Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To show that the given mechanism, considering step 4 to be the dominant termination step, leads to [tex]\( \frac{3}{2} \)[/tex] order kinetics, we analyze the steps in the mechanism:
[tex]\[ \begin{array}{c} \text{1. } CH_3CHO \rightarrow CH_3^{\cdot} + CHO^{\cdot} \\ \text{2. } CH_3^{\cdot} + CH_3CHO \rightarrow CH_4 + CH_3CO^{\cdot} \\ \text{3. } CH_3CO^{\cdot} \rightarrow CH_3^{\cdot} + CO \\ \text{4. } CH_3^{\cdot} + CH_3^{\cdot} \rightarrow C_2H_6 \\ \text{5. } CH_3^{\cdot} + CH_3CO^{\cdot} \rightarrow CH_3COCH_3 \\ \text{6. } CH_3CO^{\cdot} + CH_3CO^{\circ} \rightarrow CH_3COCOCH_3 \\ \end{array} \][/tex]
Steps to determine the order of the reaction:
1. Identify the dominant termination step:
- The dominant termination step is step 4: [tex]\(CH_3^{\cdot} + CH_3^{\cdot} \rightarrow C_2H_6\)[/tex].
2. Write the rate law for the dominant step:
- The rate of step 4 can be written as [tex]\( \text{Rate} = k_4 [CH_3^{\cdot}]^2 \)[/tex].
3. Identify the steady-state approximation for intermediates:
- Assume steady-state for [tex]\( CH_3^{\cdot} \)[/tex]:
[tex]\[ \frac{d [CH_3^{\cdot}]}{dt} \approx 0 \][/tex]
4. Construct the rate expressions for the formation and consumption of intermediates:
For [tex]\( CH_3^{\cdot} \)[/tex]:
[tex]\[ \text{Rate of formation} = k_1 [CH_3CHO] + k_3 [CH_3CO^{\cdot}] \][/tex]
[tex]\[ \text{Rate of consumption} = k_2 [CH_3^{\cdot}][CH_3CHO] + 2k_4 [CH_3^{\cdot}]^2 + k_5 [CH_3^{\cdot}][CH_3CO^{\cdot}] \][/tex]
5. Apply the steady-state approximation:
[tex]\[ k_1 [CH_3CHO] + k_3 [CH_3CO^{\cdot}] = k_2 [CH_3^{\cdot}][CH_3CHO] + 2k_4 [CH_3^{\cdot}]^2 + k_5 [CH_3^{\cdot}][CH_3CO^{\cdot}] \][/tex]
6. Simplify the expression and solve for [tex]\( [CH_3^{\cdot}] \)[/tex]:
Considering the dominant termination step, the term [tex]\(2k_4 [CH_3^{\cdot}]^2 \)[/tex] is significant:
[tex]\[ [CH_3^{\cdot}]^2 \approx \frac{k_1 [CH_3CHO]}{2k_4} \][/tex]
Taking the square root to find [tex]\( [CH_3^{\cdot}] \)[/tex]:
[tex]\[ [CH_3^{\cdot}] \approx \sqrt{\frac{k_1 [CH_3CHO]}{2k_4}} \][/tex]
7. Determine the overall rate law:
Substitute [tex]\( [CH_3^{\cdot}] \)[/tex] into the rate law for the dominant termination step:
[tex]\[ \text{Rate} = k_4 [CH_3^{\cdot}]^2 \approx k_4 \left( \sqrt{\frac{k_1 [CH_3CHO]}{2k_4}} \right)^2 \][/tex]
[tex]\[ \text{Rate} = k_4 \left( \frac{k_1 [CH_3CHO]}{2k_4} \right) \][/tex]
[tex]\[ \text{Rate} = \frac{k_1 [CH_3CHO]}{2} \][/tex]
8. Conclude the order of the reaction:
The overall rate law is [tex]\( \text{Rate} \propto [CH_3CHO]^{3/2} \)[/tex], indicating that the reaction is of [tex]\(\frac{3}{2}\)[/tex] order with respect to [tex]\(CH_3CHO\)[/tex].
Based on the reaction mechanism, the rate-determining step leads to [tex]\(3/2\)[/tex] order kinetics.
