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The Arrhenius equation models the temperature dependence of a rate constant byHere $A$ 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 Arrhenius equation.

Eliminating $A$ between two positive temperatures gives the useful predictionFor $E_a>0$, warming increases $k$. The logarithmic sensitivity is $d\ln (k/k_{\mathrm{ref}})/dT=E_a/(R{T}^2)$, 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 $E_a=40\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, versus $2.81398$ for $80\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$.

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 $(\mathrm{mol}{\mathrm{L}}^{-1}{)}^{1-m}{\mathrm{s}}^{-1}$. Thus a first-order reaction uses ${\mathrm{s}}^{-1}$, whereas a second-order reaction uses $\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$. 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.

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 $m_X=-60000/(1000R)\approx -7.21634$, giving $E_a=-(1000\mathrm{K})R{m}_X=60\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. The slope against $1/T$, by contrast, is $-7216.34\mathrm{K}$. The orange tangent at $300\mathrm{K}$ gives $E_{a,\mathrm{app}}\approx 64.9887\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$; its finite-interval slope between $280$ and $340\mathrm{K}$ gives about $65.1227\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. 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 estimateSeveral 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.^[ref]