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

step 1 For a zero -order reaction, the rate stays constant. Rate is simply equal to the rate constant.

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 follow- ing 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 (c) Calculate the time in half-lives when the integrated rate law is no longer valid, that is, when $[\mathrm{A}]_{t}=0$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 follow-
ing 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 (c) Calculate the time in half-lives when the integrated rate law is no longer valid, that is, when $[\mathrm{A}]_{t}=0$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

Integrated Rate Laws

Integrated rate laws are equations derived from the differential rate laws that relate the concentration of reactants to time. They are fundamental in chemical kinetics as they allow the prediction of how concentrations decrease or increase over time, enabling the determination of reaction rates and the comparison of different kinetic orders.

Zero-order Reaction Kinetics

Zero-order reactions are those in which the rate is independent of the concentration of the reactant. This unique behavior leads to a linear relationship between the concentration of the reactant and time, as expressed by the integrated rate law [A]? = [A]? - k*t. Understanding this kinetic order helps in analyzing systems where the reaction rate remains constant until depletion of the reactant.

Graphical Analysis in Reaction Kinetics

Graphical analysis is a key tool in understanding reaction kinetics. For zero-order reactions, plotting [A] versus time yields a straight line with a negative slope equal to the rate constant k, while a plot of reaction rate versus concentration results in a horizontal line, reflecting the concentration-independent rate. Such plots help in the visualization and validation of the kinetic order and rate laws.

Half-life in Reaction Kinetics

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half its initial value. In zero-order reactions, the half-life depends on the initial concentration and is derived from the integrated rate law; it differs significantly from first-order reactions where the half-life is constant regardless of starting concentration. This concept is crucial for predicting reaction progress over time.

Limits of Integrated Rate Laws

The integrated rate law has its domain of validity, and for a zero-order reaction, it is only applicable until the reactant concentration reaches zero. Beyond this point, the mathematical expression predicts non-physical negative concentrations. Recognizing these limits is important for correctly applying kinetic models and understanding the endpoint of the reaction.

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