For the simultaneously executing parallel first-order reactions and , if it is experimentally determ — Chemical Kinetics Chemistry Question
Question
For the simultaneously executing parallel first-order reactions $A \xrightarrow{k_1} B$ and $A \xrightarrow{k_2} C$, if it is experimentally determined that $k_1 = 2 \times 10^{-3} \text{ s}^{-1}$ and $k_2 = 6 \times 10^{-3} \text{ s}^{-1}$, what must be the uniform ratio of the concentrations of the synthesized products $[C] / [B]$ at any arbitrary given time $t$ during the reaction?
💡 Solution & Explanation
In parallel first-order reactions, the rates of formation of the products are directly proportional to their specific rate constants. $\frac{d[B]}{dt} = k_1[A]$ and $\frac{d[C]}{dt} = k_2[A]$. Integrating both from $t=0$ to $t$ gives $[B]_t = k_1 \int [A]dt$ and $[C]_t = k_2 \int [A]dt$. The integrals cancel out perfectly, resulting in the time-independent constant ratio $[C]/[B] = k_2 / k_1 = (6 \times 10^{-3}) / (2 \times 10^{-3}) = 3$.