical cell, and the equilibrium constant for a reversible reaction. Gibbs free energy also determines how much work is attainable for any given process.

Any natural process occurs spontaneously if and only if the associated change in G for the system is negative ([itex]\Delta G[itex] < 0). Likewise, a system reaches equilibrium when the associated change in G for the system is zero ([itex]\Delta G[itex] = zero). And, no spontaneous process will occur if the final value of G is positive, or, greater than zero.

It is named after American chemist Willard Gibbs.

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## Useful identities

[itex]\Delta G = \Delta H - T \Delta S \,[itex]
[itex]\Delta G = -R T \ln K \,[itex]
[itex]\Delta G = -nF \Delta E \,[itex]

and rearranging gives

[itex]nF\Delta E = RT \ln K \,[itex]

which relates the electrical potential of a reaction to the equilibrium coefficient for that reaction.

where

ΔG = change in Gibbs free energy
ΔH = change in enthalpy
T = temperature
ΔS = change in entropy
R = gas constant
ln = natural logarithm
K = equilibrium constant
n = number of electrons/mole product
ΔE = electrical potential of the reaction

## Derivation of Gibbs Free Energy

Let Stot be the total entropy of a thermally closed system. An isolated system cannot exchange heat with its surroundings. Total entropy is only defined for an isolated system, an open system has internal entropy instead.

The second law of thermodynamics states that if a process is possible, then

[itex] \Delta S_{tot} \ge 0 \,[itex]

and if [itex] \Delta S_{tot} = 0 \,[itex] then the process is reversible.

Since the heat transfer Δq vanishes for a closed system, then any reversible process will be adiabatic, and an adiabatic process is also isentropic [itex] \left( {\Delta q\over T} = \Delta S = 0 \right) \,[itex].

Now consider an open system. It has internal entropy Sint, and the system is thermally connected to its surroundings, which have entropy Sext.

The entropy form of the second law does not apply directly to the open system, it only applies to the closed system formed by both the system and its surroundings. Therefore a process is possible iff

[itex] \Delta S_{int} + \Delta S_{ext} \ge 0 \,[itex].

We will try to express the left side of this inequation entirely in terms of internal state functions. ΔSext is defined as:

[itex] \Delta S_{ext} = - {\Delta q\over T} \,[itex]

Temperature T is the same both internally and externally, since the system is thermally connected to its surroundings. Also, Δqrev is heat transferred to the system, so -Δqrev is heat transferred to the surroundings, and −ΔQ/T is entropy gained by the surroundings. We now have:

[itex] \Delta S_{int} - {\Delta q\over T} \ge 0 \,[itex]

Multiply both sides by T:

[itex] T \Delta S_{int} - \Delta q\ge 0 \,[itex]

ΔQ is heat transferred to the system; if the process is now assumed to be isobaric, then Δqp = ΔH:

[itex] T \Delta S_{int} - \Delta H \ge 0\, [itex]

ΔH is the enthalpy change of reaction (for a chemical reaction at constant pressure and temperature). Then

[itex] \Delta H - T \Delta S_{int} \le 0 \,[itex]

for a possible process. Let the change ΔG in Gibbs free energy be defined as

[itex] \Delta G = \Delta H - T \Delta S_{int} \,[itex] (1)

Notice that it is not defined in terms of any external state functions, such as ΔSext or ΔStot. Then the second law becomes:

[itex] \Delta G < 0 \,[itex] favored reaction
[itex] \Delta G = 0 \,[itex] reversible reaction
[itex] \Delta G > 0 \,[itex] disfavored reaction

Gibbs free energy G itself is defined as

[itex] G = H - T S_{int} \,[itex] (2)

but notice that to obtain equation (2) from equation (1) we must assume that T is constant.

Thus, Gibbs free energy is most useful for thermochemical processes at constant temperature and pressure: both isothermal and isobaric. Such processes do not seem to move on a P-V diagram; they do not seem to be dynamic at all. However, chemical reactions do undergo changes in chemical potential, which is a state function. Thus, thermodynamic processes are not confined to the two dimensional P-V diagram. There is at least a third dimension for n, the quantity of gas.

### Back to Entropy

If a closed system (Δqrev = 0) is at constant pressure (Δqrev = ΔH), then

[itex] \Delta H = 0 \,[itex]

Therefore the Gibbs free energy of a closed system is:

[itex] \Delta G = -T \Delta S \,[itex]

and if [itex] \Delta G \le 0 \,[itex] then this implies that [itex] \Delta S \ge 0 \,[itex], back to where we started the derivation of ΔG.

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