View source for Activation energy

A concept originating in the theory of chemical reactions. It plays an important role in [[Combustion theory|combustion theory]].

The evolution of a chemical reaction is determined by its specific-reaction rate constant, usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102501.png" />. This quantity depends mainly on the temperature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102502.png" />. For a one-step chemical reaction, its functional dependence is given empirically by the Arrhenius expression: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102503.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102504.png" /> is the universal gas constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102505.png" /> the activation energy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102506.png" /> the frequency factor for the reaction step; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102507.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102508.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a1102509.png" /> may depend weakly (for example, polynomially) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025010.png" />.

At the molecular level, a chemical reaction is a collision process between reactant molecules, from which reaction products emerge. The molecules move on a potential-energy surface, whose shape is determined by a solution of the [[Schrödinger equation|Schrödinger equation]]. A configuration of the reactant molecules corresponds to a local minimum in one region, and a configuration of the reaction products to a local minimum in another region, where the two minima are generally separated by a barrier in the potential-energy surface. At a [[Saddle point|saddle point]] on the barrier, the height of the surface above the energy of the reactant region assumes a minimum value. A collision of the reactant molecules can produce products only if the energy of the reactants (for example, their kinetic energy) exceeds this minimum height. The minimum barrier height defines the activation energy, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025011.png" />. It is the energy the reactants must acquire before they can react. In practice, the activation energy is determined experimentally, by measuring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025012.png" /> at various values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025013.png" /> and making a best straight-line fit through the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025014.png" /> versus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025015.png" />.

==Activation-energy asymptotics.==
Activation-energy asymptotics play an important role in [[Combustion theory|combustion theory]], in the study of diffusion flames.

A diffusion flame is a non-pre-mixed, quasi-steady, nearly isobaric flame, where most of the reaction occurs in a narrow zone. The structure of a diffusion flame, especially in a multi-step reaction, is very complicated, but can be analyzed in model problems by means of activation-energy asymptotics.

The small parameter in activation-energy asymptotics is the reciprocal of the Zel'dovich number. The Zel'dovich number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025016.png" />, is a non-dimensional measure of the temperature sensitivity of the reaction rate:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025017.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025018.png" /></td> </tr></table>

while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025020.png" /> denote the temperature far upstream and far downstream of the moving flame, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025021.png" /> is a reference energy. The graph of the reaction rate has a single peak, which narrows and increases in magnitude and whose location approaches the fully burnt condition (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025022.png" />) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025023.png" />.

The governing equations for the temperature and the concentration of the reaction-limiting component define a singular perturbation problem (cf. also [[Perturbation theory|Perturbation theory]]) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110250/a11025024.png" />. The regular (or outer) expansion yields the temperature and concentration profiles in the convective-diffusive zone outside the flame, and the  "singular"  (or  "inner" ) expansion the profiles in the reactive-diffusive zone inside the flame. The complete profiles are then found by the method of matched asymptotic expansions (cf. [[Perturbation theory|Perturbation theory]]).

Activation-energy asymptotics were first introduced in combustion theory by Yu.B. Zel'dovich and D.A. Frank-Kamenetskii [[#References|[a1]]]. The method was formalized in [[#References|[a2]]]. Many results are summarized in [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y.B. Zel'dovich,   D.A. Frank-Kamenetskii,   "The theory of thermal flame propagation"  ''Zhur. Fiz. Khim.'' , '''12'''  (1938)  pp. 100  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.B. Bush,   F.E. Fendell,   "Asymptotic analysis of laminar flame propagation for general Lewis numbers"  ''Comb. Sci. and Technol.'' , '''1'''  (1970)  pp. 421</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Ya.B. Zel'dovich,   G.I. Barenblatt,   V.B. Librovich,   G.M. Makhviladze,   "The mathematical theory of combustion and explosions" , Consultants Bureau  (1985)  (In Russian)</TD></TR></table>