Gibbs’ Free Energy Function

• All spontaneous processes tend to achieve a state of minimum energy and maximum entropy.
• For a system, generally it is not possible to achieve both, the minimum energy and maximum entropy simultaneously.
• There are cases where either of the two factors is favourable.
• For example, many endothermic reactions are spontaneous because the favourable entropy factor (ΔS is highly positive) outweighs the unfavourable entropy factor (ΔH is positive).
• On the other hand in some reactions the unfavourable entropy factor (ΔS_sys < 0 i.e., ΔS_sys is negative) is outweighed by the favourable enthalpy factor (ΔH is negative).
• From these observations, it becomes clear that ΔH (or, ΔE), nor ΔS_sys alone can predict whether a reaction is spontaneous or not.
• However, the quantity, ΔS_total = ΔS_sys + ΔS_surr can be used in predicting the spontaneity of a process.
• For a spontaneous process, ΔS_total should be positive.
• For chemical systems, however, it is not always possible to determine ΔS_surr.
• Under these conditions, it was felt that a new thermodynamic function is needed to predict the direction of spontaneity.

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• Gibbs’ free energy function was introduced by J. Willard Gibbs to predict the direction of spontaneity.
• This function is denoted by the symbol G.
• The Gibbs’ free energy function (G) accounts for the enthalpy and entropy changes in a system and is defined by

G = H – TS…………………………………..(i)

Where, H is the enthalpy of the system; S is the entropy of the system; T is the temperature of the system in Kelvin scale.

Since, H, T, S are state functions, hence, G is also a state function.

• The Gibbs’ free energy is defined as the amount of energy available for doing useful work under constant temperature and pressure conditions.
• The free energy function (G) as such is not very useful. Instead, the change in free energy function ΔG is very useful in predicting the state of any system.
• The ΔG is generally termed as the free energy change.

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Suppose a system is in its initial state, say state 1.

Let it be changed to another state, say state 2, under constant temperature and pressure conditions. Then, we can write,

State 1: G1 = H1 – TS1        State 2: G2 = H2 – TS2

The change in free energy ΔG is then given by

ΔG = G2 – G1 = (H2 – TS2) – (H1 – TS1) = (H2 – H1) – T (S2 – S1)

Or, ΔG = ΔH – T ΔS

This equation is known as Gibbs’ – Helmholtz equation.

Gibbs’ Free Energy Change and Spontaneity of a Process

For a system which is not isolated, the direction of spontaneity can be predicted by the total energy entropy (ΔS_total). ΔS_total is given by,

ΔS_total = ΔS_sys + ΔS_surr………………………………..(ii)

If a reaction is carried out at constant temperature and pressure, and heat equal to ΔH is given out to the surrounding, then,

ΔS_surr = qrev/T = – ΔH_sys/T…………………………….(iii)

Substituting equation (iii) in equation (ii), we get,

ΔS_total = ΔS_sys – (ΔH_sys/T)

Since, all the terms on the right refer to the system, hence the subscript sys is dropped. The above equation than can be written as

T ΔS_total = TΔS – ΔH

Or, -T ΔS_total = ΔH – TΔS………………………………..(iv)

We know that,

G = H – TS

Or, ΔG = ΔH – TΔS – SΔT

For a change conducted at constant temperature and pressure, since ΔT = 0, hence

ΔG = ΔH – TΔS…………………………………………….(v)

Comparing equation (v) and (iv) one gets,

ΔG = – TΔS_total…………………………………………..(vi)

Thus, from above equation we can say that ΔG can also be used for predicting the direction of spontaneity.

References: 

i) https://goldbook.iupac.org/terms/index/A

ii) https://link.springer.com/article/10.1557/mrs.2019.162

iii) https://chemistry.stackexchange.com/questions/23541/gibbs-free-energy-can-be-expressed-as-a-function-of-p-t-and-n-but-are-enthalpy-a

Gibbs’ Free Energy Function