= Modified Arrhenius equation
{c}
{title2=$k(T)=A_* (T/T_*)^n\exp[-E_0/(RT)]$}
The <Modified Arrhenius equation> allows a power of <thermodynamic temperature> in the <pre-exponential factor>:
$$
k(T)=A_*\left(\frac{T}{T_*}\right)^n\exp\left(-\frac{E_0}{RT}\right),
\qquad T_*>0.
$$
Here $A_*$, $E_0$, $n$, and $T_*$ are constant model parameters. Using the <dimensionless quantity> $T/T_*$ leaves $A_*$ with the same units as $k$; in the equivalent convention $BT^n\exp[-E_0/(RT)]$, $B$ instead has units of $k$ multiplied by $\mathrm K^{-n}$. The power exponent $n$ here is not an <order of reaction>. The IUPAC definition uses that equivalent $BT^n$ convention.<IUPAC Compendium of Chemical Terminology (Gold Book) by International Union of Pure and Applied Chemistry/modified Arrhenius equation>{ref}
Applying the <apparent activation energy> definition yields the original short calculation
$$
\frac{d\ln(k/k_{\mathrm{ref}})}{dT}=\frac nT+\frac{E_0}{RT^2},
\qquad E_{a,\mathrm{app}}(T)=E_0+nRT.
$$
For the orange synthetic <Arrhenius plot>, $n=2$, $T_*=300\,\mathrm K$, and $E_0=60\,\mathrm{kJ\,mol^{-1}}$. Its <pre-exponential factor> changes by $(310/300)^2\approx1.06778$ from $300$ to $310\,\mathrm K$; combined with the <exponential function>, $k(310\,\mathrm K)\approx0.231991\,\mathrm{s^{-1}}$, compared with $0.217266\,\mathrm{s^{-1}}$ for the constant-<pre-exponential factor> model. The diagram uses those models only to reveal the difference between an exponent parameter and a slope-derived <activation energy>.
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