Problem 22 - Chemistry and Chemical Reactivity

Question

The dimerization of butadiene, \(\mathrm{C}_{4} \mathrm{H}_{6},\) to form 1,5 -cyclooctadiene is a second-order process that occurs when the diene is heated. In an experiment, a sample of 0.0087 mol of \(\mathrm{C}_{4} \mathrm{H}_{6}\) was heated in a 1.0-L flask. After 600. seconds, \(21 \%\) of the butadiene had dimerized. Calculate the rate constant for this reaction.

Step-by-Step Solution

Answer

The rate constant is approximately 0.051 M⁻¹ s⁻¹.

Step 1: Understand the Problem

We need to calculate the rate constant (k) for a second-order reaction involving the dimerization of butadiene given that - initial moles of butadiene are 0.0087 mol, - volume is 1.0 L (so initial concentration is 0.0087 M), and - 21% of it dimerizes in 600 seconds.

Step 2: Calculate Initial and Final Concentrations

The initial concentration \([C_{4}H_{6}]_0 = \frac{0.0087}{1.0} = 0.0087 \, M \). After 600 s, 21% has reacted, hence unreacted concentration is \([C_{4}H_{6}] = 0.0087 \, M \times (1 - 0.21) = 0.006873 \, M\).

Step 3: Use the Second-Order Kinetics Equation

In a second-order reaction, the relationship is given by \(\frac{1}{[A]} - \frac{1}{[A]_0} = kt\).Substitute \([A]_0 = 0.0087 \, M\), \([A] = 0.006873 \, M\), and \(t = 600 \, s\).

Step 4: Calculate the Rate Constant (k)

Plug the values into the second-order rate equation: \[\frac{1}{0.006873} - \frac{1}{0.0087} = k \times 600 \].Simplifying both sides, divide by 600 to find \(k\).

Step 5: Solve for k

Calculate: \(\frac{1}{0.006873} \approx 145.53 \, M^{-1}, \quad \frac{1}{0.0087} \approx 114.94 \, M^{-1}\).So, \[\frac{1}{[A]} - \frac{1}{[A]_0} = 145.53 - 114.94 = 30.59 \].Thus,\[k = \frac{30.59}{600} \approx 0.051 \, M^{-1}\cdot s^{-1}\].

Key Concepts

rate constant calculation

Calculating the rate constant, often symbolized as \( k \), is essential in understanding the speed of chemical reactions. In a second-order reaction, the rate constant gives us insight into how quickly reactants are transformed into products over time.
Since this is a second-order reaction, the specific formula used for the calculation is:

• \( rac{1}{[A]} - rac{1}{[A]_0} = kt \)

In this equation, \([A]_0\) is the initial concentration of the reactant butadiene, and \([A]\) is its concentration after the reaction has progressed for a specific time \( t \).
This straightforward calculation requires knowledge of both the initial and remaining concentrations after a reaction period, which allows the deduction of the value for \( k\). It tells you how concentration changes over time, making it a vital component for predicting how fast reactions occur.

dimerization of butadiene

Dimerization is a chemical process where two molecules combine to form a new compound, often doubling the molecular size. In the case of butadiene, \( C_4 H_6 \), dimerization transforms it into 1,5-cyclooctadiene.
This particular reaction is fascinating because it involves the interaction of identical diene molecules at elevated temperatures. This conversion not only results in increased molecular complexity but also impacts significantly how the reaction progresses, embedding kinetic principles.
Understanding the precise nature of dimerization in butadiene is important for fields like polymer chemistry and materials science, where manipulating molecules through thermal reactions allows for the creation of new substances with desired properties.

chemical kinetics

Chemical kinetics is a branch of chemistry concerned with the rates of reaction and the steps they go through. It provides a detailed understanding of how reactions work, including the factors influencing them, such as concentration, temperature, and catalysts.
In chemical kinetics, the speed or rate at which a chemical process occurs is critical. It's not just about whether a reaction happens but how long it takes from start to finish. For example, in the dimerization of butadiene, the kinetics tells us that the rate depends on the concentration of butadiene and the temperature applied.
By studying how factors like temperature and concentration affect reaction rates, chemical kinetics helps predict and control the speed of reactions in industrial and laboratory settings alike.

second-order rate equation

The second-order rate equation is significant when understanding reactions where the rate is proportional to the square of the concentration of a single reactant or the product of two reactants’ concentrations. In our case, this applies to the dimerization of butadiene, which follows the equation:

• \( rac{1}{[A]} - rac{1}{[A]_0} = kt \)

This equation provides an insight into how the concentration of a reactant decreases over time in a second-order reaction. Second-order kinetics are typically observed in dimerization reactions and some substitution reactions.
The linearity of this relationship showcases how vital concentration changes can be monitored over time to determine reaction rates and potentially adjust conditions such as pressure or concentration to control the process efficiencies.