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Chapter 14: Problem 22

For a first-order reaction, how long will it take for the concentration of reactant to fall to one-eighth its original value? Express your answer in terms of the half-life \(\left(t_{\frac{1}{2}}\right)\) and in terms of the rate constant \(k\)

Short Answer

Expert verified
It will take three half-lives (\(3 \cdot t_{\frac{1}{2}}\)) or approximately 2.08 times the inverse of the rate constant (\(2.079 \cdot \frac{1}{k}\)) for the concentration of a reactant to drop to one-eighth its original value in a first-order reaction.

Step by step solution

01

- Understanding a First Order Reaction

For first order reactions, the rate of reaction decreases exponentially over time. We can express this as a mathematical relationship, i.e., \( N = N_0 \cdot (0.5) ^ \frac{t}{t_{\frac{1}{2}}}\) where \(N\) is the final concentration, \(N_0\) is the initial concentration, and \(t_{\frac{1}{2}}\) is the half-life.
02

Step 2- Applying the Relationship

We want the concentration to fall to one-eighth its original value, thus \( N = \frac{1}{8} N_0 \). Now we substitute \(N\) into the equation from step 1 and simplify which results in \( \frac{1}{8} = (0.5) ^ \frac{t}{t_{\frac{1}{2}}}\).
03

- Solving the Equation

We need to solve the equation from step 2 for \(t\). This involves utilising properties of logarithms. Solving the equation gives us \( t = 3 \cdot t_{\frac{1}{2}}\). This means that for a first order reaction, it will take three half-lives for the concentration of reactant to fall to one-eighth its original value.
04

- Express the Solution in Terms of the Rate Constant

For a first-order reaction, the half-life \(t_{\frac{1}{2}}\) is related to the rate constant \(k\) by the equation \(t_{1/2} = \frac{0.693}{k}\). To express our solution in terms of the rate constant, substitute this equation into our answer from step 3. We then get \(t = 3 \cdot \frac{0.693}{k} = 2.079 \cdot \frac{1}{k} \). This means that it will take approximately 2.08 times the inverse of the rate constant for the concentration of the reactant to fall to one-eighth its original value.

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Most popular questions from this chapter

The bromination of acetone is acid-catalyzed: \(\mathrm{CH}_{3} \mathrm{COCH}_{3}+\mathrm{Br}_{2} \frac{\mathrm{H}^{+}}{\text {catalyst }} \mathrm{CH}_{3} \mathrm{COCH}_{2} \mathrm{Br}+\mathrm{H}^{+}+\mathrm{Br}\) The rate of disappearance of bromine was measured for several different concentrations of acetone, bromine, and \(\mathrm{H}^{+}\) ions at a certain temperature: $$ \begin{array}{lcllc} & & & & {\text { Rate of }} \\ & & & & \text { Disappearance } \\ & {\left[\mathrm{CH}_{3} \mathrm{COCH}_{3}\right]} & {\left[\mathrm{Br}_{2}\right]} & {\left[\mathrm{H}^{+}\right]} & \text {of } \mathrm{Br}_{2}(\mathrm{M} / \mathrm{s}) \\ \hline \text { (a) } & 0.30 & 0.050 & 0.050 & 5.7 \times 10^{-5} \\ \text {(b) } & 0.30 & 0.10 & 0.050 & 5.7 \times 10^{-5} \\ \text {(c) } & 0.30 & 0.050 & 0.10 & 1.2 \times 10^{-4} \\ \text {(d) } & 0.40 & 0.050 & 0.20 & 3.1 \times 10^{-4} \\ \text {(e) } & 0.40 & 0.050 & 0.050 & 7.6 \times 10^{-5} \end{array} $$ (a) What is the rate law for the reaction? (b) Determine the rate constant.

Which of these species cannot be isolated in a reaction: activated complex, product, intermediate?

The rate constant of a first-order reaction is \(66 \mathrm{~s}^{-1}\) What is the rate constant in units of minutes?

These data were collected for the reaction between hydrogen and nitric oxide at \(700^{\circ} \mathrm{C}\) : \(2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{N}_{2}(g)\) $$ \begin{array}{cllc} \text { Experiment } & {\left[\mathrm{H}_{2}\right]} & {[\mathrm{NO}]} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\ \hline 1 & 0.010 & 0.025 & 2.4 \times 10^{-6} \\ 2 & 0.0050 & 0.025 & 1.2 \times 10^{-6} \\ 3 & 0.010 & 0.0125 & 0.60 \times 10^{-6} \end{array} $$ (a) Determine the order of the reaction. (b) Calculate the rate constant. (c) Suggest a plausible mechanism that is consistent with the rate law. (Hint: Assume the oxygen atom is the intermediate.)

Consider the reaction $$ X+Y \longrightarrow Z $$ These data are obtained at \(360 \mathrm{~K}\): $$ \begin{aligned} &\text { Initial Rate of }\\\ &\begin{array}{ccc} \text { Disappearance of } \mathrm{X}(\mathrm{M} / \mathrm{s}) & {[\mathrm{X}]} & {[\mathrm{Y}]} \\ \hline 0.147 & 0.10 & 0.50 \\ 0.127 & 0.20 & 0.30 \\ 4.064 & 0.40 & 0.60 \\ 1.016 & 0.20 & 0.60 \\ 0.508 & 0.40 & 0.30 \end{array} \end{aligned} $$ (a) Determine the order of the reaction. (b) Determine the initial rate of disappearance of \(X\) when the concentration of \(\mathrm{X}\) is \(0.30 \mathrm{M}\) and that of \(\mathrm{Y}\) is \(0.40 \mathrm{M}\)
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