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Chapter 11: Problem 96

The following reaction is second-order in A and first-order in B. $$\mathrm{A}+\mathrm{B} \longrightarrow \text { products }$$ (a) Write the rate expression. (b) Consider the following one-liter vessel in which each square represents a mole of \(\mathrm{A}\) and each circle represents a mole of \(\mathrm{B}\). What is the rate of the reaction in terms of \(k ?\) (c) Assuming the same rate and \(k\) as (b), fill the similar one-liter vessel shown in the figure with an appropriate number of circles (representing B).

Short Answer

Expert verified
Additionally, how many moles of B are needed in a vessel with 1 mole of A to maintain the same rate? Answer: The rate of the reaction with 2 moles of A and 4 moles of B is 16k. To maintain the same rate with 1 mole of A, you would need 16 moles of B.

Step by step solution

01

(a) Writing the rate expression

The given reaction is second-order in A and first-order in B. Therefore, the rate expression can be written as: $$\text{rate} = k[\mathrm{A}]^2[\mathrm{B}]$$
02

(b) Finding the rate of the reaction in terms of k

In the given one-liter vessel, we have 2 moles of A (squares) and 4 moles of B (circles). Using the rate expression, we can find the rate in terms of k: $$\text{rate} = k[\mathrm{A}]^2[\mathrm{B}] = k(2\,\text{mol})^2(4\,\text{mol}) = 16k$$ So, the rate of the reaction is \(16k\).
03

(c) Filling the one-liter vessel with an appropriate number of circles (B) for the same rate

Since we want to maintain the same rate as in part (b) and we have the same rate constant, k, we can set up an equation using the rate expression: $$16k = k[\mathrm{A}]^2[\mathrm{B}]$$ In the new scenario, there are 1 mole of A (square) in the one-liter vessel. So, you need to find how many moles of B (circles) are required to keep the same rate: $$16k = k(1\,\text{mol})^2[\mathrm{B}]$$ $$16k = k[\mathrm{B}]$$ Now, solve for the concentration of B: $$[\mathrm{B}] = \frac{16k}{k}$$ $$[\mathrm{B}] = 16\,\text{mol}$$ Therefore, in order to maintain the same rate as in part (b), you need 16 circles (moles of B) in the one-liter vessel in the new scenario.

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Most popular questions from this chapter

Write the rate expression for each of the following elementary steps: (a) \(\mathrm{NO}+\mathrm{O}_{3} \longrightarrow \mathrm{NO}_{2}+\mathrm{O}_{2}\) (b) \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) (c) \(\mathrm{K}+\mathrm{HCl} \longrightarrow \mathrm{KCl}+\mathrm{H}\)

Diethylhydrazine reacts with iodine according to the following equation: $$\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}(\mathrm{NH})_{2}(l)+\mathrm{I}_{2}(a q) \longrightarrow\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{~N}_{2}(l)+2 \mathrm{HI}(a q)$$ The rate of the reaction is followed by monitoring the disappearance of the purple color due to iodine. The following data are obtained at a certain temperature. $$ \begin{array}{cccc} \hline \text { Expt. } & {\left[\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}(\mathrm{NH})_{2}\right]} & {\left[\mathrm{I}_{2}\right]} & \text { Initial Rate }(\mathrm{mol} / \mathrm{L} \cdot \mathrm{h}) \\ \hline 1 & 0.150 & 0.250 & 1.08 \times 10^{-4} \\ 2 & 0.150 & 0.3620 & 1.56 \times 10^{-4} \\ 3 & 0.200 & 0.400 & 2.30 \times 10^{-4} \\ 4 & 0.300 & 0.400 & 3.44 \times 10^{-4} \\ \hline\end{array}$$ (a) What is the order of the reaction with respect to diethylhydrazine, iodine, and overall? (b) Write the rate expression for the reaction. (c) Calculate \(k\) for the reaction. (d) What must \(\left[\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2}(\mathrm{NH})_{2}\right]\) be so that the rate of the reaction is \(5.00 \times 10^{-4} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{h}\) when \(\left[\mathrm{I}_{2}\right]=0.500 ?\)

For the first-order thermal decomposition of ozone $$\mathrm{O}_{3}(g) \longrightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)$$ \(k=3 \times 10^{-26} \mathrm{~s}^{-1}\) at \(25^{\circ} \mathrm{C}\). What is the half-life for this reaction in years? Comment on the likelihood that this reaction contributes to the depletion of the ozone layer.

For a first-order reaction \(a \mathrm{~A} \longrightarrow\) products, where \(a \neq 1\), the rate is \(-\Delta[\mathrm{A}] / a \Delta t\), or in derivative notation, \(-\frac{1}{a} \frac{d[\mathrm{~A}]}{d t} .\) Derive the integrated rate law for the first-order decomposition of \(a\) moles of reactant.

Sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) hydrolyzes into glucose and fructose. The hydrolysis is a first-order reaction. The half-life for the hydrolysis of sucrose is \(64.2\) min at \(25^{\circ} \mathrm{C}\). How many grams of sucrose in \(1.25 \mathrm{~L}\) of a \(0.389 \mathrm{M}\) solution are hydrolyzed in \(1.73\) hours?
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