Activation energy, usually denoted (E_a), is a quantity in chemical kinetics that describes how strongly a reaction’s rate constant depends on temperature. It is commonly pictured as the energy barrier that reactants must overcome during a chemical reaction. Strictly, however, the experimentally defined Arrhenius activation energy and the height of a microscopic energy barrier are distinct quantities that need not be identical. IUPAC therefore defines activation energy operationally through the temperature dependence of the rate coefficient. (goldbook.iupac.org)
Definition and units
The general definition is
[ E_a(T)=RT^2\frac{d\ln k}{dT} =-R\frac{d\ln k}{d(1/T)}, ]
where (k) is the reaction rate constant, (T) is absolute temperature, and (R) is the molar gas constant. This derivative defines a local temperature sensitivity: activation energy can vary with temperature rather than being an invariant property of a reaction. A positive value means that (k) increases with temperature; a negative value means that it decreases over the interval considered. (goldbook.iupac.org)
Molar activation energies are commonly reported in kilojoules per mole, with joules per mole used in equations involving SI values of (R). Temperature must be expressed in kelvins. An activation energy quoted per individual molecular event must instead be paired with the Boltzmann constant, (k_B), rather than (R). Consistency between molar and molecular quantities is essential. (openstax.org)
The Arrhenius equation
Many reactions approximately follow the Arrhenius equation over a limited temperature range:
[ k=A\exp\left(-\frac{E_a}{RT}\right). ]
Here (A) is the pre-exponential factor, with the same units as (k). In the simplest treatment, both (A) and (E_a) are temperature-independent. The exponential function makes the rate constant particularly sensitive to temperature when (E_a) is large relative to (RT). The equation can describe both individual reaction steps and composite reactions, although its parameters have different mechanistic interpretations in those cases. (moureu.iupac.org)
In collision theory, (A) incorporates collision frequency and favorable orientation, while the exponential factor represents the energetic restriction on successful collisions. Increasing temperature increases the fraction of energetic encounters; it does not simply lower a fixed barrier. At a given temperature, a smaller activation energy produces a larger rate constant only if other factors, including (A), are comparable. (openstax.org)
Molecular interpretation
A reaction involves rearrangement of chemical bonds and the relative positions of atoms within or between molecules. On a simplified potential-energy diagram, reactants and products occupy lower-energy regions separated by a maximum along the reaction pathway. The corresponding transition state is not a stable, isolable intermediate. In a multidimensional description, the relevant barrier is associated with a saddle point on a potential-energy surface. (openstax.org)
This picture explains why energetically favorable reactions can nevertheless proceed slowly. The energy difference between products and reactants is not the same as the barrier separating them. Moreover, an electronic potential-energy barrier, a thermally averaged activation energy, and a free-energy barrier describe different aspects of the process. Their numerical values should not be treated as interchangeable without specifying the theoretical model and experimental conditions. (goldbook.iupac.org)
Relation to activation thermodynamics
Transition-state theory relates rates to the thermodynamic cost of reaching a transition-state ensemble. Its activation quantities include enthalpy of activation, (\Delta H^\ddagger), entropy of activation, (\Delta S^\ddagger), and Gibbs energy of activation:
[ \Delta G^\ddagger =\Delta H^\ddagger-T\Delta S^\ddagger. ]
For a unimolecular reaction, a conventional form of the Eyring equation is
[ k=\kappa\frac{k_BT}{h} \exp\left(-\frac{\Delta G^\ddagger}{RT}\right), ]
where (h) is the Planck constant and (\kappa) is a transmission coefficient. For reactions involving multiple reactant molecules, appropriate standard-state factors must be included. (comp.chem.umn.edu)
Under the usual unimolecular assumptions, including a temperature-independent transmission coefficient, differentiating this expression gives (E_a=\Delta H^\ddagger+RT). Thus Arrhenius activation energy is related to, but differs from, activation enthalpy. Activation Gibbs energy additionally incorporates entropy and depends on the specified standard state. (iupac.qmul.ac.uk)
Experimental determination
Activation energy is commonly determined by measuring rate constants at several controlled temperatures. Taking logarithms gives
[ \ln k=\ln A-\frac{E_a}{R}\frac{1}{T}. ]
If Arrhenius behavior holds, plotting (\ln k) against (1/T) yields a straight line with slope (-E_a/R). A linear regression estimates the slope and intercept. Two measurements also permit the calculation
[ E_a= R\frac{\ln(k_2/k_1)}{1/T_1-1/T_2}. ]
Several temperatures allow departures from linearity to be detected, whereas two points alone cannot establish that the relationship remains linear throughout an interval. (openstax.org)
For a multistep reaction mechanism, the measured value may be an overall or apparent activation energy rather than the barrier of one elementary step. The reported temperature range and the definition of the measured rate coefficient are consequently important parts of the result. (moureu.iupac.org)
Catalysis and applications
Catalysis accelerates reactions by providing an alternative mechanism, commonly represented by a pathway with lower activation barriers. A catalyst participates in individual steps and is regenerated in the overall process. Enzymes provide biological examples, while solid catalysts supply reactive surfaces where adsorption, transformation, and product desorption occur. Catalyzed and uncatalyzed pathways can have different intermediates and different numbers of steps. (openstax.org)
Arrhenius-type activation energies also characterize processes outside ordinary chemical reactions, including diffusion and material degradation. In reliability engineering, temperature-dependent failure models use activation energies to relate accelerated tests to lower-temperature conditions. Such extrapolation presupposes that the same failure mechanism remains operative; a change of mechanism can invalidate a single Arrhenius description. (itl.nist.gov)