Free Energy Change and Equilibrium Constant

Free Energy Change

Gibbs free energy, or free energy ( we can consider as a kind of potential energy), is the maximum amount of energy of a system that can perform some useful work in a physicochemical system. It is equal to the sum of entropy and the product of temperature in a closed system.
Change in free energy (ΔG) can indicate the direction of chemical reaction under two conditions: constant pressure and constant temperature. It can be written as-
G = H − TS
where, H is Enthalpy, T is the Temperature and S is the entropy of the system
or, G = U + PV − TS (As H = U + PV)
or, dG = dU + PdV + VdP − TdS − SdT
or, dG = dq + VdP − TdS − SdT     (As,dU = dU + PdV)
or, dG = TdS + VdP − TdS − SdT     (As,dS = dq/T)
or, dG = VdP − SdT
This equation shows the variation of free energy with temperature and pressure if system undergoes a reversible change. If change in free energy (ΔG) is equal to zero, then the reaction will not occur i.e. reaction is in equilibrium.
If change in free energy (ΔG) is greater than zero, the reaction will shift towards right.
If change in free energy (ΔG) is lesser than zero, the reaction will shift towards left.
If change in free energy (ΔG) is positive, the reaction will occur spontaneously.
If change in free energy (ΔG) is negative, the reaction will not occur spontaneously.

Equilibrium Constant

Chemical equilibrium for a reaction is characterized by its equilibrium constant (K_eq). The value of K_eq is determined by its free energy change under standard conditions, a quantity called the standard free energy change (ΔG°) for that reaction.
Equilibrium constant is the ratio of the concentration of products to the concentration of reactants.
Let us consider a reaction which is in equilibrium state
A + B ⇌ C + D
In this equation, the equilibrium constant-
K_eq = [C][D] / [A][B] K_c

Relation Between Free Energy Change and Equilibrium Constant

Free energy change of the reaction in any state, ΔG (when equilibrium has not been attained) is related to the standard free energy change of the reaction, ΔG° (which is equal to the difference in the free energies of formation of the products and reactants both in their standard states) according to the equation.
ΔG = ΔG° + RT InQ
Where Q is the reaction quotient.
At equilibrium-
∆G = 0 and Q become equal to the equilibrium constant.
Hence the equation becomes-
ΔG° = –RT InK_eq
ΔG° = –2.303 RT logK_eq
K_eq = e^−ΔG°/RT
The above equation gives the relationship between standard Gibbs energy change for the reaction and its equilibrium constant.

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