The decomposition of azomethane, (CH3)2 N2, to nitrogen and ethane gases is a first-order react

step 1 In this question, we're given the decomposition of azomethane and we're told that it's a first -

The decomposition of azomethane, (CH3)2 N2, to nitrogen and...

The decomposition of azomethane, $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2},$ to nitrogen and ethane gases is a first-order reaction, $$ \left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2}(g) \rightarrow \mathrm{N}_{2}(g)+\mathrm{C}_{2} \mathrm{H}_{6}(g) $$ At a certain temperature, a 29 -mg sample of azomethane is reduced to $12 \mathrm{mg}$ in $1.4 \mathrm{~s}$. (a) What is the rate constant $k$ for the decomposition at that temperature? (b) What is the half-life of the decomposition? (c) How long will it take to decompose $78 \%$ of the azomethane?

The decomposition of azomethane, $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2},$ to nitrogen and ethane gases is a first-order reaction,
$$
\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2}(g) \rightarrow \mathrm{N}_{2}(g)+\mathrm{C}_{2} \mathrm{H}_{6}(g)
$$
At a certain temperature, a 29 -mg sample of azomethane is reduced to $12 \mathrm{mg}$ in $1.4 \mathrm{~s}$.
(a) What is the rate constant $k$ for the decomposition at that temperature?
(b) What is the half-life of the decomposition?
(c) How long will it take to decompose $78 \%$ of the azomethane?

Key Concepts

Percent Decomposition

Percent decomposition refers to the fraction or percentage of the initial reactant that has been transformed into products over time. In kinetic problems, it is used to determine the time required for a certain proportion of reactant to decompose, applying the integrated rate law to assess the extent of reaction progress.

Integrated Rate Law

The integrated rate law for a first order reaction links the concentration of the reactant at any time to its initial concentration through the equation ln([A]?/[A]) = kt. This equation is fundamental for analyzing kinetic data, calculating the rate constant, and predicting how long it will take for a given change in concentration.

Half-Life

The half-life of a reaction is the time required for the concentration of a reactant to fall to half of its original value. For first order reactions, the half-life is constant regardless of the initial concentration and is calculated using the formula t?/? = ln(2)/k, making it a useful parameter for characterizing the kinetics.

First Order Reaction

A first order reaction is one where the rate depends directly on the concentration of a single reactant. The rate law for a first order reaction is expressed as rate = k[reactant], meaning that the reaction rate is proportional to the current concentration of the reactant, leading to exponential decay over time.

Rate Constant

The rate constant, denoted as k, is a proportionality factor in the rate law that determines the speed of the reaction. In first order kinetics, k has units of inverse time and its magnitude directly influences how quickly the concentration of the reactant decreases.

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