Eyring Equation

Eyring Equation

Eyring Equation, developed by Henry Eyring in 1935, is based on transition state theory and is used to describe the relationship between reaction rate and temperature. It is similar to the Arrhenius Equation, which also describes the temperature dependence of reaction rates. However, whereas Arrhenius Equation can be applied only to gas-phase kinetics, the Eyring Equation is useful in the study of gas, condensed, and mixed-phase reactions that have no relevance to the collision model.
The equation of this theory of absolute reaction rate can be expressed in terms of thermodynamical functions instead of partition functions. Eyring and his coworkers derived the thermodynamical formulations of reaction reate.
Consider the following reaction-

where [AB]^* is the activated complex and K^* is the equilibrium constant between the reactants and activated complex.
From classical mechanics, the energy of vibration is given by-
RT/N_A (or k_BT where k_B is the Boltzmann constant) whereas from quantum mechanics, it is given by hν so that-
hν = RT/N_A
ν = RT/N_Ah.
The vibrational frequency ν is the rate at which the activated complex molecules move across the energy barrier. Thus, the rate constant k₂ can be identified with ν.
The reaction rate is given by-
Rate = -d[A]/dt = Kk₂[AB]^*
         = k(k_BT/h)[AB]^*          -----Equation-1
where the factor K, called the transmission coefficient, is a measure of the probability that a molecule, once it passes over the barrier, will keep on going ahead and not return. The value of K is taken to be unity; it is thus omitted from the rate expression. The concentration of the activated complex, [AB]^*, can be obtained by writing the equilibrium expression-
K^* = [AB]^*/[A][B]
whence, [AB]^* = K^*[A][B]          -----Equation-2
From equation-1 and equation-2, we have-
-d[A]/dt = (k_BT/h).K^*[A][B]
Thus the rate constant K₂ may be expressed as-
K₂ = (k_BT/h).K^*          -----Equation-3
The equilibrium constant K^* may be expressed in terms of (ΔG^o)^*, called standard Gibbs free energy of activation.
(ΔG^o)^* = − RTlnK^*          -----Equation-4
and (ΔG^o)^* = (ΔH^o)^* − T(ΔS^o)^*          -----Equation-5
From equation-3 and equation-4, we have-
K^* = e^{−(ΔGo)*/RT}          -----Equation-6
or, K^* = e^(ΔSo)*/R. e^{− (ΔHo)*/RT}          -----Equation-7
Now from equation-3 and equation-7, we have-
K₂ = (k_BT/h).e^(ΔSo)*/R. e^{− (ΔHo)*/RT}          -----Equation-8
the equation-8 is known as Eyring equation.

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