Arrhenius Equation? [closed]








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I'm a bit confused. So we have an Arrhenius equation, which gives the dependence of the rate constant of a chemical reaction on the absolute temperature. What will happen to the value of the rate constant when the temperature (T) is 0, when the activationn energy is 0, and when the temperature (T) is extremely high? Thank you in advance!




Leaving the maths aside we can think of what is actually happening. The high temperature case is the easiest to explain since here the reactant molecules have a *huge* amount of energy and so the activation energy is irrelevant and the reaction proceeds with a rate constant independent of temperature and this is the 'pre-exponential' term A. 


The low temperature and low activation cases are trickier and is only really answered by noting the the Arrhenius equation is only an empirical equation and is not applicable when the activation energy is very small or the temperature very low. 


When the activation energy is low the reaction proceeds with a measured rate constant determined by how quickly the reactant species can diffuse together, since when they meet reaction is effectively instantaneous and faster than diffusion. The reaction is called 'diffusion limited'. If the reactant species are held apart as may be done in some electron transfer reactions then from the Marcus theory of electron transfer reactions the rate constant has a maximum value when the activation energy is zero, of about $10^{13}\mathrm{s}^{-1}$ and this is the 'pre exponential' value.


As we approach zero K the reactants will become solidified so that different conditions apply here as species cannot diffuse together on any sensible time scale. The Arrhenius equation does not apply here.


If you study Statistical Mechanics and Quantum Theory then it is possible to derive 'Arrhenius type' equations from a sound basis. 


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What you are asking is simple maths:
$$
k = A \cdot \exp\left(-\frac{E\_a}{RT}\right)
$$


As can be seen this equation is not defined for $T=\pu{0 K}$. However if one inspects the limit of the function then it is obvious that $k$ will converge towards $k=0$. Therefore one can say that at zero kelvin no reaction will take place. 


For very high temperatures one can see that $k$ will converge towards $k=A$. 


If the activation energy is zero then the rate constant will be $k=A$ for all temperatures (except $T=\pu{0 K}$).


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