6) At $500^{\circ} \mathrm{C}$, cyclopropane $\left(\mathrm{C}_{3} \mathrm{H}_{6}\right)$ rearranges to propene $\left(\mathrm{CH}_{3}-\mathrm{CH}=\mathrm{CH}_{2}\right)$. The reaction is 1st order with a rate constant $k=6.7 \times 10^{-4} \mathrm{~s}^{-1}$.
a. Write the rate equation, and determine the rate of reaction when the conversion $x$ is 0 and 0.5 , given that the initial concentration of
$$
\left[3.35 \times 10-5 \mathrm{~mol} \mathrm{I}^{-1} \mathrm{~s}^{-1} ; 1.675 \times 10^{-5} \mathrm{~mol} \mathrm{I}^{-1} \mathrm{~s}^{-1}\right]
$$
b. Integrate the rate equation and determine:
i. The concentration of cyclopropane after 30 minutes
$$
[0.015 \mathrm{M}]
$$
ii. The time taken for the concentration of cyclopropane to fall to $0.001 \mathrm{~mol}^{2} \mathrm{litr}^{-1}$.
$$
[5839 \mathrm{~s},(97 \text { minutes)] }
$$
iii. The fractional conversion of cyclopropane and the concentration of propene after 50 minutes.
[0.866; $0.0433 \mathrm{M}]$

Question

Accepted Answer

∻Solution∻ ‖ 1 ⋮ **Write the rate equation**: For a first-order reaction, the rate equation is given by the formula $rate = k[A]$, where $k$ is the rate constant and $[A]$ is the concentration of the reactant. ‖ ‖ 2 ⋮ **Determine the rate of reaction at conversion $x = 0$**: When $x = 0$, the concentration of cyclopropane is at its initial value. The rate of reaction is then $rate = k[A]_0 = (6.7 imes 10^{-4} ext{ s}^{-1})([A]_0)$. ‖ ‖ 3 ⋮ **Determine the rate of reaction at conversion $x = 0.5$**: When $x = 0.5$, the concentration of cyclopropane is half of its initial value. The rate of reaction is then $rate = k[A]_{0.5} = (6.7 imes 10^{-4} ext{ s}^{-1})(\frac{[A]_0}{2})$. ‖ ‖ 4 ⋮ **Integrate the rate equation to find concentration after 30 minutes**: The integrated rate equation for a first-order reaction is $\ln([A]_0/[A]) = kt$. Solving for $[A]$ after 30 minutes (1800 seconds) gives the concentration of cyclopropane. ‖ ‖ 5 ⋮ **Determine the time for concentration to fall to $0.001 ext{ M}$**: Using the integrated rate equation, solve for $t$ when $[A] = 0.001 ext{ M}$. ‖ ‖ 6 ⋮ **Calculate the fractional conversion and concentration of propene after 50 minutes**: The fractional conversion is given by $x = 1 - [A]/[A]_0$. The concentration of propene is equal to the initial concentration of cyclopropane minus its concentration after 50 minutes. ‖ ∻1 Answer∻ ⚹ The rate of reaction at conversion $x = 0$ is $3.35 imes 10^{-5} ext{ mol L}^{-1} ext{s}^{-1}$ and at conversion $x = 0.5$ is $1.675 imes 10^{-5} ext{ mol L}^{-1} ext{s}^{-1}$. ⚹ ∻2 Answer∻ ⚹ The concentration of cyclopropane after 30 minutes is $0.015 ext{ M}$. ⚹ ∻3 Answer∻ ⚹ The time taken for the concentration of cyclopropane to fall to $0.001 ext{ M}$ is 5839 seconds (97 minutes). ⚹ ∻4 Answer∻ ⚹ The fractional conversion of cyclopropane after 50 minutes is 0.866, and the concentration of propene is $0.0433 ext{ M}$. ⚹ ∻Key Concept∻ ⚹ The rate equation for a first-order reaction relates the rate of reaction to the concentration of the reactant and the rate constant. ⚹ ∻Explanation∻ ⚹ The rate of a first-order reaction decreases as the concentration of the reactant decreases, which is reflected in the rate of reaction at different conversion levels. ⚹◊What is the overall reaction equation for the rearrangement of cyclopropane to propene at 500°C⍭ Generate me a similar question◊

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