Temperature dependence of reaction rates
Reaction speed often increases with temperature because more collisions have enough energy to overcome an energy
barrier. The Effect of Temperature on Reaction Rates (Arrhenius Equation) uses measured rate constants to
compute activation energy and the pre-exponential factor from a straight-line fit.
Core definitions and essential formulas
Many reactions follow the Arrhenius relationship between the rate constant \(k\) and absolute temperature \(T\):
\[
k = A\,e^{-E_a/(RT)}.
\]
Here \(A\) is the pre-exponential factor, \(E_a\) is the activation energy, and \(R\) is the gas constant.
Temperature must be in kelvins. Taking the natural logarithm gives a linear form used for graphing and regression:
\[
\ln k = -\frac{E_a}{R}\,\frac{1}{T} + \ln A.
\]
In a plot of \(y=\ln k\) versus \(x=1/T\), the slope is \(m=-E_a/R\) and the intercept is \(b=\ln A\). From fitted
values, \(E_a=-mR\) and \(A=e^{\,b}\).
Larger \(E_a\) means stronger temperature sensitivity: a small increase in \(T\) produces a larger relative change
in \(k\). Larger \(A\) shifts \(k\) upward at all temperatures and is associated with collision frequency and
orientation effects. Typical units: \(E_a\) in \(\mathrm{J\cdot mol^{-1}}\) (or \(\mathrm{kJ\cdot mol^{-1}}\)),
\(T\) in \(\mathrm{K}\), and \(k\) in whatever units match the chosen rate law (such as \(\mathrm{s^{-1}}\) for
first order or \(\mathrm{L\cdot mol^{-1}\cdot s^{-1}}\) for second order). When the calculator outputs a best-fit
line, slope/intercept, \(E_a\), and \(A\), the key check is whether the \(\ln k\) versus \(1/T\) trend is close to
linear over the provided temperature range.
Common pitfalls
- Using Celsius directly instead of converting to kelvins.
- Mixing units for \(E_a\) (J/mol vs kJ/mol) without consistent reporting.
- Using \(\log_{10}\) instead of \(\ln\) without the proper conversion factor.
- Including data from different mechanisms or phases, which breaks linear Arrhenius behavior.
Micro example
If the fitted slope is \(m=-6000\ \mathrm{K}\), then
\[
E_a = -mR = 6000(8.314)=4.99\times 10^{4}\ \mathrm{J\cdot mol^{-1}}\approx 49.9\ \mathrm{kJ\cdot mol^{-1}}.
\]
Use this tool when multiple \((T,k)\) measurements are available under the same mechanism and concentration
conditions. Avoid applying it across wide temperature ranges where the reaction order changes; a next-step concept
is transition-state analysis or separating regimes with different activation energies.
