The integrated rate law for the zero-order reaction A ⟶B is [A]t=[A]0-k t . (a) Sketch the follo

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The integrated rate law for the zero-order reaction A ⟶B is...

The integrated rate law for the zero-order reaction $\mathrm{A} \longrightarrow \mathrm{B}$ is $[\mathrm{A}]_{t}=[\mathrm{A}]_{0}-k t .$ (a) Sketch the following plots: (i) rate versus $[\mathrm{A}]_{t}$ and (ii) $[\mathrm{A}]_{t}$ versus $t$. (b) Derive an expression for the half-life of the reaction. (c) Calculate the time in half-lives when the integrated rate law is no longer valid, that is, when $[\mathrm{A}]_{t}=0$.

The integrated rate law for the zero-order reaction $\mathrm{A} \longrightarrow \mathrm{B}$ is $[\mathrm{A}]_{t}=[\mathrm{A}]_{0}-k t .$
(a) Sketch the following plots: (i) rate versus $[\mathrm{A}]_{t}$ and (ii) $[\mathrm{A}]_{t}$ versus $t$. (b) Derive an expression for the half-life of the reaction. (c) Calculate the time in half-lives when the integrated rate law is no longer valid, that is, when $[\mathrm{A}]_{t}=0$.

Key Concepts

Half-Life Derivation

The half-life of a reaction, defined as the time required for the concentration of a reactant to decrease to half of its initial value, is derived differently for various reaction orders. In a zero-order reaction, the half-life depends on the initial concentration; by substituting [A]t = ½[A]0 into the integrated rate law, one obtains t1/2 = [A]0/(2k). This derivation is an important concept in kinetics because it highlights the unique dependence of the half-life on the initial concentration for zero-order processes, unlike in first-order reactions where the half-life is constant.

Graphical Analysis in Reaction Kinetics

Graphical representation in reaction kinetics involves plotting data to visually assess the kinetic behavior of reactions. In the context of a zero-order reaction, plotting [A]t versus time yields a straight line with a slope of -k, while plotting the reaction rate versus [A]t results in a horizontal line, since the rate is constant and independent of the reactant concentration. These plots are crucial for verifying the reaction order and for determining the rate constant experimentally.

Limitations of Integrated Rate Laws

Integrated rate laws typically assume ideal reaction conditions and continuous validity until complete consumption of the reactant. However, the practical application of these laws fails when the reactant concentration reaches zero since the mathematical form of the rate law becomes invalid beyond this time. Understanding the boundaries and limitations of integrated rate laws is critical in chemical kinetics to apply the models correctly and interpret experimental data, especially when approaching the end of a reaction.

Zero-Order Reaction Kinetics

Zero-order kinetics refers to reactions where the rate is independent of the concentration of the reactant. In such reactions, the rate at which the concentration decreases is constant over time, leading to a linear dependency between the reactant concentration and time. This concept is fundamental in kinetics and distinguishes zero-order behavior from first- or second-order reactions where the rate depends on the concentration of one or more reactants.

Integrated Rate Law

The integrated rate law provides a relation describing how the concentration of a reactant changes over time. For a zero-order reaction, the integrated rate law is written as [A]t = [A]0 - kt, which is a linear equation. This form allows for the prediction and analysis of concentration profiles throughout the reaction course and is essential for understanding reaction kinetics and determining kinetic parameters.

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