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:

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*

*and*

_{a}*A*of a reaction are experimentally determined.

The Arrhenius equation is based on the van’t Hoff equation: . Consider a reversible elementary reaction , where *k** _{1 }*and

*k*

*are the rate constants of the forward and reverse reactions respectively. At equilibrium, and therefore, or . Substituting in the van’t Hoff equation,*

_{2}If we define ,

Eq40 is best interpreted using a potential energy graph:

The definition of activation energy, , results in eq39 when integrated. Taking the natural logarithm on both sides of eq39,

Eq41 is a linear function with dependent variable and independent variable , so that a plot of versus gives a gradient of and vertical intercept of .

###### 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 , 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 .

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 is usually much bigger than

*xRT*, and for

*A*,

*T*is dwarfed by the term .

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