Arrhenius's concept of activation energy Arrhenius argued that for reactants to transform into products, they must first acquire a minimum amount of energy, called the
activation energy {{tmath|E_\mathrm{a} }}. At an absolute temperature , the fraction of molecules that have a kinetic energy greater than {{tmath|E_\mathrm{a} }} can be calculated from
statistical mechanics. This fraction increases with temperature because molecular energies follow a Maxwell–Boltzmann distribution, which broadens as rises and increases the proportion of molecules with kinetic energies equal to or greater than the activation energy. 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. From a physical perspective, activation energy represents an energy barrier that must be overcome for reactant molecules to reach a transition state. Increasing temperature raises the fraction of molecules with sufficient kinetic energy to overcome this barrier, which explains the strong temperature dependence of reaction rates described by the Arrhenius equation. The calculations for reaction rate constants involve an energy averaging over a
Maxwell–Boltzmann distribution with {{tmath|E_\mathrm{a} }} as lower bound and so are often of the type of
incomplete gamma functions, which turn out to be proportional to {{tmath|\textstyle \exp{\frac{-E_\mathrm{a} }{RT} } }}.
Collision theory 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 {{tmath|E_\mathrm{a} }}. The number of binary collisions between two unlike molecules per second per unit volume is found to be z_{AB} = N_A N_B d_{AB}^2 \sqrt\frac{8 \pi k_\mathrm{B}T}{ \mu_{AB}} , where and are the number densities of and , {{tmath|d_{AB} }} is the average diameter of and , is the temperature which is multiplied by the
Boltzmann constant {{tmath|k_\mathrm{B} }} to convert to energy, and {{tmath|\mu_{AB} }}is the
reduced mass of and . The rate constant is then calculated as {{tmath|1= \textstyle k = z_{AB}\exp{\frac{-E_\mathrm{a} }{RT} } }}, so that the collision theory predicts that the pre-exponential factor is equal to the collision number {{tmath|z_{AB} }}. However for many reactions this agrees poorly with experiment, so the rate constant is written instead as {{tmath|1= \textstyle k = \rho z_{AB}\exp{\frac{-E_\mathrm{a} }{RT} } }}. Here 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.
Transition state theory The
Eyring equation, another Arrhenius-like expression, appears in the "
transition state theory" of chemical reactions, formulated by
Eugene Wigner,
Henry Eyring,
Michael Polanyi and
M. G. Evans in the 1930s. The Eyring equation can be written: k = \frac{k_\mathrm{B}T}{h} e^{-\frac{\Delta G^\ddagger}{RT}} = \frac{k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger}{R}}e^{-\frac{\Delta H^\ddagger}{RT}}, where \Delta G^\ddagger is the
Gibbs energy of activation, \Delta S^\ddagger is the
entropy of activation, \Delta H^\ddagger is the
enthalpy of activation, k_\mathrm{B} is the
Boltzmann constant, and h is the
Planck constant. At first sight this looks like an exponential multiplied by a factor that is in temperature. However, free energy is itself a temperature-dependent quantity. The free energy of activation \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 . 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.
Limitations of the idea of Arrhenius activation energy Both the Arrhenius activation energy and the rate constant 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 and . 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 . Macroscopic measurements of and 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. Another situation where the explanation of the Arrhenius equation parameters falls 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. There are deviations from the Arrhenius law during the
glass transition in all classes of glass-forming matter. 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. == See also ==