= Apparent activation energy
{title2=$E_{a,\mathrm{app}}(T)=-R\,d\ln(k/k_{\mathrm{ref}})/d(1/T)$}
An <apparent activation energy> reports the <temperature> sensitivity of an observed <rate constant>, without assuming that it equals one microscopic <reaction energy barrier>. The <chain rule> converts the empirical <activation energy> definition into the local slope of an <Arrhenius plot>:
$$
E_{a,\mathrm{app}}(T)=-R\frac{d\ln(k/k_{\mathrm{ref}})}{d(1/T)}.
$$
The fixed reference $k_{\mathrm{ref}}$ keeps the <natural logarithm> dimensionless. A fitted straight-line slope estimates this quantity over the chosen <temperature> interval; curvature means that one number does not describe every local slope.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry/activation energy>{ref}
For the explicit model $k(T)=A(T)\exp[-E_0/(RT)]$ with constant exponent parameter $E_0$, differentiation gives
$$
E_{a,\mathrm{app}}(T)=E_0+RT^2\frac{d}{dT}\ln\frac{A(T)}{A(T_*)},
$$
where $T_*$ is fixed. The second term demonstrates why a microscopic interpretation needs more than a slope. If $E_0$ were itself allowed to depend on <temperature>, differentiation would also contribute $-T\,dE_0/dT$; simply inserting a variable $E_0(T)$ and naming it the measured <activation energy> is generally incorrect.
An instructive <transition state theory> example for an effective <first-order reaction> is
$$
k(T)=\frac{k_{\mathrm B}T}{h}\exp\left(\frac{\Delta S^{\ddagger\circ}}{R}\right)\exp\left(-\frac{\Delta H^{\ddagger\circ}}{RT}\right).
$$
Here $k_{\mathrm B}$ and $h$ are the <Boltzmann constant> and <Planck constant>; $\Delta S^{\ddagger\circ}$ is the <entropy of activation>, and $\Delta H^{\ddagger\circ}$ the <enthalpy of activation>. With those two quantities constant over the chosen range and successful passage assumed, direct differentiation gives $E_{a,\mathrm{app}}=\Delta H^{\ddagger\circ}+RT$, while the <Gibbs energy of activation> is $\Delta G^{\ddagger\circ}=\Delta H^{\ddagger\circ}-T\Delta S^{\ddagger\circ}$. These are three distinguishable quantities. The stated <first-order reaction> and fixed <standard states> matter when comparing these formulas with measurements.<Glossary of Terms Used in Physical Organic Chemistry (IUPAC Recommendations 1994): E entries>{ref}
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