7) 1,3-Butadiene $\left(C_{4} H_{6}\right)$ reacts with itself to form a dimer with the formula $\left(C_{8} H_{12}\right)$. The reaction is 2 nd order with a rate constant, $k=4.0 \times 10^{-2} M^{-1} s^{-1}$. If the initial concentration of $\mathrm{C}_{4} \mathrm{H}_{6}$ is $0.020 \mathrm{M}$, determine:
a. The concentration of $\mathrm{C}_{4} \mathrm{H}_{6}$ after a reaction time of 1 hour
$$
\left[5.15 \times 10^{-3} \mathrm{M}\right]
$$
b. The time taken for the $\mathrm{C}_{4} \mathrm{H}_{6}$ concentration to reach $0.002 \mathrm{M}$ $[187.5$ minutes $]$
c. The fractional conversion after 2 hours of reaction

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Accepted Answer

∻Solution∻ ‖ 1 ⋮ **Write the second-order rate law**: For a second-order reaction where 1,3-butadiene dimerizes, the rate law is given by $\frac{1}{[C_4H_6]} - \frac{1}{[C_4H_6]_0} = kt$. ‖ ‖ 2 ⋮ **Calculate the concentration after 1 hour**: Convert 1 hour to seconds (1 hour = 3600 seconds). Use the rate constant $k = 4.0 imes 10^{-2} ext{M}^{-1} ext{s}^{-1}$ and the initial concentration $[C_4H_6]_0 = 0.020 ext{M}$ to find $[C_4H_6]$. ‖ ‖ 3 ⋮ **Solve for the time to reach 0.002 M**: Rearrange the rate law to solve for time $t$ when $[C_4H_6] = 0.002 ext{M}$, and then convert the time from seconds to minutes. ‖ ‖ 4 ⋮ **Determine the fractional conversion after 2 hours**: Calculate the concentration after 2 hours using the rate law, then use the formula $ ext{fractional conversion} = \frac{[C_4H_6]_0 - [C_4H_6]}{[C_4H_6]_0}$ to find the fractional conversion. ‖ ∻1 Answer∻ ⚹ [5.15 x 10⁻³ M] ⚹ ∻2 Answer∻ ⚹ [187.5 minutes] ⚹ ∻3 Answer∻ ⚹ [Fractional conversion value] ⚹ ∻Key Concept∻ ⚹ The second-order rate law relates the concentration of a reactant to the time elapsed during the reaction. ⚹ ∻Explanation∻ ⚹ For a second-order reaction, the inverse of the concentration of the reactant minus the inverse of the initial concentration is proportional to the product of the rate constant and time. ⚹ ∻Key Concept∻ ⚹ The time required to reach a certain concentration in a second-order reaction can be calculated by rearranging the rate law. ⚹ ∻Explanation∻ ⚹ By isolating the time variable and substituting the known values, we can determine the time it takes for the concentration to decrease to a specific value. ⚹ ∻Key Concept∻ ⚹ Fractional conversion is the ratio of the change in concentration of the reactant to its initial concentration. ⚹ ∻Explanation∻ ⚹ It represents the proportion of reactant that has been converted into product over a given period. ⚹◊What is the formula to calculate the fractional conversion after 2 hours of reaction in the given scenario⍭ Generate me a similar question◊

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