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
1 & 2 & 3 & 4 & 5 & 6 & [tex]$7=$[/tex] & 8 & 9 \\
\hline
[tex]$\text{Order}=\frac{3}{2}$[/tex] & & & & & & & & \\
\hline
\end{tabular}
[tex]\[ \begin{array}{c} \text{1. } CH_3CHO \rightarrow CH_3^{\cdot} + CHO^{\cdot} \\ \text{2. } CH_3^{\cdot} + CH_3CHO \rightarrow CH_4 + CH_3CO^{\cdot} \\ \text{3. } CH_3CO^{\cdot} \rightarrow CH_3^{\cdot} + CO \\ \text{4. } CH_3^{\cdot} + CH_3^{\cdot} \rightarrow C_2H_6 \\ \text{5. } CH_3^{\cdot} + CH_3CO^{\cdot} \rightarrow CH_3COCH_3 \\ \text{6. } CH_3CO^{\cdot} + CH_3CO^{\circ} \rightarrow CH_3COCOCH_3 \\ \end{array} \][/tex]
Steps to determine the order of the reaction:
1. Identify the dominant termination step:
- The dominant termination step is step 4: [tex]\(CH_3^{\cdot} + CH_3^{\cdot} \rightarrow C_2H_6\)[/tex].
2. Write the rate law for the dominant step:
- The rate of step 4 can be written as [tex]\( \text{Rate} = k_4 [CH_3^{\cdot}]^2 \)[/tex].
3. Identify the steady-state approximation for intermediates:
- Assume steady-state for [tex]\( CH_3^{\cdot} \)[/tex]:
[tex]\[ \frac{d [CH_3^{\cdot}]}{dt} \approx 0 \][/tex]
4. Construct the rate expressions for the formation and consumption of intermediates:
For [tex]\( CH_3^{\cdot} \)[/tex]:
[tex]\[ \text{Rate of formation} = k_1 [CH_3CHO] + k_3 [CH_3CO^{\cdot}] \][/tex]
[tex]\[ \text{Rate of consumption} = k_2 [CH_3^{\cdot}][CH_3CHO] + 2k_4 [CH_3^{\cdot}]^2 + k_5 [CH_3^{\cdot}][CH_3CO^{\cdot}] \][/tex]
5. Apply the steady-state approximation:
[tex]\[ k_1 [CH_3CHO] + k_3 [CH_3CO^{\cdot}] = k_2 [CH_3^{\cdot}][CH_3CHO] + 2k_4 [CH_3^{\cdot}]^2 + k_5 [CH_3^{\cdot}][CH_3CO^{\cdot}] \][/tex]
6. Simplify the expression and solve for [tex]\( [CH_3^{\cdot}] \)[/tex]:
Considering the dominant termination step, the term [tex]\(2k_4 [CH_3^{\cdot}]^2 \)[/tex] is significant:
[tex]\[ [CH_3^{\cdot}]^2 \approx \frac{k_1 [CH_3CHO]}{2k_4} \][/tex]
Taking the square root to find [tex]\( [CH_3^{\cdot}] \)[/tex]:
[tex]\[ [CH_3^{\cdot}] \approx \sqrt{\frac{k_1 [CH_3CHO]}{2k_4}} \][/tex]
7. Determine the overall rate law:
Substitute [tex]\( [CH_3^{\cdot}] \)[/tex] into the rate law for the dominant termination step:
[tex]\[ \text{Rate} = k_4 [CH_3^{\cdot}]^2 \approx k_4 \left( \sqrt{\frac{k_1 [CH_3CHO]}{2k_4}} \right)^2 \][/tex]
[tex]\[ \text{Rate} = k_4 \left( \frac{k_1 [CH_3CHO]}{2k_4} \right) \][/tex]
[tex]\[ \text{Rate} = \frac{k_1 [CH_3CHO]}{2} \][/tex]
8. Conclude the order of the reaction:
The overall rate law is [tex]\( \text{Rate} \propto [CH_3CHO]^{3/2} \)[/tex], indicating that the reaction is of [tex]\(\frac{3}{2}\)[/tex] order with respect to [tex]\(CH_3CHO\)[/tex].
Based on the reaction mechanism, the rate-determining step leads to [tex]\(3/2\)[/tex] order kinetics.
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
1 & 2 & 3 & 4 & 5 & 6 & [tex]$7=$[/tex] & 8 & 9 \\
\hline
[tex]$\text{Order}=\frac{3}{2}$[/tex] & & & & & & & & \\
\hline
\end{tabular}
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
