Diff Eq applied to A+B->C chemical reaction

I'm taking Applied Diff Eq and was presented with the following problem. I'm not really into applications, and chemistry isn't really my strongest subject, but the class is required and the instructor really loves them. Anyway, I just wanted to check to make sure I did this problem correctly.

Suppose A and B represent two substances that can combine to form a new substance C. Suppose we have a container with a solution containing low concentrations of substances A and B, and A and B molecules react only when they happen to come close to each other. If a(t) and b(t) represent the amount of A and B in the solution, respectively, then the chance that a molecule of A is close to a molecule of B at time t is proportional to the product $\displaystyle a(t)\cdot b(t)$. Hence the rate of reaction of A and B to form C is proportional to ab. Suppose C precipitates out of the solution as soon as it is formed, and the solution is always kept well mixed.

Describe an experiment you could perform to determine an approximate value for the reaction-rate parameter $\displaystyle \alpha$ in the system:

$\displaystyle \begin{array}{c}

\frac{da}{dt}=-\alpha ab\\

\frac{db}{dt}=-\alpha ab\end{array} $

Include the calculations you would perform using the data from your experiment to determine the parameter.

So here's my little experiment:

- Start with $\displaystyle a(0)=M$ and $\displaystyle b(0)=N$
- Let $\displaystyle c(t)$ represent the amount of precipitate at time $\displaystyle t$.
- Let $\displaystyle L=c(1)$
- $\displaystyle (d) \begin{array}{c}

\frac{da}{dt}\approx-L\\

\frac{db}{dt}\approx-L\end{array} , so \begin{array}{c}

a(1)\approx M-L\\

b(1)\approx N-L\end{array}$. - And therefore, $\displaystyle \alpha\approx\frac{L}{(M-L)(N-L)}$

My final answer has to be in the form of a python program that will actually find $\displaystyle \alpha$, but I need to make sure that this is the correct approach.