How do you calculate the concentrations of multiple reactants in a certain point in time? For $\ce{A + B -&gt; C + D}$, given that $v = k [\ce{A}] [\ce{B}]$. I know the derivation for $\ce{A-&gt;B + C}$ , being $\ln[\ce{A}]=\ln[\ce{A}]_0−νkt$, but I'd like to know how you can, given a point in time, calculate the concentrations of A and B in the first equation, given the initial concentrations.
$$\frac{dA}{dt}=-kAB$$
$$\frac{dB}{dt}=-kAB$$subject to initial conditions $A=A_0$ and $B=B_0$ at time t = 0.  Let x = number of moles of A that react = number of moles of B that react.  Then,$$\frac{d(A_0-x)}{dt}=-k(A_0-x)(B_0-x)$$and$$\frac{d(B_0-x)}{dt}=-k(A_0-x)(B_0-x)$$Therefore, from both equations  $$\frac{dx}{dt}=k(A_0-x)(B_0-x)$$ subject to the initial condition x = 0 at t = 0.  This differential equation can readily be integrated using separation of variables and partial fractions.