Parallel 1st Order Reaction Rates Consider some general parallel first order process, where $\ce{A -&gt; B}$ and $\ce{A -&gt; C}$ (rate constants $k_1$ and $k_2$ respectively)
The overall rate equation of this process is said to be $-\frac{\mathrm d[A]}{\mathrm dt} = k_1[\mathrm A] + k_2[\mathrm A]$
I get the impression that this is derived from the addition of each individual rate equation, i.e. the addition of $-\frac{\mathrm d[\mathrm A]}{\mathrm dt} = k_1[\mathrm A]$ and $-\frac{\mathrm d[\mathrm A]}{\mathrm dt} = k_2[\mathrm A]$ would result in the overall rate equation shown above.
However, if this is the case, then why isn’t the rate equation actually $-\frac{2\,\mathrm d[\mathrm A]}{\mathrm dt} = k_1[\mathrm A] + k_2[\mathrm A]$? I mean, you’re adding two $-\frac{\mathrm d[\mathrm A]}{\mathrm dt}$ terms together; and so considering basic algebra, surely this would result in $-\frac{2\,\mathrm d[\mathrm A]}{\mathrm dt}$?
Or do differentials just not work in this way, and I’m simply confusing myself?
There is no need adding the number $2$ in front of the definition of the rate. So the rate equation is :
$$-d[A]/dt = k_1 [A] + k_2[A] = (k_1 + k_2)[A]$$