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Arrhenius Equation

As per the book (Nivaldo J. Tro)

The Frequency Factor: The number of approaches to the activation barrier per unit time.

The Exponential factor: Number between 0 and 1 that represents the fraction of molecules that have enough energy to make it over the activation barrier on a given approach. The exponential factor is the fraction of approaches that are actually successful and result in the product.

Orientation Factor

Collision Frequency: It is the number of collisions that occurs per unit time.

Orientation Factor: Usually between 0 and 1, which represents the fraction of collisions with an orientation that allows the reaction to occur

I am unable to distinguish the between the exponential factor and orientation factor. Aren't they saying the same thing? How do they differ from each other?

Another doubt is what does the number of approaches mean? i.e. number of approaches taken by a single reactant per sec, or all reactant per sec.

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The orientation factor is really an empirical add-on fix as way of trying to make the Arrhenius equation seem more reasonable. Most collisions between molecules do not lead to reaction, which is why the activation energy is there, but even when a molecule has obtained enough energy to react ( by random collisions in solution or vapour) we may know from its structure that it can only react from one side or at one end, then an orientation factor is added.

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The two are not quite saying the same thing—the orientation factor is a parameter that holds information about the likelihood the two molecules required for reaction are arranged in a way where they will able to react at all, whereas the exponential factor encodes, as you describe, the percentage of particles that have enough energy to actually engage in the reaction.

Perhaps an analogy will help here: Let's assume that we are playing a game of darts, but our game of darts will just be "who can hit the most bullseyes?" The process of hitting a bullseye will represent a successful reaction occurring.

If we stand, say, ten feet away from the dart board, then there will be some minimum energy that one has to throw the darts at in order for it to even be possible to hit the board. If I've broken my dart-throwing arm, it is very possible that I never throw a dart with enough energy to reach the board, and all my darts scatter on the ground. I need to increase the velocity of my throw in order to reach an energy where hitting the dartboard is even possible. This represent our exponential factor where the percentage of darts I throw that can even reach the dartboard are counted. Similarly, two particles need to have adequate energy for the reaction process to occur whatsoever. The only way I could get a bullseye without having enough energy would be for me to stand much closer to the dartboard, but that breaks the rules of the game! If you are curious why this parameter is exponential with temperature is related to a property in statistical mechanics where the distribution of particle kinetic energies is related to the exponential of each energy-pair's ratio.

Back to darts. Just because some of my darts reach the dartboard, this doesn't mean that they are all bullseyes. Only a fraction of the darts I throw that reach the dartboard will hit the bullseye—the arrangement of the dart relative to the dartboard it is colliding with determines whether it is a bullseye or a miss. The fraction of bullseyes I hit out of all the darts that reach the dartboard is represented by our orientation factor. Some reactions don't require super specific arrangements, and this would be like having a very large bullseye; others need very specific arrangements, and those would be like having a very tiny bullseye.

The collision frequency is, unsurprisingly, just how fast we throw the darts. The orientation factor and exponential factor basically act as fractions that determine what percent of collisions are successful. If we throw 20 darts per second, but only 16 make it to the board and only 25% of those are bullseyes, then our bullseye rate is $k = (20$ darts s$^{-1})(\frac{16}{20}$ dartboard hits$)(\frac{1}{4}$ bullseyes per dartboard hit$) = 4$ bullseyes per second. Similarly, if we have 20 collisions per second, but only 16 have sufficient energy to react, and only 25% of those with sufficient energy are oriented correctly, our reaction rate is $k = (20$ collisions s$^{-1})(\frac{16}{20}$ energetic successes$)(\frac{1}{4}$ correct orientations per energetic success$) = 4$ reactions per second.

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