= Activation energy
{title2=$E_a$}
{wiki}
= Arrhenius activation energy
{c}
{synonym}
A small increase in <temperature> can produce a large increase in a <reaction rate>. In the simplest <Arrhenius equation> model, warming from $300$ to $310\,\mathrm K$ increases a <rate constant> with $E_a=80\,\mathrm{kJ\,mol^{-1}}$ by about $2.81$ times, even though the absolute <thermodynamic temperature> rises by only about $3.33\%$. This sensitivity helps explain why cooling can slow unwanted <chemical reactions> and why a hot process can become difficult to control. The comparison assumes the same <rate law> and <reactant> <concentrations>; the <activation energy> characterizes $k(T)$, rather than every change in the measured <reaction rate>.
The physical motivation is the distribution of molecular <energies>. Warming changes the <Boltzmann distribution>, increasing the relative population of energetically costly configurations. A <transition state> can therefore become much more accessible than the modest change in average <kinetic energy> might suggest. The <exponential function> in an <Arrhenius equation> captures such sensitivity in a simple model; it is not a universal formula for the fraction above every barrier in a <Maxwell-Boltzmann distribution>.<Chemistry 2e by Paul Flowers and others/12.5 Collision Theory>[§12.5, Activation Energy and the Arrhenius Equation: energy distributions and temperature sensitivity]{ref}
<Activation energy> quantifies how strongly a <rate constant> depends on <thermodynamic temperature>. For a positive <rate constant> $k(T)$, the empirical definition is
$$
E_a(T)=RT^2\frac{d}{dT}\ln\frac{k(T)}{k_{\mathrm{ref}}},
$$
where $R$ is the <molar gas constant> and $k_{\mathrm{ref}}>0$ is a fixed reference with the same units as $k$. The ratio is a <dimensionless quantity>, keeping the <natural logarithm> meaningful without changing its <derivative>. Thus $E_a$ is a molar <energy>, commonly in $\mathrm{J\,mol^{-1}}$ or $\mathrm{kJ\,mol^{-1}}$. The <Arrhenius equation> makes $E_a$ constant over its assumed range. This rate-based definition, and the need for care when identifying it with a threshold on a <potential energy surface>, are stated in the <IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry/activation energy>.
In a simple barrier picture, reaching a <transition state> costs <energy> relative to the starting <reactants>. The height of that barrier is different from the difference between the <reactants> and <reaction products>. An energetically favourable endpoint can still be reached slowly. An <energy profile (chemistry)> makes the distinction visible, while an <Arrhenius plot> tests temperature sensitivity from <rate constants>. A measured $E_a$ need not equal a barrier read from an arbitrarily chosen diagram.
For a computed example, assume a <first-order reaction> has $k(300\,\mathrm K)=0.100\,\mathrm{s^{-1}}$, constant $E_a=80\,\mathrm{kJ\,mol^{-1}}$, and a temperature-independent <pre-exponential factor>. The <Arrhenius equation> gives
$$
\frac{k(310\,\mathrm K)}{k(300\,\mathrm K)}
=\exp\left[\frac{80000}{8.314462618}\left(\frac1{300}-\frac1{310}\right)\right]
\approx2.81398.
$$
Hence $k(310\,\mathrm K)\approx0.281398\,\mathrm{s^{-1}}$. Its <half-life> falls from $\ln2/0.100\approx6.93147\,\mathrm s$ to $\ln2/0.281398\approx2.46322\,\mathrm s$. These are chosen model parameters, not measurements of an identified <chemical reaction>. The exponent uses <kelvin>, not unconverted Celsius temperatures, and $80000\,\mathrm{J\,mol^{-1}}$ must match $R$ in $\mathrm{J\,mol^{-1}\,K^{-1}}$.
\Image[/activation-energy-potential-barriers.png]
{title=Synthetic <potential-energy profiles>: starting levels determine directional <reaction energy barriers>}
{id=image-activation-energy-potential-barriers}
{description=Original chosen profiles have <reactant> level $0$, <reaction product> level $-20\,\mathrm{kJ\,mol^{-1}}$, and peak levels $80$ and $50\,\mathrm{kJ\,mol^{-1}}$. Forward <reaction energy barriers> are $80$ and $50$; reverse <reaction energy barriers> are $100$ and $70\,\mathrm{kJ\,mol^{-1}}$. The <reaction coordinate> is not time. These curves are illustrative functions, not laboratory measurements or calculated molecular <potential energy surfaces>; their heights do not by themselves specify <Arrhenius activation energies>.}
A microscopic <reaction energy barrier> belongs to a specified path on a <potential energy surface>. An experimental <activation energy> belongs to a specified <rate constant> and its <temperature> dependence. A multi-step <reaction mechanism> can change which step controls the observed conversion, and a temperature-dependent <pre-exponential factor> contributes to the observed <activation energy>. Even the <Gibbs energy of activation> of <transition state theory> is a different quantity: it incorporates <entropy> as well as <enthalpy> and depends on the <standard states>. These distinctions prevent a fitted <Arrhenius plot> from being mistaken for a direct microscopic energy measurement.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry/activation energy>{ref}
<Catalysis> illustrates the separation between a path and its endpoints. At the same <temperature> and with the same <reactants>, <reaction products>, and <standard states>, a regenerated <catalyst> changes the available <reaction mechanism> and <rate constants>, while the overall standard change in <Gibbs free energy> and the <equilibrium constant> remain the same:
$$
K=\exp\left(-\frac{\Delta_rG^\circ}{RT}\right).
$$
For a reversible <first-order reaction> pair $\mathrm A\rightleftharpoons\mathrm B$, consistency at <chemical equilibrium> requires $k_f/k_r=K$. A model that multiplies only $k_f$ by a catalytic speedup while holding $k_r$ fixed would change this ratio and misrepresent <catalysis>. This simple ratio applies to that specified <rate law>; a multi-step catalytic <reaction mechanism> needs its own rate equations. A <potential-energy profile> alone cannot determine $K$, because its endpoint difference is not generally the standard change in <Gibbs free energy>.<Chemistry 2e by Paul Flowers and others/13.3 Shifting Equilibria: Le Châtelier’s Principle>[§13.3, Effect of a Catalyst: faster equilibration without changing equilibrium composition]{ref}
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