# Derive the relation between ΔG° and equilibrium constant (K) for the reaction - aA_bB ⇌ cC+dD. - Chemistry

Derive the relation between DeltaG^@and equilibrium constant (K) for the reaction -

aA_bB ⇌ cC+dD.

#### Solution

The free energy (G) of any substance at a temperature T is represented as

G=G^@+RTln[

A+B ⇌ C+D

GA, GB, GC and GD are standard free energies

G_A=G_A^0+RTln[A]

G_B=G_B^0+RTln[B]

G_C=G_c^0+RTln[C]

G_D=G_D^0+RTln[D]

:.DeltaG=sumG_"product"-sumG_"reactant"

=[G_C+G_D]-[G_A+G_B]

={G_C^0+RTln[C]+G_D^0+RTLn[D]}-{G_A^0+RTln[A]+G_B^0+RTln[B]}

{(G_C^0+G_D^0)-(G_A^0+G_B^0)}+{RTln[C]+RTln[D]-RTln[A]+RTln[B]}

if DeltaG^0={(G_C^0+G_D^0)-(G_A^0+G_B^0)}

:.DeltaG=DeltaG^0+(RTln[C]xx[D]-RTln[A]xx[B])

DeltaG=DeltaG^0+RTln

or DeltaG=DeltaG^0+RTlnQ

Q=(["product"])/(["reactant"])

Q=K=([C]xx[D])/([A]xx[B])

Hence from above equation

DeltaG=DeltaG^0+RTlnk

since at equillibrium ΔG=0

:.O=DeltaG^0+RTlnK

:.DeltaG^0=-RTlnK

or

:.DeltaG^0=-2.303RTlog_10K

Concept: Chemical Thermodynamics and Energetic - Equilibrium Constant
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