The rate equation for the decomposition of N2 O5 (giving NO2 and O2 ) is Rate =k[N2 O5] . The va

step 1 Question number 25 is both a half -life question and an integrated rate law equation. First, to

The rate equation for the decomposition of N2 O5 (giving NO2...

The rate equation for the decomposition of $\mathrm{N}_{2} \mathrm{O}_{5}$ (giving $\mathrm{NO}_{2}$ and $\mathrm{O}_{2}$ ) is Rate $=k\left[\mathrm{N}_{2} \mathrm{O}_{5}\right] .$ The value of $k$ is $6.7 \times 10^{-5} \mathrm{s}^{-1}$ for the reaction at a particular temperature. (a) Calculate the half-life of $\mathrm{N}_{2} \mathrm{O}_{5}$. (b) How long does it take for the $\mathrm{N}_{2} \mathrm{O}_{5}$ concentration to drop to one tenth of its original value?

The rate equation for the decomposition of $\mathrm{N}_{2} \mathrm{O}_{5}$ (giving $\mathrm{NO}_{2}$ and $\mathrm{O}_{2}$ ) is Rate $=k\left[\mathrm{N}_{2} \mathrm{O}_{5}\right] .$ The value of $k$ is $6.7 \times 10^{-5} \mathrm{s}^{-1}$ for the reaction at a particular temperature.
(a) Calculate the half-life of $\mathrm{N}_{2} \mathrm{O}_{5}$.
(b) How long does it take for the $\mathrm{N}_{2} \mathrm{O}_{5}$ concentration to drop to one tenth of its original value?

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Key Concepts

First-Order Reaction

A first-order reaction is one in which the rate of reaction is directly proportional to the concentration of a single reactant. In these reactions, the rate can be expressed as Rate = k[A], where k is the first-order rate constant and [A] is the concentration of the reactant. The distinguishing feature of first-order kinetics is that the half-life of the reaction remains constant regardless of the starting concentration.

Integrated Rate Law

The integrated rate law for a first-order reaction relates the concentration of the reactant at any time to its initial concentration using the equation ln([A]t/[A]0) = -kt. This relationship allows for the calculation of the time it takes for the reactant's concentration to change by a specific factor, such as decreasing to half (half-life) or to one tenth of its original value, by rearranging the equation and solving for time.

Half-Life Calculation

The half-life of a reaction is the time required for the concentration of a reactant to drop to half of its initial value. For first-order reactions, the half-life (t1/2) is given by the formula t1/2 = ln(2)/k. This property is particularly useful because it implies that the half-life is independent of the initial concentration, meaning that every successive half-life period will reduce the concentration by an additional half.

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