10.5
Consecutive Reactions
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.
Consecutive reactions involve a sequence where the product of a preceding reaction becomes the reactant for the subsequent one. In a simple scheme, A transforms into B, which further reacts to form C, with rate constants k1 and k2, respectively. This concept is evident in the radioactive decay series. Assuming an initial state with only A present, the conservation of matter leads to three coupled differential equations, determining the concentrations of A, B, and C over time.
The rate of change for B is influenced by two opposing effects: an increase from A and a decrease from the formation of C. The resulting equations for A, B, and C at time t are derived through integration, reflecting their dependence on both rate constants and the differences between them.
When the second reaction is slower (k1 ≫ k2), there's an initial buildup of B, gradually transforming into C over time. Conversely, if the second reaction is faster (k1 ≪ k2), B rapidly converts to C, with minimal accumulation of B. This behavior is governed by the relative magnitudes of the rate constants k₁ and k₂, reflecting purely kinetic control rather than equilibrium.
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.
From Chapter 10:
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