= Arrhenius equation
{c}
{title2=$k(T)=A\exp[-E_a/(RT)]$}
{wiki}
The <Arrhenius equation> models the <temperature> dependence of a <rate constant> by
$$
k(T)=A\exp\left(-\frac{E_a}{RT}\right),\qquad A>0,\quad T>0.
$$
Here $A$ is the <pre-exponential factor>, $E_a$ the <activation energy>, $R$ the <molar gas constant>, and $T$ the absolute <thermodynamic temperature>. In its original form, $A$ and $E_a$ are held constant over the range being modeled. It is an empirical relationship, not a universal rule for every <reaction mechanism>. See the <IUPAC Compendium of Chemical Terminology (Gold Book) by International Union of Pure and Applied Chemistry/Arrhenius equation>.
Eliminating $A$ between two positive <temperatures> gives the useful prediction
$$
\ln\frac{k(T_2)}{k(T_1)}
=\frac{E_a}{R}\left(\frac1{T_1}-\frac1{T_2}\right).
$$
For $E_a>0$, warming increases $k$. The logarithmic sensitivity is $d\ln(k/k_{\mathrm{ref}})/dT=E_a/(RT^2)$, so there is no universal doubling per ten degrees. The same $10\,\mathrm K$ rise from $300$ to $310\,\mathrm K$ gives a factor $1.67749$ for $E_a=40\,\mathrm{kJ\,mol^{-1}}$, versus $2.81398$ for $80\,\mathrm{kJ\,mol^{-1}}$.
\Image[/arrhenius-temperature-sensitivity.png]
{title=Temperature sensitivity and straight-line <Arrhenius plots> for two illustrative <activation energies>}
{description=Both models are anchored at $k(300\,\mathrm K)=0.100\,\mathrm{s^{-1}}$, so they have different <pre-exponential factors>. Left: the higher <activation energy> produces a larger relative change with <temperature>. Right: the <natural logarithm> of the dimensionless ratio $k(T)/k(300\,\mathrm K)$ is linear in inverse <temperature>. These exact model curves are not experimental data.}
The useful simplification is that a strong change in <reaction rate> becomes a straight-line relation after changing variables. Measurements at several <temperatures> can estimate the slope and test whether one pair of constants describes the chosen range. Curvature can indicate a failure of that approximation; fitting a line does not uniquely establish a microscopic <reaction mechanism>.
Svante Arrhenius's 1889 paper on <sucrose> conversion through <catalysis> by a <chemical acid> compared the strong observed <temperature> dependence with the much smaller increase in collision frequency. He discussed van't Hoff's earlier <temperature> relations and proposed a small population of hypothetical active <sucrose>, replenished from ordinary <sucrose>, to interpret the dependence. The inspected translated excerpt derives an inverse-<temperature> relation in its equations (6)–(7) and then applies it to the <reaction rate>. That historical hypothesis is distinct from a modern <transition state> description. See the <On the Reaction Velocity of the Inversion of Cane Sugar by Acids>. The excerpt omits parts of the original paper; it does not provide a complete original apparatus description or dataset.
The <pre-exponential factor> must have the same units as the <rate constant>, since the <exponential function> supplies a <dimensionless quantity>. For a <rate law> $v=k[\mathrm A]^m$, both $A$ and $k$ have units $(\mathrm{mol\,L^{-1}})^{1-m}\,\mathrm{s^{-1}}$. Thus a <first-order reaction> uses $\mathrm{s^{-1}}$, whereas a <second-order reaction> uses $\mathrm{L\,mol^{-1}\,s^{-1}}$. Calling $A$ a frequency factor does not give it frequency units for every <order of reaction>. Its value includes the physical and geometrical factors allowed by the chosen <reaction mechanism>; equal <activation energies> need not mean equal <reaction rates>.
\Image[/arrhenius-apparent-energy-and-prefactor.png]
{title=Synthetic <Arrhenius plots>: a constant <pre-exponential factor> and a temperature-dependent <pre-exponential factor>}
{id=image-arrhenius-apparent-energy-and-prefactor}
{description=Both chosen <first-order reaction> models have $k(300\,\mathrm K)=0.100\,\mathrm{s^{-1}}$ and the same exponent parameter $E_0=60\,\mathrm{kJ\,mol^{-1}}$. Blue uses a constant <pre-exponential factor>; orange multiplies it by $(T/300\,\mathrm K)^2$. All markers are synthetic evaluations, not observations. Left: <natural logarithm> of $k/k(300\,\mathrm K)$ against $1000\,\mathrm K/T$. Right: the corresponding <apparent activation energies>, $60$ and $60+2RT/1000$ in $\mathrm{kJ\,mol^{-1}}$. The dashed orange tangent at $300\,\mathrm K$ has numerical slope $-7.81634$ on the displayed horizontal scale.}
To read the blue <Arrhenius plot>, define the <dimensionless quantity> $X=1000\,\mathrm K/T$ and $y=\ln[k/k(300\,\mathrm K)]$. Its numerical slope is $m_X=-60\,000/(1000R)\approx-7.21634$, giving $E_a=-(1000\,\mathrm K)Rm_X=60\,\mathrm{kJ\,mol^{-1}}$. The slope against $1/T$, by contrast, is $-7216.34\,\mathrm K$. The orange tangent at $300\,\mathrm K$ gives $E_{a,\mathrm{app}}\approx64.9887\,\mathrm{kJ\,mol^{-1}}$; its finite-interval slope between $280$ and $340\,\mathrm K$ gives about $65.1227\,\mathrm{kJ\,mol^{-1}}$. The distinction between a local <derivative> and a fitted interval matters even with perfectly noise-free synthetic evaluations.
Two measured positive <rate constants> give a two-point estimate
$$
E_{a,\mathrm{fit}}=R\frac{\ln[k(T_2)/k(T_1)]}{1/T_1-1/T_2}.
$$
Several <temperatures> let <linear regression> test the constant-parameter approximation: inspect the <regression residuals>, keep the same <rate law> and composition convention, and account for uncertainty in both $k$ and $T$. A straight line over a short range supports a useful local approximation; it does not uniquely identify a <reaction mechanism>. A fixed change of units or of $k_{\mathrm{ref}}$ shifts the intercept without changing the inferred <activation energy>.<Chemistry 2e by Paul Flowers and others/12.5 Collision Theory>[§12.5, Example 12.13 and alternate two-point approach: estimating activation energy from a slope]{ref}
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