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

The answer is

$$k\big(M - (N - k)\big)\theta$$

for $k = 0, 1, 2, \dots, N$.

I am unsure about what the expression $k\big(M - (N - k)\big)$ 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 $\mathrm{B}$ 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 $k$.

andselisk
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icobes
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1 Answers1

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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?

Henry
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  • 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 00:02
  • 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 06:16