# Serial reactions or consecutive reactions: rate vs time

Serial reactions, or consecutive reactions, are two or more reactions in which the product of the first reaction becomes the reactant in the next. The simplest case of a serial reaction involves a reagent A turns into B which in turn, again in the reaction environment, turns into P. In the simplest case, all reactions are irreversible reactions of the first order, so we can be write

$$\mathrm{A} \xrightarrow{k_1} \mathrm{B} \xrightarrow{k_2} \mathrm{P}$$

the rates of these reactions will be

$$\begin{equation*} \begin{cases} r_\mathrm{A} = -k_1\ c_\mathrm{A} \\ r_\mathrm{B} = k_1\ c_\mathrm{A} -\ k_2\ c_\mathrm{B} \\ r_\mathrm{P} = k_2\ c_\mathrm{B} \\ \end{cases} \end{equation*}$$

Where $$k$$ are the kinetics constants, and $$c$$ is the concentration of the various substances. Plotting $$r = f (t)$$, I get this

The maximum rate of P formation is reached when $$r_\mathrm{B}$$ is zero. My hypothesis is that the maximum rate of formation of P must be reached when the concentration of B is maximum, therefore at the minimum of the $$r_\mathrm{B}(t)$$ curve.. Is my guess right, or is the graph right?

• I hope this would be helpful. Jul 22 at 18:30
• Remember that a condition for a maximum of a differentiable function is the first derivative is zero. So rB must be zero for B having maximal concentration. Jul 23 at 7:31

Simple anwser.

Your answer is correct. The concentration of species B reach a maximum when the rate of formation of the intermediate must equals the rate of dissapeance.

Since $$r_B$$ can be written as

$$r_B = \frac{d|B|}{dt} = k_1|A| - k_2|B|$$

this function has an stationary point at $$t_{max}$$ when $$r_B = 0$$. $$r_B$$ is positive for $$t < t_{max}$$ and negative after reaching the maximum $$t > t_{max}$$.

This result is intuitive if you consider that below $$t_{max}$$ the concentration of B increases ($$r_B > 0$$) and drecreases above it $$t_{max}$$.

The rate equations have analytic solutions and are easily integrated numerically. But you must be awere that the analitical solutions of the rate equations differ for $$k_1 = k_2$$ and $$k_1 \neq k_2$$.

You can find a detailed discussion of this problem in the textbooks:

• J. H. Espenson. Chemical Kinetics and Reaction Mechanisms, second edition. Mc Graw-Hill (1995).
• J. I. Steinfeld, J. S. Francisco, W. L. Hase. Chemical Kinetics and Dynamics, second ed., Prentice Hall (1999).