Arrhenius equation

Arrhenius argued that for reactants to transform into products, they must first acquire a minimum amount of energy, called the activation energy E_a. At an absolute temperature T, the fraction of molecules that have a kinetic energy greater than E_a can be calculated from statistical mechanics. The concept of activation energy explains the exponential nature of the relationship, and in one way or another, it is present in all kinetic theories.

The calculations for reaction rate constants involve an energy averaging over a Maxwell–Boltzmann distribution with ${\displaystyle E_{\rm {a}}}$ as lower bound and so are often of the type of incomplete gamma functions, which turn out to be proportional to ${\displaystyle e^{\frac {-E_{\rm {a}}}{RT}}}$.

One approach is the collision theory of chemical reactions, developed by Max Trautz and William Lewis in the years 1916–18. In this theory, molecules are supposed to react if they collide with a relative kinetic energy along their line of centers that exceeds E_a. The number of binary collisions between two unlike molecules per second per unit volume is found to be[10]

$${\displaystyle z_{AB}=N_{\rm {A}}d_{AB}^{2}{\sqrt {\frac {8\pi k_{\rm {B}}T}{\mu _{AB}}}},}$$

where N_A is the Avogadro constant, d_AB is the average diameter of A and B, T is the temperature which is multiplied by the Boltzmann constant k_B to convert to energy units, and μ_AB is the reduced mass.

The rate constant is then calculated as ${\displaystyle k=z_{AB}e^{\frac {-E_{\rm {a}}}{RT}},}$ so that the collision theory predicts that the pre-exponential factor is equal to the collision number z_AB. However for many reactions this agrees poorly with experiment, so the rate constant is written instead as ${\displaystyle k=\rho z_{AB}e^{\frac {-E_{\rm {a}}}{RT}},}$. Here ${\displaystyle \rho }$ is an empirical steric factor, often much less than 1.00, which is interpreted as the fraction of sufficiently energetic collisions in which the two molecules have the correct mutual orientation to react.[10]

The Eyring equation, another Arrhenius-like expression, appears in the "transition state theory" of chemical reactions, formulated by Wigner, Eyring, Polanyi and Evans in the 1930s. The Eyring equation can be written:

$${\displaystyle k={\frac {k_{\rm {B}}T}{h}}e^{-{\frac {\Delta G^{\ddagger }}{RT}}}={\frac {k_{\rm {B}}T}{h}}e^{\frac {\Delta S^{\ddagger }}{R}}e^{-{\frac {\Delta H^{\ddagger }}{RT}}},}$$

where ${\displaystyle \Delta G^{\ddagger }}$ is the Gibbs energy of activation, ${\displaystyle \Delta S^{\ddagger }}$ is the entropy of activation, ${\displaystyle \Delta H^{\ddagger }}$ is the enthalpy of activation, ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant, and ${\displaystyle h}$ is Planck's constant.[11]

At first sight this looks like an exponential multiplied by a factor that is linear in temperature. However, free energy is itself a temperature dependent quantity. The free energy of activation ${\displaystyle \Delta G^{\ddagger }=\Delta H^{\ddagger }-T\Delta S^{\ddagger }}$ is the difference of an enthalpy term and an entropy term multiplied by the absolute temperature. The pre-exponential factor depends primarily on the entropy of activation. The overall expression again takes the form of an Arrhenius exponential (of enthalpy rather than energy) multiplied by a slowly varying function of T. The precise form of the temperature dependence depends upon the reaction, and can be calculated using formulas from statistical mechanics involving the partition functions of the reactants and of the activated complex.

Both the Arrhenius activation energy and the rate constant k are experimentally determined, and represent macroscopic reaction-specific parameters that are not simply related to threshold energies and the success of individual collisions at the molecular level. Consider a particular collision (an elementary reaction) between molecules A and B. The collision angle, the relative translational energy, the internal (particularly vibrational) energy will all determine the chance that the collision will produce a product molecule AB. Macroscopic measurements of E and k are the result of many individual collisions with differing collision parameters. To probe reaction rates at molecular level, experiments are conducted under near-collisional conditions and this subject is often called molecular reaction dynamics.[12]

Another situation where the explanation of the Arrhenius equation parameters fall short is in heterogeneous catalysis, especially for reactions that show Langmuir-Hinshelwood kinetics. Clearly, molecules on surfaces do not "collide" directly, and a simple molecular cross-section does not apply here. Instead, the pre-exponential factor reflects the travel across the surface towards the active site.[13]

There are deviations from the Arrhenius law during the glass transition in all classes of glass-forming matter.[14] The Arrhenius law predicts that the motion of the structural units (atoms, molecules, ions, etc.) should slow down at a slower rate through the glass transition than is experimentally observed. In other words, the structural units slow down at a faster rate than is predicted by the Arrhenius law. This observation is made reasonable assuming that the units must overcome an energy barrier by means of a thermal activation energy. The thermal energy must be high enough to allow for translational motion of the units which leads to viscous flow of the material.