10.5
Consecutive Reactions
Consecutive reactions involve a sequence where the product of a preceding reaction becomes the reactant for the subsequent one. In a simple scheme, A…
Consecutive reactions, like a radioactive decay series, involve a sequence where reactant A transforms into product B, which then forms C, with first-order rates r1 and r2.
The rate of change of A’s concentration is the rate of decay of A, and for C, it is the rate of decay of B, whereas the rate of change in B’s concentration depends on the formation rates of B and C.
Integrating the first-order rate law for A, with an initial concentration of [A]0, yields the concentration of A at time t. Substituting and solving the resulting differential equation provides B’s concentration at an elapsed time t.
According to the conservation of matter, the sum of A, B, and C concentrations at any time equals the [A]0. Rearranging and solving it results in C’s concentration.
The expressions for B and C are complicated as they depend on differences in the rate constants.
When the second reaction is slower, there is an initial buildup of B which eventually transforms into C. But if it is faster, B rapidly converts to C with minimal accumulation.
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Q1: What is a consecutive reaction and how does it differ from a single-step reaction?
A consecutive reaction is a sequence where the product of one reaction becomes the reactant for the next. In the scheme A → B → C, reactant A transforms into product B, which then forms C. Unlike single-step reactions, consecutive reactions involve multiple rate constants (k1 and k2) and produce intermediate products that accumulate or deplete depending on relative reaction rates.
Q2: How does the rate of change of intermediate B depend on the overall reaction sequence?
The rate of change for B is influenced by two opposing effects: an increase from A's transformation into B and a decrease from B's conversion to C. The net rate depends on both rate constants k1 and k2. When k1 is much larger than k2, B accumulates initially before slowly converting to C.
Q3: What role does the conservation of matter play in consecutive reactions?
Conservation of matter states that the sum of concentrations of A, B, and C at any time equals the initial concentration [A]0. This principle generates three coupled differential equations that determine the concentrations of all species over time, ensuring mass is neither created nor destroyed throughout the reaction sequence.
Q4: How do rate constants k1 and k2 affect the accumulation of intermediate B?
When k1 is much greater than k2 (k1 ≫ k2), the second reaction is slower, causing an initial buildup of B that gradually transforms into C. Conversely, when k1 is much less than k2 (k1 ≪ k2), B rapidly converts to C with minimal accumulation. This behavior reflects kinetic control governed by the relative magnitudes of the rate constants.
Q5: Why are the mathematical expressions for B and C concentrations more complex than for A?
The expressions for B and C depend on differences between rate constants k1 and k2, making them more complex than A's first-order decay. B's concentration reflects both formation from A and loss to C, while C's concentration depends on the accumulated product from B. These coupled dependencies require integration of differential equations.
Q6: How is the concentration of A determined in a consecutive reaction?
The concentration of A at time t is determined by integrating the first-order rate law with initial concentration [A]0. Since A only decays into B and does not form from any other species, its concentration follows simple exponential decay: [A]t = [A]0 exp(-k1t), independent of the second reaction rate.
Q7: What is an example of a consecutive reaction in nature?
A radioactive decay series exemplifies consecutive reactions, where an unstable nucleus decays into an intermediate product, which then decays into a final stable product. Each decay step follows first-order kinetics with distinct rate constants, demonstrating how intermediate products accumulate and deplete based on the relative decay rates.