# What is the reaction rate constant of an effective reaction?

Consider the reactions

$$\ce{A + B ->[k_1] C }$$ and

$$\ce{C ->[k_2] E + F}$$

with reaction rate constants $k_1$ and $k_2$. I know that the two reactions can be written as follows

$$\ce{A +B->E +F}$$

with a rate constant $k_3$. How does $k_1$ and $k_2$ relate to $k_3$? Does it matter how many species are in the reactants (such as if the first reaction was simply $\ce{A->C}$)?

It is not clear that they are related. It is simpler to consider the equivalent scheme $A \rightarrow C \rightarrow E$ with rate constnats $k_1,\, k_2$ then the decay of A is $A_t/A_0= \exp(-k_1t)$ the amount of C rises and falls; it is $$\displaystyle C_t/A_0 = \frac{k_1}{k_2-k_1}(\exp(-k_1t)-\exp(-k_2t))$$ and E is $$E/A_0 = 1-A_t-C_t=1-\exp(-k_1t)-\frac{k_1}{k_2-k_1}(\exp(-k_1t)-\exp(-k_2t))$$
When $k_2$ is very large and much bigger than $k_1$ then $\exp(-k_2t) \rightarrow 0$ and $\displaystyle E/A_0= 1-(1+\frac{k_1}{k_2})\exp(-k_1t)$ and in the opposite limit $k_1 \gg k_2$ then $E/A_0= 1-\exp(-k_2t)$.
• Can you explain how do you come with the first equation which describes the concentration of $[C_t]$ in terms of $k_1$ and $k_2$ as a function of time?(it will be a complete answer you describe that too.) Apr 28, 2019 at 15:25
• Write down $dC_t/dt =k_1A_t-k_2C_t$ and substitute for $A_t$ as given and solve the equation. At $t=0,\,C_0=0$. Apr 29, 2019 at 6:49