/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 97 A reaction of the form $$aA \lon... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 97

A reaction of the form $$aA \longrightarrow Products$$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 \(36.1 \,\mathrm{s}\).

Step by step solution

01

Calculate the reaction rate constant

The slope of the given plot equals the negative value of the reaction rate constant, so: \[k = -(-7.35 \times 10^{-3})\] \[k = 7.35 \times 10^{-3} \,\mathrm{s^{-1}}\]
02

Determine the concentration of A at 22.9% completion

To find the concentration of A at \(22.9\%\) completion, we'll multiply the initial concentration with the remaining percentage: \[[\mathrm{A}] = [\mathrm{A}]_{0} \times \left(1 - \frac{22.9}{100}\right)\] \[[\mathrm{A}] = 0.0100 \,\mathrm{M} \times \left(1- 0.229\right)\] \[[\mathrm{A}] = 0.0100 \,\mathrm{M} \times 0.771\] \[[\mathrm{A}] = 0.00771 \,\mathrm{M}\]
03

Calculate the time required for 22.9% completion

Now, we can insert the values of \(k, [\mathrm{A}]_{0}\), and \([\mathrm{A}]\) into the formula for the first-order reaction, and solve for the time, \(t\): \[\ln\frac{[\mathrm{A}]_{0}}{[\mathrm{A}]} = kt\] \[\ln\frac{0.0100}{0.00771} = (7.35 \times 10^{-3}) t\] \[0.265 = 7.35 \times 10^{-3}t\] Now we'll solve for \(t\): \[t = \frac{0.265}{7.35 \times 10^{-3}}\] \[t = 36.1 \,\mathrm{s}\] So, it takes 36.1 seconds for the reaction to reach \(22.9\%\) completion.

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Most popular questions from this chapter

The activation energy of a certain uncatalyzed biochemical reaction is \(50.0 \space\mathrm{kJ} / \mathrm{mol} .\) In the presence of a catalyst at \(37^{\circ} \mathrm{C}\) the rate constant for the reaction increases by a factor of \(2.50 \times 10^{3}\) as compared with the uncatalyzed reaction. Assuming the frequency factor \(A\) is the same for both the catalyzed and uncatalyzed reactions, calculate the activation energy for the catalyzed reaction.

Consider a reaction of the type aA \(\longrightarrow\) products, in which the rate law is found to be rate \(=k[\mathrm{A}]^{3}\) (termolecular reactions are improbable but possible). If the first half-life of the reaction is found to be \(40 .\) s, what is the time for the second half-life? Hint: Using your calculus knowledge, derive the integrated rate law from the differential rate law for a termolecular reaction: $$\text { Rate }=\frac{-d[\mathrm{A}]}{d t}=k[\mathrm{A}]^{3}$$

The reaction $$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}+\mathrm{OH}^{-} \longrightarrow\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}+\mathrm{Br}^{-}$$ in a certain solvent is first order with respect to \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CBr}\) and zero order with respect to OH \(^{-} .\) In several experiments, the rate constant \(k\) was determined at different temperatures. 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 \(25^{\circ} \mathrm{C}\)

For the reaction \(A+B \rightarrow C\), explain at least two ways in which the rate law could be zero order in chemical A.

Consider the following statements: "In general, the rate of a chemical reaction increases a bit at first because it takes a while for the reaction to get "warmed up.' After that, however, the rate of the reaction decreases because its rate is dependent on the concentrations of the reactants, and these are decreasing." Indicate everything that is correct in these statements, and indicate everything that is incorrect. Correct the incorrect statements and explain.
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