Chapter 15: Problem 17
Explain the meaning of the orientation factor in the collision model.
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Chapter 15: Problem 17
Explain the meaning of the orientation factor in the collision model.
These are the key concepts you need to understand to accurately answer the question.
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Get started for freeAnthropologists can estimate the age of a bone or other sample of or- ganic matter by its carbon-14 4 content. The carbon-14 in a living organ- ism is constant until the organism dies, after which carbon-14 decays with first-order kinetics and a half-life of 5730 years. Suppose a bone from an ancient human contains 19.5\(\%\) of the \(\mathrm{C}-14\) found in living or- ganisms. How old is the bone?
The desorption of a single molecular layer of \(n\) -butane from a single crystal of aluminum oxide is found to be first order with a rate constant of 0.128\(/ \mathrm{s}\) at 150 \(\mathrm{K}\) . \begin{equation} \begin{array}{l}{\text { a. What is the haff-life of the desorption reaction? }} \\ {\text { b. If the surface is initially completely covered with } n \text { -butane at }} \\ {150 \mathrm{K}, \text { how long will it take for } 25 \% \text { of the molecules to desorb? For }} \\ {50 \% \text { to desorb? }}\\\\{\text { c. If the surface is initially completely covered, what fraction will remain }} \\ {\text { covered after } 10 \text { s? After } 20 \mathrm{s?}}\end{array} \end{equation}
A reaction in which \(\mathrm{A}, \mathrm{B},\) and C react to form products is first order in \(\mathrm{A},\) second order in \(\mathrm{B},\) and \(\mathrm{zero}\) order in \(\mathrm{C}\) . \begin{equation} \begin{array}{l}{\text { a. Write a rate law for the reaction. }} \\ {\text { b. What is the overall order of the reaction? }} \\ {\text { c. By what factor does the reaction rate change if }[A] \text { is doubled (and the }} \\\ {\text { other reactant concentrations are held constant)? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { d. By what factor does the reaction rate change if }[\mathrm{B}] \text { is doubled (and the }} \\ {\text { other reactant concentrations are held constant)? }} \\ {\text { e. By what factor does the reaction rate change if }[\mathrm{C}] \text { is doubled (and the }} \\\ {\text { other reactant concentrations are held constant)? }}\\\\{\text { f. By what factor does the reaction rate change if the concentrations of }} \\\ {\text { all three reactants are doubled? }}\end{array} \end{equation}
The first-order integrated rate law for a reaction \(\mathrm{A} \longrightarrow\) products is derived from the rate law using calculus. \begin{equation} \begin{aligned} \text { Rate } &=k[\mathrm{A}] \quad \text { (first-order rate law) } \\ \text { Rate } &=\frac{d[\mathrm{A}]}{d t} \\\ \frac{d[\mathrm{A}]}{d t} &=-k[\mathrm{A}] \end{aligned} \end{equation} The equation just given is a first-order, separable differential equa- tion that can be solved by separating the variables and integrating: \begin{equation} \begin{aligned} \frac{d[\mathrm{A}]}{[\mathrm{A}]} &=-k d t \\\ \int_{[\mathrm{A}]_{0}}^{[\mathrm{A]}} \frac{d[\mathrm{A}]}{[\mathrm{A}]} &=-\int_{0}^{t} k d t \end{aligned} \end{equation} In the integral just given, \([\mathrm{A}]_{0}\) is the initial concentration of \(\mathrm{A} . \mathrm{We}\) then evaluate the integral: \begin{equation} \begin{aligned}[\ln [\mathrm{A}]]_{[\mathrm{A}]_{0}}^{[\mathrm{Al}} &=-k[t]_{0}^{t} \\ \ln [\mathrm{A}]-\ln [\mathrm{A}]_{0} &=-k t \end{aligned} \end{equation} \begin{equation} \ln [\mathrm{A}]=-k t+\ln [\mathrm{A}]_{0}(\text { integrated rate law }) \end{equation} \begin{equation} \begin{array}{l}{\text { a. Use a procedure similar to the one just shown to derive an inte- }} \\ {\text { grated rate law for a reaction } A \longrightarrow \text { products, which is one-half- }} \\ {\text { order in the concentration of } A \text { (that is, Rate }=k[A]^{1 / 2} )}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. Use the result from part a to derive an expression for the half-life }} \\ {\text { of a one-half-order reaction. }}\end{array} \end{equation}
What are the units of \(k\) for each type of reaction? \begin{equation} \begin{array}{l}{\text { a. first-order reaction }} \\ {\text { b. second- order reaction }} \\ {\text { c. } \text { zero-order reaction }}\end{array} \end{equation}
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