# Gibb’s Free Energy [Gibb's Function]

In order to define this term, let us consider a process taking place isothermally and reversibly at constant pressure. There will be a volume change say $\Delta V$. The maximum work obtained by it will not be amount of energy available for doing useful work. From the total amount of work, some part of the work is used to perform the mechanical work or pressure volume work of expansion or contraction against the atmospheric pressure. This work will be equal to $P\Delta V$.

Hence the whole of work given by the process will be made up of two parts:

(a)    The mechanical work of expansion or contraction.

(b)   The work other than mechanical work is known as the net work $(W_{net})$. The net work includes all other forms of work energy performed by the system on the surroundings which can be applied to useful work. Thus net work done can be given by $W_{net} = W_{max} = H_{max}- P \Delta V$ …..(1)

According to first law of thermodynamics, $dE = dQ - dW_{max}$ $or \hspace{3mm} dE = TdS - D W_{max} \hspace{4mm}( \because dQ = TdS)$ ….(2)

Substituting the value of $W_{max}$ from (1) in (2), we get $dE =TdS- d(W_{net} + P \Delta V) \\[3mm] = TdS- dW_{net}- d(PV) \\[3mm] OR, d w_{net} = -d E + TdS- d(PV) \\[3mm] or, W_{net} = =-d (E- TS + PV)$

On integration, it gives $W_{net} = -\Delta (E- TS + PV)$….(3)

Thus $W_{net}$ is determined by the initial and the final states of the process and is equal to the decrease in (E — TS +PV). This combination of properties is called Gibb’s free energy and is represented by G.

I.e.

G = E – TS + PV

Or,  G = E + PV – TS

Or,  G = H – TS[ $\because H = E + PV$]

This is known as Gibb’s Helmholtz equation.

The Gibb’s free energy is a better criterion for spontaneity over total entropy change because Gibb’s free energy considers only the entropy change of the system and not the surroundings. Thus we can write above equation as: $\Delta G = \Delta H - T \Delta S$

Where $\Delta G = G_{products} - G_{reactants}$

The following three values are possible for $\Delta G$:

(i) $\Delta G$ is positive: It means the reaction is non-Spontaneous.

(ii) $\Delta G$ is negative: It means the reaction is spontaneous.

(iii) $\Delta G$  is zero: It means the reaction is in equilibrium state.

The standard free energy change, $\Delta G^0$,(E.g. a gas is at 1 atm pressure and concentration of all reactants and products in solution is 1 M) is also related with equilibrium constant as: $\Delta G^0_{reaction} = -2.303 RT \log K$

Where all the terms have usual meaning.

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