Arrhenius equation

The Arrhenius equation was conceived by Svante Arrhenius, a Swedish chemist, in 1889.

It relates the rate constant, k, of a reaction with the temperature of the reaction, T, as follows:

$k=Ae^{-\frac{E_a}{RT}}\; \; \; \; \; \; \; \; 39$

where E_a is the activation energy of the reaction, R is the universal gas constant and A is the pre-exponential factor. The values of E_a and A of a reaction are experimentally determined.

The Arrhenius equation is based on the van’t Hoff equation: $\Delta_rH^o=RT^2\frac{dlnK}{dT}$ . Consider a reversible elementary reaction $A+B\; \; \begin{matrix} k_1\\\rightleftharpoons \\ k_2 \end{matrix}\; \; P$ , where k₁and k₂ are the rate constants of the forward and reverse reactions respectively. At equilibrium, $rate_1=rate_2$ and therefore, $k_1[A][B]=k_2[P]$ or $\frac{k_1}{k_2}=\frac{[P]}{[A][B]}=K$ . Substituting $K=\frac{k_1}{k_2}$ in the van’t Hoff equation,

$\Delta_rH^o=RT^2\frac{dlnk_1}{dT}-RT^2\frac{dlnk_2}{dT}$

If we define $RT^2\frac{dlnk}{dT}=E_a$ ,

$\Delta_rH^o=E_{a1}-E_{a2}\; \; \; \; \; \; \; \; 40$

Eq40 is best interpreted using a potential energy graph:

The definition of activation energy, $E_a=RT^2\frac{dlnk}{dT}$ , results in eq39 when integrated. Taking the natural logarithm on both sides of eq39,

$lnk=lnA-\frac{E_a}{RT}\; \; \; \; \; \; \; \; 41$

Eq41 is a linear function with dependent variable $lnk$ and independent variable $\frac{1}{T}$ , so that a plot of $lnk$ versus $\frac{1}{T}$ gives a gradient of $-\frac{E_a}{R}$ and vertical intercept of $lnA$ .

Question

Are the activation energy E_a and pre-exponential factor A of a reaction independent of temperature?

Answer

For the Arrhenius equation to be applicable, a plot of lnk versus 1/T must, with strong linear correlation, produce a straight line when values of k at various T are substituted in eq41. This implies that the activation energy E_a and pre-exponential factor A of a reaction are constants and therefore independent of temperature. In fact, the Arrhenius equation works reasonably well for many reactions over a temperature range of about 100 K. However, deviations from the equation do occur for some other reactions.

A more rigorous approach to analyse the relation between k and T using the transition state theory (TST) reveals the temperature-dependence of the activation energy of a reaction, with $E_a=\Delta^{\ddagger}H^o+xRT$ , where x = 1 for unimolecular gas-phase reactions and x = 2 for bimolecular gas-phase reactions. The TST also shows that the pre-exponential factor is dependent on temperature, where $A=\frac{\kappa e^xkRT}{p^oh}^2e^{\frac{\Delta^{\ddagger}S^o}{R}}$ .

So why does the Arrhenius equation work for so many reactions when both the activation energy and pre-exponential factor of a reaction are temperature-dependent? For E_a, the value $\Delta^{\ddagger}H^o$ is usually much bigger than xRT, and for A, T²is dwarfed by the term $\frac{\kappa e^xkR}{p^oh}e^{\frac{\Delta^{\ddagger}S^o}{R}}$ .

Arrhenius equation

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