This equation is based on the concept of the existence of a transition state or an activated complex an intermediate state in all chemical reaction.
According to Eyring and Polanyi equation,
(1) For any reaction to occur, the reactant molecules having sufficient energy should approach each other to form a short lived unstable intermediate (AB ≠ ) which is known as activated complex.
(2) This activated complex is having excess energy, much higher than the energy of reactants and products.
(3) This activated complex is in thermodynamic equilibrium with the reactants.
(4) Let us consider the reactants A, B obey second order kinetics.
A + B κ > C [n=2]
According to Eyring,
A + B ⇔AB ≠ > C
The activated complex have all the properties of the normal molecule in every respect but it differs from other molecule in the number of transitional and Vibrational degree of freedom of movement.
(5) The normal molecule has a definite number of translational and vibrational degree of freedom depending on the number of atoms in the molecule.
Since the activated complex has high energy, it's constantly vibrating and due to it's vibration one of it's vibrational degree of freedom, is last and it's converted into translational degree of freedom.
Therefore the activated complex has one vibrational degree of freedom lesser and one translational degree of freedom more than the normal molecule.
(6) The rate of a reaction depends on, the frequency of vibration of AB ≠ and the concentration of AB ≠ .
Rate ∝ ʋ⁰
Rate ∝ [ AB ≠]
Rate ∝ ʋ [ AB ≠]
Rate = ʋ [ AB ≠ ] ...........(1)
(7) The frequency of vibration depends on the temperature of reaction.
ʋ ∝ T
ʋ = κB T
h
Rate = κB T [AB ≠] ..........(2)
h
(8) Since , the formation of AB ≠ from the reactant is involved in thermodynamic equilibrium, we can apply the law of mass action.
K ≠ = [ AB≠ ]
[A] [B]
[ AB ≠ ] = k ≠ [ A] [ B ] ...............(3)
Substituting (3) in (2)
Rate = κB T k≠ [ A ] [ B ] .........(4)
h
(9) Since the reaction under study obey the second order kinetics the rate law equation (11) Will be,
Rate ∝ [ A ] [ B ]
Rate = κ [ A ] [ B ] ..............(5)
Comparing equation (4) and (5)
κ [ A ] [ B ] = κB T k≠ [ A ] [ B ]
h
κ = κB T k≠ .............(6)
h
(10) In thermodynamics, the vant half Isotherm is,
∆G ≠ = − RTlnk ≠
lnk ≠ = − ∆G ≠
RT
k ≠ = e −∆G ≠ ..................(7)
RT
Substituting equation (7) in (6)
κ = κB T e −∆G ≠ .......(8)
h RT
(11) According to Gibb's Helmholtz equation,
∆G ≠ = ∆H ≠ − T ∆S ≠
Therefore ,
κ = κB T e ( − ∆H ≠ + T ∆S ≠ )
h RT
κ = κB T e - ∆G ≠ e ∆S ≠ ......(9)
h RT RT
Equations (8) and (9) are known as Eyring equations.
Here,
1) κ − Rate constant for the reaction
2) κB - Boltz mann's constant
3) h - plank's constant
4) T - Temperature of the reaction
5) R - Universal gas constant
6) ∆G ≠ − free energy change for activation
7) ∆H ≠ − Enthalpy change for activation
8) ∆S ≠ − Entropy change for activation
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