Question

Cyclopropane, \(\mathrm{C}_{3} \mathrm{H}_{6}\), is a gas used as a general anesthetic. It undergoes a slow molecular rearrangement to propylene. At a certain temperature, the following data were obtained relating concentration and rate: $$\begin{array}{cc}\text { Initial Concentration of } & \text { Initial Rate of Formation } \\\\\text { Cyclopropane }\left(\mathrm{mol} \mathrm{L}^{-1}\right) & \text {of Propylene }\left(\mathrm{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}\right) \\\0.050 & 2.95 \times 10^{-5} \\\0.100 & 5.90 \times 10^{-5} \\\0.150 & 8.85 \times 10^{-5}\end{array}$$ What is the rate law for the reaction? What is the value of the rate constant, with correct units?

Short answer

The rate law for the reaction is \( Rate = k \times [\text{Cyclopropane}] \), and the rate constant \( k \) is \( 5.90 \times 10^{-4} \text{s}^{-1} \).

Step-by-step solution

Analyze the given data

Examine the concentration and rate data provided for cyclopropane and identify how changes in concentration affect the rate of formation of propylene.

Determine the order of reaction

Since doubling and tripling the concentration of cyclopropane doubles and triples the rate of formation of propylene, respectively, the reaction is first order with respect to cyclopropane.

Write the rate law

The rate law for a first-order reaction is given by: \( Rate = k \times [\text{Cyclopropane}] \)

Calculate the rate constant, k

Use the data from any trial to calculate k. For example, using the first trial data \( 2.95 \times 10^{-5} = k \times 0.050 \), which gives \( k = \frac{2.95 \times 10^{-5}}{0.050} \)

Determine the units for the rate constant

Since the reaction is first-order, the units for k are \( \text{s}^{-1} \)

Key concepts

Reaction Order

Understanding the concept of reaction order is pivotal in the field of chemical kinetics as it provides insight into how the concentration of reactants affects the rate of the reaction. In simple terms, the reaction order with respect to a given reactant is the exponent to which its concentration term in the rate equation is raised. It reveals the relationship between the concentration of reactants and the rate at which they react.

For instance, if the rate of reaction doubles when the concentration of a reactant doubles, the reaction is said to be first order with respect to that reactant. This means the exponent is 1. If the rate quadruples, it's second order, and the exponent is 2. In the context of our cyclopropane example, since the rate of formation of propylene changes proportionally with the changes in cyclopropane concentration, we deduce that the reaction is first order with respect to cyclopropane.

To pin down the reaction order, watching how the rate responds to varying concentration levels is the key. In doing so, you'll be able to determine how substances interact on a molecular level, which can help in developing efficient synthetic pathways in industrial processes, preserving food, or even drug formulation.

Rate Constant Calculation

The rate constant is a coefficient that provides the link between the concentration of reactants and the rate of the reaction. It is unique for every reaction at a given temperature. Calculating the rate constant is a critical step in deciphering the kinetics of a reaction.

To calculate the rate constant, also symbolized by 'k', you need to have a form of the rate law, as well as some experimental data that fits into that law. Once the reaction order is determined, you use the rate law equation and solve for 'k' using algebraic methods.

In our example with cyclopropane, after establishing the reaction is first order, the rate law is expressed as: \( Rate = k \times [\text{Cyclopropane}] \). With the rate and concentration data provided, we can plug in the numbers and isolate 'k'. It's important to ensure that the units are consistent so that 'k' has the correct units. For a first-order reaction, the units for 'k' are typically s\textsuperscript{-1} or min\textsuperscript{-1}, indicating how fast the concentration of the reactant decreases over time.

Chemical Kinetics

Chemical kinetics is the study of rates of chemical processes and the factors affecting them. It's about understanding how different variables such as concentration, temperature, and catalysts influence the speed at which reactions occur. This field is fundamental in both theoretical contexts, such as reaction mechanism proposals, and practical applications like industrial reactor design.

In chemical kinetics, you'll come across crucial concepts, such as the activation energy, which is the minimum energy that reacting species must have for the reaction to occur. Catalysts can lower this energy barrier, thereby increasing the reaction rate without being consumed in the process.

Another key point is temperature's role in reaction rates. Generally, as temperature increases, so does the rate of reaction, described quantitatively by the Arrhenius equation. Kinetic studies thus enable chemists to predict reaction behavior, optimize conditions for desired product yields, and control reaction timings, which is incredibly important for safety in chemical manufacturing.