9.1 Consecutive reactions and their kinetics

Consecutive reactions are a series of chemical transformations where one reaction's product becomes the next reaction's reactant. This concept is crucial in understanding complex chemical processes like radioactive decay and metabolic pathways. It's all about how molecules change step-by-step.

Analyzing consecutive reactions involves rate equations and concentration-time profiles. These tools help identify the rate-determining step and predict how reactants, intermediates, and products behave over time. Understanding these dynamics is key to controlling and optimizing multi-step chemical processes.

Consecutive Reactions

Consecutive reactions in chemical systems

• Consecutive reactions involve a series of chemical reactions where the product of one reaction serves as the reactant for the subsequent reaction (e.g., radioactive decay series, enzymatic reactions in metabolic pathways)
• Reactions occur sequentially, with each step depending on the completion of the previous step
• Examples of consecutive reactions in chemical systems include:
  • Radioactive decay series (uranium-238 to lead-206 through alpha and beta decays)
  • Enzymatic reactions in metabolic pathways (citric acid cycle with enzyme-catalyzed steps)
  • Multistep organic synthesis reactions (synthesis of complex molecules from simpler precursors)

Rate equations for consecutive reactions

• Consider a general consecutive reaction: $A \stackrel{k_1}{\longrightarrow} B \stackrel{k_2}{\longrightarrow} C$
  • $A$, $B$, and $C$ represent the reactant, intermediate, and final product, respectively
  • $k_1$ and $k_2$ are the rate constants for the first and second steps
• Rate equations for the reactant ($A$), intermediate ($B$), and final product ($C$):
  • $\frac{d[A]}{dt} = -k_1[A]$
  • $\frac{d[B]}{dt} = k_1[A] - k_2[B]$
  • $\frac{d[C]}{dt} = k_2[B]$
• Kinetic behavior of reactants and intermediates:
  • Reactant $A$ concentration decreases exponentially with time, following first-order kinetics
  • Intermediate $B$ concentration initially increases, reaches a maximum, and then decreases as it is consumed in the second step
  • Final product $C$ concentration increases with time, approaching a constant value as the reaction proceeds to completion

Concentration-time profiles analysis

• The rate-determining step is the slowest step in a consecutive reaction, controlling the overall rate of product formation
• Analyzing concentration-time profiles helps identify the rate-determining step:
  • If $k_1 \ll k_2$, the first step is rate-determining, and intermediate $B$ concentration remains low throughout the reaction
  • If $k_1 \gg k_2$, the second step is rate-determining, and intermediate $B$ concentration builds up before being consumed
• Characteristics of concentration-time profiles for different rate-determining steps:
  • First step rate-determining: $[A]$ decreases slowly, $[B]$ remains low, and $[C]$ increases slowly
  • Second step rate-determining: $[A]$ decreases rapidly, $[B]$ builds up and then decreases, and $[C]$ increases slowly

Kinetics problem-solving for consecutive reactions

• The steady-state approximation assumes that the intermediate concentration remains constant during the majority of the reaction
  • This approximation is valid when the rate of formation and consumption of the intermediate are equal
• Applying the steady-state approximation to the general consecutive reaction:
  1. Set $\frac{d[B]}{dt} = 0$, which gives $k_1[A] - k_2[B] = 0$
  2. Solve for $[B]$: $[B] = \frac{k_1[A]}{k_2}$
• Substitute the steady-state concentration of $B$ into the rate equation for $C$:
  • $\frac{d[C]}{dt} = k_2[B] = k_2\left(\frac{k_1[A]}{k_2}\right) = k_1[A]$
• The rate of formation of the final product $C$ depends only on the initial reactant $A$ concentration and the first step rate constant, $k_1$