# Gibbs free energy(G or F)

Gibbs free energy is the property of the system whose decrease is the measure of the maximum external work available during the transformation of system reversibly (or maximum reversible work that a thermodynamic system can perform) at constant pressure and temperature. Free energy is a state function so, depend only on initial and final state of the system.

If S is the entropy of a system at T^{o}K and H is its enthalpy then, Gibbs free energy is mathematically expressed as-

G = H - TS

On differentiation we get,

ΔG = ΔH − TΔS − SΔT

we know that- ΔH = ΔE + PΔV + VΔP

so, ΔG = ΔE + PΔV + VΔP − TΔS − SΔT

or, ΔG = ΔA + PΔV + VΔP − SΔT (as ΔE − TΔS = ΔA (work function))

at constant pressure and temperature-

or, ΔG = ΔA + PΔV

or, ΔG = −W_{max} + PΔV (as W = −ΔA from 1st law of thermodynamics)

or, −ΔG = W_{max} − PΔV

PΔV is the work done due to expansion against a constant pressure. So, decrease in free energy accompanying a process which occurs at constant pressure and temperature is the maximum work obtained from the system other than PΔV. Hence it is also called Net Work.

So, Net Work = −ΔG

## Variation of Free Energy with Temperature and Pressure

We know that-

F = H − TS -----(equation-1)

or, F = E + PV − TS (as H = E + PV)

differentiating this equation we get-

dF = dE + PdV + VdP − TdS − SdT

or, dF = dq + VdP − TdS − SdT (as dq = dE + PdV)

or, dF = TdS + VdP − TdS − SdT (as dq_{rev}/T = dS)

or, dF = VdP − SdT -----(equation-2)

**at constant Temperature-**

or, dF = VdP

or, (dF/dP)_{T} = V -----(equation-3)

**at constant Pressure-**

or, dF = − SdT

or, (dF/dT)_{P} = − S -----(equation-4)

## Significance of Gibbs Free Energy

### Spontaneity of Reactions

If ΔG < 0, the process occurs spontaneously.

If ΔG > ΔG, the process occurs non-spontaneous.

If ΔG = 0, the system is at equilibrium.

### Maximum Work

The Gibbs free energy change gives the maximum amount of non-expansion work (work other than that due to volume changes) that can be extracted from a closed system.

### Phase Transitions

The Gibbs free energy is useful for understanding phase transitions. At equilibrium between phases, the Gibbs free energy is minimized and equal for the phases in transition.

### Equilibrium

At equilibrium, the Gibbs free energy is minimized or zero so, there is no net change in the concentrations of reactants and products.

ΔG = − RTlnKl

The above equation shows the relationship between Gibbs free energy to the equilibrium constant K.

### Electrochemistry

In electrochemistry, Gibbs free energy changes are related to the electrical work done by galvanic cells.

ΔG = −nFE

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