# Gibbs Free Energy

## 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 at constant pressure and temperature and is 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)

and at constant Pressure-

or, dF = − SdT

or, (dF/dT)

_{P}= − S -----(equation-4)

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