# Determining rate constants for parallel reactions from experimental data

I have a question regarding the rates of parallel reactions. I have gathered experimental data for the overall reaction (i.e. conversions and selectivities and specific times) but due to the two reactions running parallel to each other for this reaction means I can't use a simple integrated rate plot.

Say for

$$\ce{A ->[\large k_1] B} \\ \ce{A ->[\large k_2] C}$$

By substitution and integration, I have found that:

$$[A](t) = [A](0) \mathrm{exp}(-(k_1+k_2)t)$$

and then the following equations can be derived in terms of the products formed:

$$\frac{d[B]}{dt} = k_1 [A](0) \mathrm{exp}(-(k_1+k_2)t)$$

which gives

$$[B](t) = \frac{k_1[A](0)}{k_1+k_2}(1-\mathrm{exp}(-(k_1+k_2)t)$$

when integrated. Also,

$$[C](t) = \frac{k_2[A](0)}{k_1+k_2}(1-\mathrm{exp}(-(k_1+k_2)t)$$

is the equation for the other parallel reaction.

My initial thoughts were to plot the graphs (as I know the values for $[C]$, $[B]$, and $[A]$) and form some simultaneous equations, but this is an exponential plot and so i'm not sure how this would work.. or if I could even plot it, as I don't know $k_1$ and $k_2$ - that is what I need to determine.

• To clarify, you know the values of [A](t) and either [B](t) or [C](t) for some value of $t$ other than zero? – Tyberius Mar 25 '17 at 22:51

One way you could get $k_1$ and $k_2$ is to first find their sum using your equation for $[A](t)$ at a known $t$.
Once you have done that, you can solve for $k_1$ using your equation for $[B](t)$ at a known $t$, since you now have the denominator of your first factor. You could alternatively find $k_2$ using $[C](t)$.
Once you have $k_1$, then $k_2$ is easily found from the sum.