Given a chemical equation of the form

then


In equilibrium,


For a chemical reaction

the law of mass action gives the rate as

where brackets denote concentrations and the reverse reaction is assumed to be negligible. Denote initial values with a 0 subscript and define



Since the amounts remaining of A and B after N molecules of C have been formed are and , we can rewrite (7) as


This is a nonlinear differential equation, but it can be separated and directly integrated

This integral is of the form

for q < 0, where , , and . The "discriminant" q is




Plugging the values for our problem into (13) gives

where C is a constant. Simplifying,

In order to find the value of C, we use the initial condition that , so plugging into (17) gives

Multiplying (17) through and exponentiating gives

Plugging (18) into (19) then gives

Solving for N,



gives the answer

Checking, we see that does indeed give 0, and gives .