Collision Theory: Deriving Rate Equation Confusion

[Yuqing]

I was reading a derivation of the rate equation from collision theory and there is one thing which confuses me a bit. In the derivation we assume that a particle will collide with every particle within its "collision volume" but it seems to me that this is a gross overestimate of the actual number of collisions. My reasoning is that if the particle successfully reacts during the first collision, then it will have only made 1 collision, and similarly with successive collisions. So shouldn't we instead be using an expected number of collisions rather than just saying the particle will collides with everything it has access to. This problem is then emphasized because we next multiple the number of collisions by the number of particles which seems to just blow up the error further. Am I getting confused on something here?

 

[Ygggdrasil]

Collision theory is predicting the initial rate of reaction before a significant amount of reactant has reacted. Obviously as the concentration of reactant decreases as it is depleted by the reaction, the rate of reaction will decrease.

 

[Yuqing]

I realize that, but the problem I'm describing appears to be different. Suppose a particle makes 1000 collisions per second and that 10% of these are "successful" collisions (i.e. reactive). In theory, a single particle can only make 1 successful collision since it'll have reacted and will be unable to perform additional reactions. But this theory seems to suggest that there will be 100 successful collisions, that is a single particle can produce 100 reactant molecules.

 

[Ygggdrasil]

Think of it from the point of view of an ensemble. Say you have one million particles. Now we should consider the following questions to think about the rate of reaction: (1) after x amount of time, how many of these particles will have collided with another particle? and (2) what fraction of those colliding particles will successfully react? Here the rate of collision of 1000 collisions per second does not mean that we're assuming the particle will have 1000 collisions. It means we expect uncreated particles to collide once approximately every millisecond.