91Ó°ÊÓ

Cyclopentadiene \(\left(\mathrm{C}_{5} \mathrm{H}_{6}\right)\) reacts with itself to form dicyclopentadiene \(\left(\mathrm{C}_{10} \mathrm{H}_{12}\right) .\) A \(0.0400 \mathrm{M}\) solution of \(\mathrm{C}_{5} \mathrm{H}_{6}\) was monitored as a function of time as the reaction \(2 \mathrm{C}_{5} \mathrm{H}_{6} \longrightarrow \mathrm{C}_{10} \mathrm{H}_{12}\) proceeded. The following data were collected: $$ \begin{array}{rl} \text { Time (s) } & {\left[\mathrm{C}_{5} \mathrm{H}_{6}\right](M)} \\ \hline 0.0 & 0.0400 \\ 50.0 & 0.0300 \\ 100.0 & 0.0240 \\ 150.0 & 0.0200 \\ 200.0 & 0.0174 \end{array} $$ Plot \(\left[\mathrm{C}_{5} \mathrm{H}_{6}\right]\) versus time, \(\ln \left[\mathrm{C}_{5} \mathrm{H}_{6}\right]\) versus time, and \(1 /\left[\mathrm{C}_{5} \mathrm{H}_{6}\right]\) versus time. What is the order of the reaction? What is the value of the rate constant?

Step by step solution

01

Plot the Concentration versus Time

First, we must create a table with the given data to plot the concentration of cyclopentadiene, \(\left[\text{C}_5\text{H}_6\right]\), versus time: $$ \begin{array}{rl} \text { Time (s) } & {\left[\mathrm{C}_{5} \mathrm{H}_{6}\right](M)} \\\ \hline 0.0 & 0.0400 \\\ 50.0 & 0.0300 \\\ 100.0 & 0.0240 \\\ 150.0 & 0.0200 \\\ 200.0 & 0.0174 \end{array} $$ Based on the plotted graph, determine if it shows a straight line. If so, the reaction would be zeroth-order.

02

Plot the \(\ln[\text{C}_5\text{H}_6]\) versus Time

Next, we must create a table with the given data for \(\ln\left[\text{C}_5\text{H}_6\right]\) versus time: $$ \begin{array}{rl} \text { Time (s) } & {\ln \left[\mathrm{C}_{5} \mathrm{H}_{6}\right]} \\\ \hline 0.0 & \ln(0.0400) \\\ 50.0 & \ln(0.0300) \\\ 100.0 & \ln(0.0240) \\\ 150.0 & \ln(0.0200) \\\ 200.0 & \ln(0.0174) \end{array} $$ Based on the plotted graph, determine if it shows a straight line. If so, the reaction would be first-order.

03

Plot the \(1/\left[\text{C}_5\text{H}_6\right]\) versus Time

Lastly, create a table with the given data for \(1/\left[\text{C}_5\text{H}_6\right]\) versus time: $$ \begin{array}{rl} \text { Time (s) } & {1 \/\ /\ /\left[\mathrm{C}_{5} \mathrm{H}_{6}\right]} \\\ \hline 0.0 & 1/\left(0.0400\right) \\\ 50.0 & 1/\left(0.0300\right) \\\ 100.0 & 1/\left(0.0240\right) \\\ 150.0 & 1/\left(0.0200\right) \\\ 200.0 & 1/\left(0.0174\right) \end{array} $$ Based on the plotted graph, determine if it shows a straight line. If so, the reaction would be second-order.

04

Determine the Order of the Reaction

By analyzing the three graphs, whichever graph shows the most linear relationship is the order of the reaction. Whichever graph is linear: - Concentration versus time: zeroth-order - \(\ln\left[\text{C}_5\text{H}_6\right]\) versus time: first-order - \(1/\left[\text{C}_5\text{H}_6\right]\) versus time: second-order

05

Find the Rate Constant

Once you have determined the order of the reaction, you can find the rate constant using the equation for the reaction rate as follows: - For zeroth-order: Rate = k - For first-order: Rate = k[\(\text{C}_5\text{H}_6\)] - For second-order: Rate = k[\(\text{C}_5\text{H}_6\)]^2 Calculate the slope of the linear relationship from the determined graph, and use that value to find the rate constant, k. The order of the reaction and the rate constant value will give the complete information about the reaction kinetics.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reaction Order

Understanding the reaction order is crucial in determining how the concentration of reactants affects the reaction rate. In this exercise, you are asked to test different graphs to see which one is a straight line, identifying whether it's zeroth, first, or second-order. This is because each order has a characteristic plot that results in a linear relationship.
This linear relationship helps us understand how changes in concentration influence the rate:

• Zeroth-order: The rate is independent of concentration, resulting in a linear plot of concentration versus time.
• First-order: The rate is directly proportional to the concentration, with a linear plot of \ln\[ \left[ \text{C}_5\text{H}_6 \right] \] versus time.
• Second-order: The rate is proportional to the square of the concentration, giving a linear plot of \frac{1}{\left[ \text{C}_5\text{H}_6 \right]}\ versus time.

Determining the order is the first step in understanding how the reaction progresses.

Rate Constant

The rate constant, denoted as \( k \), is a crucial component of the rate equation. Once the reaction order is known, it's possible to calculate the rate constant from the slope of the linear plot corresponding to that order.
The formula for the rate equation varies depending on the order:

• Zeroth-order: \( \text{Rate} = k \)
• First-order: \( \text{Rate} = k[\text{C}_5\text{H}_6] \)
• Second-order: \( \text{Rate} = k[\text{C}_5\text{H}_6]^2 \)

Finding \( k \) not only provides insight into the speed of the reaction but also helps compare the reaction conditions.

Graphical Analysis

Graphical analysis is a method to determine the order of a reaction by plotting experimental data in different forms. In this case, you plotted three graphs:

• Concentration vs. time
• \( \ln \left[ \text{C}_5\text{H}_6 \right] \) vs. time
• \( \frac{1}{\left[ \text{C}_5\text{H}_6 \right]} \) vs. time

By observing which graph results in a straight line, the order of the reaction is identified. This approach visually illustrates how the concentration of reactants relates to the rate of reaction over time, providing a clear path to understanding the kinetics involved.

Concentration versus Time

Plotting concentration versus time is the initial step in analyzing reaction kinetics. This graph could indicate a zeroth-order reaction if it forms a straight line. It shows how the reactant decreases over time, visualizing the rate at which it transforms into products.
For other orders, this plot will curve due to the relationship between concentration and rate. By examining different forms of the data, we can deduce the specific patterns and align them with known mathematical models to determine the reaction order.

Reaction Mechanisms

Reaction mechanisms describe the step-by-step process by which reactants become products. Understanding the kinetics and reaction order provides clues about these mechanisms and the nature of the intermediates.
By analyzing reaction orders and constants, chemists infer details about the elementary steps in a mechanism, helping to build a comprehensive view of how a reaction occurs. Recognizing these intricate processes aids in predicting how changes in conditions might affect the reaction's pathway and efficiency, offering insights into potential improvements or applications.