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Chapter 12: Problem 97

A reaction of the form \(\mathrm{aA} \longrightarrow\) Products gives a plot of \(\ln [\mathrm{A}]\) versus time (in seconds), which is a straight line with a slope of \(-7.35 \times 10^{-3}\). Assuming \([\mathrm{A}]_{0}=\) \(0.0100 M\), calculate the time (in seconds) required for the reaction to reach \(22.9 \%\) completion.

Short Answer

Expert verified
The time required for the reaction to reach 22.9% completion is approximately \(321.83\,\text{seconds}\).

Step by step solution

01

Recall the integrated rate law for a first-order reaction

For a first-order reaction, the integrated rate law is given by: \[ \ln \frac{[\mathrm{A}]}{[\mathrm{A}]_0} = -kt \] Where, ln[A] is the natural logarithm of the concentration of A at time t, [A]_0 is the initial concentration of A, k is the rate constant, and t is the time.
02

Calculate the rate constant (k) using the given slope

Since the plot of ln[A] versus time is a straight line, the slope equals the negative rate constant (-k). We are given the slope as -7.35 × 10^{-3}. Thus, \[ k = - (-7.35 \times 10^{-3}) = 7.35 \times 10^{-3} \]
03

Calculate the concentration of A at 22.9% completion

We are asked to find the time when the reaction is 22.9% complete. In other words, we must find the concentration of A when 22.9% of A has reacted. Calculate this percentage of the initial concentration of A: \[ [\mathrm{A}] = [\mathrm{A}]_0 - ([\mathrm{A}]_0 \times 0.229) = 0.0100 - (0.0100 \times 0.229) = 0.0100 - 0.00229 = 0.00771\,\mathrm{M} \]
04

Use the integrated rate law and calculated values to find the time

Now, we have the rate constant (k) and the concentration of A at 22.9% completion ([A] = 0.00771 M). We can plug these values into the integrated rate law equation to find the time (t) when the reaction is 22.9% complete: \[ \ln \frac{0.00771}{0.0100} = -(7.35 \times 10^{-3})t \] Solve for t: \[ t = \frac{\ln \frac{0.00771}{0.0100}}{-(7.35 \times 10^{-3})} = \frac{\ln 0.771}{-7.35 \times 10^{-3}} \approx 321.83\,\mathrm{seconds} \] So, it takes approximately 321.83 seconds for the reaction to reach 22.9% completion.

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Most popular questions from this chapter

Consider the general reaction $$ \mathrm{aA}+\mathrm{bB} \longrightarrow \mathrm{cC} $$ and the following average rate data over some time period \(\Delta t\) : $$ \begin{aligned} -\frac{\Delta \mathrm{A}}{\Delta t} &=0.0080 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \\ -\frac{\Delta \mathrm{B}}{\Delta t} &=0.0120 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \\ \frac{\Delta \mathrm{C}}{\Delta t} &=0.0160 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \end{aligned} $$ Determine a set of possible coefficients to balance this general reaction.

Define what is meant by unimolecular and bimolecular steps. Why are termolecular steps infrequently seen in chemical reactions?

The reaction $$ \left(\mathrm{CH}_{\mathrm{3}}\right)_{3} \mathrm{CBr}+\mathrm{OH}^{-} \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}+\mathrm{Br}^{-} $$ in a certain solvent is first order with respect to \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}\) and zero order with respect to \(\mathrm{OH}^{-}\). In several experiments, the rate constant \(k\) was determined at different temperatures. \(\mathrm{A}\) plot of \(\ln (k)\) versus \(1 / T\) was constructed resulting in a straight line with a slope value of \(-1.10 \times 10^{4} \mathrm{~K}\) and \(y\) -intercept of 33.5. Assume \(k\) has units of \(\mathrm{s}^{-1}\). a. Determine the activation energy for this reaction. b. Determine the value of the frequency factor \(A\). c. Calculate the value of \(k\) at \(25^{\circ} \mathrm{C}\).

Consider the hypothetical reaction $$ \mathrm{A}+\mathrm{B}+2 \mathrm{C} \longrightarrow 2 \mathrm{D}+3 \mathrm{E} $$ where the rate law is $$ \text { Rate }=-\frac{\Delta[\mathrm{A}]}{\Delta t}=k[\mathrm{~A}][\mathrm{B}]^{2} $$ An experiment is carried out where \([\mathrm{A}]_{0}=1.0 \times 10^{-2} \mathrm{M}\), \([\mathrm{B}]_{0}=3.0 M\), and \([\mathrm{C}]_{0}=2.0 M .\) The reaction is started, and after \(8.0\) seconds, the concentration of \(\mathrm{A}\) is \(3.8 \times 10^{-3} \mathrm{M}\). a. Calculate the value of \(k\) for this reaction. b. Calculate the half-life for this experiment. c. Calculate the concentration of A after \(13.0\) seconds. d. Calculate the concentration of \(\mathrm{C}\) after \(13.0\) seconds.

A first-order reaction is \(75.0 \%\) complete in \(320 . \mathrm{s}\). a. What are the first and second half-lives for this reaction? b. How long does it take for \(90.0 \%\) completion?
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