Describe Wynne-Jones and Eyring Treatment for the Thermodynamic Formulation of Reaction Rates.
The Wynne-Jones and Eyring treatment provides a bridge between the kinetic approach of Transition State Theory (TST) and classical thermodynamics. It allows us to express the rate of a chemical reaction in terms of standard thermodynamic functions like enthalpy, entropy, and Gibbs free energy of activation.
1. The Fundamental Eyring Equation
According to Transition State Theory, a reaction proceeds through an unstable intermediate called the activated complex (denoted by the double dagger symbol, ‡). The rate of reaction depends on the concentration of this complex and the frequency at which it converts into products.
The rate constant ($k$) is expressed as:
$$k = \frac{k_B T}{h} K^\ddagger$$
Where:
- $k_B$: Boltzmann constant
- $T$: Absolute temperature
- $h$: Planck’s constant
- $K^\ddagger$: Equilibrium constant for the formation of the activated complex
2. Thermodynamic Relationship
Wynne-Jones and Eyring related the equilibrium constant $K^\ddagger$ to the standard Gibbs free energy of activation ($\Delta G^\ddagger$) using the standard thermodynamic relation:
$$\Delta G^\ddagger = -RT \ln K^\ddagger$$ Rearranging for $K^\ddagger$: $$K^\ddagger = e^{-\frac{\Delta G^\ddagger}{RT}}$$
Substituting this into the Eyring equation:
$$k = \frac{k_B T}{h} e^{-\frac{\Delta G^\ddagger}{RT}}$$
3. Enthalpy and Entropy of Activation
Since $\Delta G^\ddagger = \Delta H^\ddagger - T\Delta S^\ddagger$, we can further expand the rate equation:
$$k = \frac{k_B T}{h} e^{\left(\frac{T\Delta S^\ddagger - \Delta H^\ddagger}{RT}\right)}$$
This gives the final Wynne-Jones and Eyring formulation:
$$k = \frac{k_B T}{h} e^{\frac{\Delta S^\ddagger}{R}} e^{-\frac{\Delta H^\ddagger}{RT}}$$
4. Comparison with Arrhenius Parameters
By comparing the Eyring equation with the Arrhenius equation ($k = A e^{-E_a/RT}$), we can define the pre-exponential factor ($A$) and activation energy ($E_a$) in thermodynamic terms:
| Parameter | Thermodynamic Equivalent |
|---|---|
| Activation Energy ($E_a$) | $\Delta H^\ddagger + RT$ (for gases) |
| Frequency Factor ($A$) | $\frac{e \cdot k_B T}{h} e^{\Delta S^\ddagger / R}$ |
Significance of Entropy of Activation ($\Delta S^\ddagger$): A positive $\Delta S^\ddagger$ suggests the activated complex is more disordered than the reactants (often seen in unimolecular dissociations), while a negative value suggests a more "tight" or ordered transition state (common in bimolecular associations).