Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

A chemical solution contains N molecules of type A and M molecules of type B. An irreversible reaction occurs between type A and type B molecules in which they bond to form a new compound AB. Suppose that in any small time interval of length h, any particular unbounded A molecule will react to any particular unbounded B molecule with probability theta * h + o(h) where theta is a reaction rate. Let X(t) denote the number of unbounded A molecules at time t. Model X(t) as a pure death process by specifying parameters.

The answer is k * ( M - (N-k) ) * theta for k = 0, 1, 2, ... , N

I am unsure about what the expression k * ( M - (N-k) ) represents and would appreciate if someone could explain the rationale behind it to me. The way I approached this question was I took k to represent the number of A molecules remaining. If we want P (X(t) = k), it is equivalent to saying N - k molecules died. If you subtract that from M, that is the remaining number of B molecules remaining to react with. Thus, multiplying that by theta should give you the rate. However, the solution does not match my rationale and adds in a k.

Thanks for the help!

share|improve this question

1 Answer 1

up vote 1 down vote accepted

You have identified that if there are more unreacted molecules of B present then the next reaction is likely to take place sooner.

Can you say anything similar about the impact of unreacted molecules of A?

share|improve this answer
    
The fewer unreacted molecules of A, the more unlikely the next reaction will take place. However, I do not see how this statement leads to the solution? –  icobes Mar 24 '11 at 23:30
    
Perhaps there might be a proportionality to $k$ as there is to $M-(N-k)$? –  Henry Mar 24 '11 at 23:56
    
I am assuming it is a linear death process and that is why you multiply by k. Is that statement correct? How would I go about identifying something as a pure death process where the death rate is just mew vs a linear death process where the death rate is n*mew? I can't make the distinction between the two. Is there anything in the question that might point to it being the case? –  icobes Mar 25 '11 at 0:02
    
In the question you have the phrase ... any particular unbounded A molecule will react to any particular unbounded B molecule ... and you need to deal the particular twice, once for A and once for B, so you need to multiply by the remaining number of As and the remaining number of Bs. –  Henry Mar 25 '11 at 6:16

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.