# Arrhenius plot

In chemical kinetics, an Arrhenius plot displays the logarithm of a reaction rate constant, (${\displaystyle \ln(k)}$, ordinate axis) plotted against inverse temperature (${\displaystyle 1/T}$, abscissa). Arrhenius plots are often used to analyze the effect of temperature on the rates of chemical reactions. For a single rate-limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy and the pre-exponential factor can both be determined.

Example:
Nitrogen dioxide decay
2 NO2 → 2 NO + O2
Direct plot:
k against T
Arrhenius plot: ln(k) against 1/T.

The Arrhenius equation can be given in the form:

${\displaystyle k=Ae^{-E_{a}/RT}}$

or alternatively

${\displaystyle k=Ae^{-E_{a}/k_{B}T}}$

The only difference is the energy units: the former form uses energy/mole, which is common in chemistry, while the latter form uses energy directly, which is common in physics. The different units are accounted for in using either the gas constant ${\displaystyle R}$ or the Boltzmann constant ${\displaystyle k_{B}}$.

The former form can be written equivalently as:

${\displaystyle \ln(k)=\ln(A)-{\frac {E_{a}}{R}}\left({\frac {1}{T}}\right)}$
Where:
${\displaystyle k}$ = Rate constant
${\displaystyle A}$ = Pre-exponential factor
${\displaystyle E_{a}}$ = Activation energy
${\displaystyle R}$ = Gas constant
${\displaystyle T}$ = Absolute temperature, K

When plotted in the manner described above, the value of the true y-intercept (at ${\displaystyle x=1/T=0}$) will correspond to ${\displaystyle \ln(A)}$, and the slope of the line will be equal to ${\displaystyle -E_{a}/R}$. The values of y-intercept and slope can be determined from the experimental points using simple linear regression with a spreadsheet.

The pre-exponential factor, A, is an empirical constant of proportionality which has been estimated by various theories which take into account factors such as the frequency of collision between reacting particles, their relative orientation, and the entropy of activation.

The expression ${\displaystyle e^{-E_{a}/RT}}$ represents the fraction of the molecules present in a gas which have energies equal to or in excess of activation energy at a particular temperature. In almost all practical cases, ${\displaystyle E_{a}\gg RT}$, so that this fraction is very small and increases rapidly with T. In consequence, the reaction rate constant k increases rapidly with temperature T, as shown in the direct plot of k(T). (Mathematically, at very high temperatures so that ${\displaystyle E_{a}\ll RT}$, k would level off and approach A as a limit, but this case does not occur under practical conditions.)

## Worked example

Based on the red "line of best fit" plotted in the graph given above:

Let y = ln(k[10−4 cm3 mol−1 s−1])
Let x = 1/T[K]

y = 4.1 at x = 0.0015
y = 2.2 at x = 0.00165

Slope of red line = (4.1 - 2.2) / (0.0015 - 0.00165) = -12,667

Intercept [y-value at x=0] of red line = 4.1 + (0.0015 x 12667) = 23.1

Inserting these values into the form above:

${\displaystyle \ln(k)=\ln(A)-{\frac {E_{a}}{R}}\left({\frac {1}{T}}\right)}$

yields:

${\displaystyle \ln(k)=23.1-12,667(1/T)}$
Plot of k = e^23.1 * e^(-12,667/T)
${\displaystyle k=e^{23.1}\cdot e^{-12,667/T}}$

as shown in the plot at the right.

${\displaystyle k=1.08\times 10^{10}\cdot e^{-12,667/T}}$

for:

k in 10−4 cm3 mol−1 s−1
T in K

Substituting for the quotient in the exponent of ${\displaystyle e}$:

-Ea / R = -12,667 K
approximate value for R = 8.31446 J K−1  mol−1

The activation energy of this reaction from these data is then:

Ea = R x 12,667 K = 105,300 J mol−1 = 105.3 kJ mol−1.