Eyring equation

General form

The general form of the Eyring–Polanyi equation somewhat resembles the Arrhenius equation: $${\displaystyle k=\frac{\kappa k_{\mathrm{B}}T}{h}{e}^{-\frac{\mathrm{\Delta }{G}^{‡}}{RT}}}$$ where ${\displaystyle k}$ is the rate constant, ${\displaystyle \mathrm{\Delta }{G}^{‡}}$ is the Gibbs energy of activation, ${\displaystyle \kappa }$ is the transmission coefficient, ${\displaystyle k_{\mathrm{B}}}$ is the Boltzmann constant, ${\displaystyle T}$ is the temperature, and ${\displaystyle h}$ is the Planck constant.

The transmission coefficient ${\displaystyle \kappa }$ is often assumed to be equal to one as it reflects what fraction of the flux through the transition state proceeds to the product without recrossing the transition state. So, a transmission coefficient equal to one means that the fundamental no-recrossing assumption of transition state theory holds perfectly. However, ${\displaystyle \kappa }$ is typically not one because (i) the reaction coordinate chosen for the process at hand is usually not perfect and (ii) many barrier-crossing processes are somewhat or even strongly diffusive in nature. For example, the transmission coefficient of methane hopping in a gas hydrate from one site to an adjacent empty site is between 0.25 and 0.5. Typically, reactive flux correlation function (RFCF) simulations are performed in order to explicitly calculate ${\displaystyle \kappa }$ from the resulting plateau in the RFCF. This approach is also referred to as the Bennett-Chandler approach, which yields a dynamical correction to the standard transition state theory-based rate constant.

It can be rewritten as: $${\displaystyle k=\frac{\kappa k_{\mathrm{B}}T}{h}{e}^{\frac{\mathrm{\Delta }{S}^{‡}}{R}}{e}^{-\frac{\mathrm{\Delta }{H}^{‡}}{RT}}}$$

One can put this equation in the following form: $${\displaystyle \ln \frac{k}{T}=\frac{-\mathrm{\Delta }{H}^{‡}}{R}\cdot \frac{1}{T}+\ln \frac{\kappa k_{\mathrm{B}}}{h}+\frac{\mathrm{\Delta }{S}^{‡}}{R}}$$ where:

• ${\displaystyle k}$ = reaction rate constant
• ${\displaystyle T}$ = absolute temperature
• ${\displaystyle \mathrm{\Delta }{H}^{‡}}$ = enthalpy of activation
• ${\displaystyle R}$ = gas constant
• ${\displaystyle \kappa }$ = transmission coefficient
• ${\displaystyle k_{\mathrm{B}}}$ = Boltzmann constant = R/N_A, N_A = Avogadro constant
• ${\displaystyle h}$ = Planck constant
• ${\displaystyle \mathrm{\Delta }{S}^{‡}}$ = entropy of activation

If one assumes constant enthalpy of activation, constant entropy of activation, and constant transmission coefficient, this equation can be used as follows: A certain chemical reaction is performed at different temperatures and the reaction rate is determined. The plot of ${\displaystyle \ln (k/T)}$ versus ${\displaystyle 1/T}$ gives a straight line with slope ${\displaystyle -\mathrm{\Delta }{H}^{‡}/R}$ from which the enthalpy of activation can be derived and with intercept ${\displaystyle \ln (\kappa k_{\mathrm{B}}/h)+\mathrm{\Delta }{S}^{‡}/R}$ from which the entropy of activation is derived.