Problem 80 - Chemistry and Chemical Reactivity

Question

A Two molecules of the unsaturated hydrocarbon 1,3-butadiene \(\left(\mathrm{C}_{4} \mathrm{H}_{6}\right)\) form the "dimer" \(\mathrm{C}_{8} \mathrm{H}_{12}\) at higher temperatures. $$2 \mathrm{C}_{4} \mathrm{H}_{6}(\mathrm{g}) \longrightarrow \mathrm{C}_{8} \mathrm{H}_{12}(\mathrm{g})$$ Use the following data to determine the order of the reaction and the rate constant, \(k\). (Note that the total pressure is the pressure of the unreacted \(\mathrm{C}_{4} \mathrm{H}_{6}\) at any time and the pressure of the \(\mathrm{C}_{8} \mathrm{H}_{12} .\)) $$\begin{array}{cl}\hline \text { Time (min) } & \text { Total Pressure (mm Hg) } \\\\\hline 0 & 436 \\\3.5 & 428 \\\11.5 & 413 \\\18.3 & 401 \\\25.0 & 391 \\\32.0 & 382 \\\41.2 & 371 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The reaction is first-order; calculate \( k \) using the slope of \( \ln(436 - 2x) \) vs. time plot.

Step 1: Understanding the Reaction

We have a reaction where 2 moles of 1,3-butadiene form a dimer. The equation is \(2 \mathrm{C}_{4} \mathrm{H}_{6}(\mathrm{g}) \rightarrow \mathrm{C}_{8} \mathrm{H}_{12}(\mathrm{g})\). We are given pressures at different times to determine the reaction order and rate constant.

Step 2: Expressing Pressure in Terms of Concentration

Since pressure is directly proportional to concentration, we express the initial concentration as \( [\mathrm{C}_{4} \mathrm{H}_{6}]_0 = 436 \) mm Hg. The change in pressure, \( x \), can be found at any time \( t \), where \([\mathrm{C}_{4} \mathrm{H}_{6}]_t = 436 - 2x\) and \([\mathrm{C}_{8} \mathrm{H}_{12}]_t = x\).

Step 3: Calculating Change in Pressure

At each time, calculate the change in pressure \( x \) using \( x = \frac{436 - p_t}{2} \), where \( p_t \) is the total pressure at time \( t \).

Step 4: Checking Reaction Order

Consider different reaction orders:- **Zero-order:** The plot of total pressure vs. time will be linear.- **First-order:** The plot of \( \ln([\text{concentration}]) \) vs. time will be linear.- **Second-order:** The plot of \( \frac{1}{[\text{concentration}]} \) vs. time will be linear.We assess each plot to determine linearity.

Step 5: Plotting and Calculating for First-Order Reaction

For a first-order reaction plot \( \ln(436 - 2x) \) against time. Calculate \( \ln(436 - 2x) \) for each time point.

Step 6: Evaluating Plot for First-Order Reaction

After plotting, if the plot is linear, it confirms a first-order reaction. If not, check the other orders.

Step 7: Calculating Rate Constant for First-Order Reaction

For a first-order reaction, the rate constant \( k \) is calculated from the slope of the linear plot of \( \ln([\text{concentration}]) \) vs. time: \( \text{slope} = -k \).

Key Concepts

Reaction Order

Chemical reactions occur at different rates, and to understand these rates, we need to determine the reaction order. The reaction order provides insight into how the concentration of reactants affects the speed of the reaction. It can be zero, first, or second order, among others, depending on how the rate changes with varying concentrations.
When you plot concentration data against time for a reaction, the shape of the graph can indicate the order:

• Zero-order: Shows a straight line when plotting total pressure (or concentration) against time.
• First-order: If you plot the natural logarithm of pressure (or concentration) against time, and it is a straight line, the reaction is first-order.
• Second-order: A plot featuring the inverse of the concentration against time being straight indicates a second-order reaction.

Understanding the reaction order is essential because it tells us the direct relationship between concentration changes and reaction speed, ultimately aiding in predicting how long a given reaction will take under certain conditions.

Rate Constant

The rate constant, often denoted as \(k\), is a fundamental component of the rate equation for a chemical reaction. It is a measure of how quickly a reaction proceeds and is influenced by factors like temperature and the presence of a catalyst.
The rate constant's units will differ depending on the order of the reaction, which is a clue to its role in connecting concentration with reaction rate. For example:

• In a zero-order reaction, the units for \(k\) are concentration/time, such as \( ext{M/s}\).
• For a first-order reaction, \(k\) has units of \(1/ ext{time}\), such as \( ext{s}^{-1}\).
• In a second-order reaction, the rate constant units are \(1/( ext{concentration} imes ext{time})\), like \( ext{M}^{-1} ext{s}^{-1}\).

Finding \(k\) involves analyzing the slope of the line from reaction order plots. For first-order equations, it's derived from the slope of the natural log plot, providing significant information on the reaction speed at any given temperature, thus critically aiding in predicting reaction behavior.

1,3-Butadiene Dimerization

1,3-Butadiene dimerization is an interesting chemical process where two molecules of 1,3-butadiene combine to form a larger molecule called a "dimer," specifically \( ext{C}_8 ext{H}_{12}\). This reaction is noteworthy in industrial and laboratory settings because it happens under specific conditions, often at higher temperatures.
During the dimerization process:

• The system shifts from two smaller molecules \( (2 ext{C}_4 ext{H}_6) \) to a single larger one \( ( ext{C}_8 ext{H}_{12}) \), emphasizing the efficiency of atom use.
• The reaction involves a change in pressure because the number of moles of gas decreases, affecting the total pressure seen in the closed system.
• This dimerization aids in understanding broader organic reactions and mechanisms, especially those involving conjugated systems like dienes.

Studying such reactions helps chemists explore how chemical structures influence outputs and is crucial for synthesizing complex molecules in industries ranging from pharmaceuticals to polymers.