[kelvinLTR]

Since the simple collision model equation is derived from a second order elementary reaction which gives you

    k=Pπ(R_a+R_b)²e^-E_a/RT   and
    k=Ae^-E_a/RT in Arrhenius equation

is it okay to use Arrhenius equation if we do not know if a given reaction is elementary second order reaction or not

[Mr.A]

for a elementary reaction the dimensions of the Arrhenius constant is [N]^-[L³⁺]³⁺[T]^-

but as we know when the order of the reaction changes, the rate constant k and the dimensions of it varies and that is going to be the dimensions of the Arrhenius constant of that reaction. So what I think is the Arrhenius constant in a elementary 2nd order reaction is just a constant for a 2nd order reaction (dimensionally). isn't it?

[Corribus]

Yes, Arrhenius equation works for more than just 2nd order reactions.  The similarities between the (general) Arrhenius expression and the collision model are something similar to a coincidence, even if they just yield some molecular insight into reaction dynamics.  (The Arrhenius equation is also similar to rate equations derived from transition state theory.)  Well, maybe coincidence isn't the right word, but Arrhenius is a semiempirical equation, not anything derived from first principles.  We might expect the preexponential factor to mean different things for different types of reactions - after all, it has different units for differnet types of reactions.  That is works for such a broad array of reactions might be surprising, but probably shouldn't be. If you think about it, the Arrhenius equation is fairly simple - rate is exponentially related to an activation energy, scaled by a constant of proportionality, A.  The unifying thing is the activation energy, a concept that is ubiquitous among all molecular kinetic theories and makes a sort of obvious sense - almost all reactions, no matter how they proceed, require a certain amount of input energy to go, and the rate is going to be related to the proportion of molecules that have meet these energetic standards at any given time (which is in turn related to the temperature).  The "A" factor takes into account everything else that might impact the reaction rate, and should be expected to fold in very different criteria based on the actual mechanism involved.   

There are reactions that do NOT exhibit Arrhenius behavior, but these reactions are those which do not have conventional activation energies - such as those which proceed through quantum tunnelling mechanisms and so forth, where reactions can proceed even without having enough energy to surmount the energetic barrier required for the reaction to take place.  Non-Arrhenius behavior, in other words, is usually related to Ea, not A.