The Arrhenius Equation and Temperature Dependence

Taking natural logarithms produces a straight-line relationship in the variables ln k and 1/T. The slope is -E_a/R, so E_a = -R times slope, and the intercept is ln A. Note that 1/T on the horizontal axis means higher temperatures lie to the left.

Read the horizontal axis label before using its slope. For x = 1/T in K⁻¹, slope = −E_a/R. If x = 1000/T, then ln k = [−E_a/(1000R)]x + ln A, so E_a = −1000R × slope. A graph scaling mistake changes the activation energy by a factor of 1000 even if the line was fitted perfectly. In the usual positive-E_a case the slope is negative because larger 1/T means a lower temperature and a smaller k.

Subtracting the linear equation at two temperatures eliminates ln A entirely, leaving a relationship between the two rate constants, the two temperatures and E_a. Any one unknown can be found from the other four.

To predict k₂, rearrange before calculating: k₂ = k₁ exp[(E_a/R)(1/T₁ − 1/T₂)]. With E_a = 50.0 kJ/mol, T₁ = 300 K and T₂ = 320 K, the exponent is (50000/8.314)(1/300 − 1/320) = 1.253. Thus k₂/k₁ ≈ 3.50. The positive exponent agrees with heating increasing k in this positive-E_a model. The ratio is unitless only if both rate constants use the same time and concentration units. A Celsius difference may equal a kelvin difference, but reciprocal Celsius temperatures cannot replace reciprocal kelvin temperatures.

Two reactions can share the same rate at one temperature yet respond very differently to heating: the one with the larger E_a has the steeper ln k against 1/T line and gains far more from a temperature rise. When an experimental plot curves, suspect a change of mechanism over the temperature range rather than a failure of the equation.

A curved Arrhenius plot can signal changing mechanism, changing apparent parameters or other experimental effects; it is a reason to investigate rather than fit one line blindly. Extrapolation far beyond the measured range is uncertain. Comparing temperature sensitivity at equal initial k assumes comparable, approximately constant E_a and A over that range. A larger E_a alone does not guarantee a larger k at one temperature because A also matters.

In k = A e^-Ea/RT, A sets the multiplicative scale of the rate constant for a given Arrhenius model. If two reactions have the same E_a but different A values, their k ratio equals their A ratio at every common temperature. A embodies collision frequency and orientation within the model; it should not be mistaken for activation energy or concentration.

A has the same units as k, and represents more than a bare count of collisions in many reactions. A catalyst changes the mechanism and can change both apparent E_a and A. For the simple positive-barrier model, lowering E_a with A fixed raises k exponentially; this comparison is useful but is not a rule that every catalyst leaves A untouched. Complex mechanisms can even show an apparent negative E_a; introductory questions normally specify ordinary positive-E_a Arrhenius behavior.