Two chemicals in solution react together to form a compound: one unit of compound is formed from $a$ units of chemical $A$ and $b$ units of chemical $B$, with $a + b = 1$. Assume the concentration is low enough that the rate of reaction is directly proportional to the amount of chemical $A$ present, and directly proportional to the amount of chemical $B$ present. For example, doubling the amount of $A$ and tripling the amount of $B$ would increase the rate of reaction by a factor of 6.
Find the amount produced $x$ after $t$ seconds, up to a constant of proportionality.
I attempted to solve this using differential equations, but got a nonsensical result. Can you help me find my error?
(These other math.SE posts are relevant but didn't answer my question.)
Let $A_0$ be the initial amount of chemical $A$ and $B_0$ of $B$. Define $$\alpha = \frac {A_0}{a} \\ \beta = \frac {B_0} {b}.$$ Then the rate of change of the amount of the compound $x$ is $$\begin{align*} \frac {dx} {dt} &= k(a\alpha -ax)(b\beta -bx) \\ &= kab(x - \alpha)(x - \beta). \end{align*}$$
Therefore, $$t = \frac {1} {kab} \int \frac {1}{(x -\alpha)(x - \beta) }\, dx.$$
First, consider the case of $\alpha = \beta$ (that is, the two reactants are in ideal proportion). Then $$t = \frac {1}{kab}\frac {1}{\alpha - x} + C \\ x = \alpha - \frac {1}{kabt} + C$$ with $C$ capturing the initial amount of product (presumably zero).
As expected, as $t \to \infty, x \to \alpha + C.$ But this solution also claims that as $t \to 0, x \to - \infty$, which is of course impossible!
Where is my error?
If you start from
$$t = \frac {1}{kab}\frac {1}{\alpha - x} + C, $$ you get
$$ x = \alpha - \frac {1}{kab(t-C)}.$$
Then, I am sure that if you compute $C$ explicitly, you find that it’s negative, so that the fraction does not blow up at $t=0$ or any positive time.