Problem 97 - Chemistry and Chemical Reactivity

Question

The reaction cyclopropane \(\rightarrow\) propene occurs on a platinum metal surface at \(200^{\circ} \mathrm{C}\). (The platinum is a catalyst.) The reaction is first-order in cyclopropane. Indicate how the following quantities change (increase, decrease, or no change) as this reaction progresses, assuming constant temperature. (a) [cyclopropane] (b) Ipropene] (c) [catalyst] (d) the rate constant, \(k\) (e) the order of the reaction (f) the half-life of cyclopropane

Step-by-Step Solution

Answer

(a) Decrease, (b) Increase, (c) No change, (d) No change, (e) No change, (f) No change.

Step 1: Understand Reaction Order

The reaction is first-order in cyclopropane, which means the rate of reaction depends linearly on the concentration of cyclopropane. The rate equation is: \(\text{Rate} = k \cdot [\text{cyclopropane}]\), where \(k\) is the rate constant.

Step 2: Analyze Change in Cyclopropane Concentration

As the reaction progresses, cyclopropane is converted into propene. Thus, the concentration of cyclopropane \([\text{cyclopropane}]\) will decrease over time.

Step 3: Analyze Change in Propene Concentration

Since cyclopropane is transforming into propene, the concentration of propene \([\text{propene}]\) will increase over time.

Step 4: Analyze Change in Catalyst Concentration

The catalyst, platinum, facilitates the reaction but does not get consumed. Therefore, the concentration of the catalyst remains unchanged.

Step 5: Analyze Change in Rate Constant

The rate constant \(k\) for a reaction is determined by temperature and is specific to the reaction. Since the temperature remains constant, the rate constant \(k\) does not change.

Step 6: Analyze Change in Reaction Order

The order of the reaction is determined by the stoichiometry of the reaction, which in this case remains first-order in cyclopropane. Thus, the reaction order does not change.

Step 7: Analyze Change in Half-life of Cyclopropane

For a first-order reaction, the half-life \(t_{1/2}\) is independent of the initial concentration and is given by \(t_{1/2} = \frac{\ln(2)}{k}\). Since \(k\) remains constant with constant temperature, the half-life does not change.

Key Concepts

Reaction Kinetics

Reaction kinetics is a fascinating field of chemistry that explains how fast or slow a chemical reaction occurs. The speed of a reaction can depend on several factors such as temperature, pressure, and the presence of a catalyst. Since these factors influence how reactants transform into products, understanding reaction kinetics is crucial in predicting and controlling the outcome of reactions in both industrial and laboratory settings.

Catalysts play a special role in reaction kinetics. They increase the rate of a reaction without being consumed in the process. For example, platinum acts as a catalyst in the reaction of cyclopropane turning into propene. At constant temperature, the reaction speed might remain steady, but the transformation is made faster by the presence of the catalyst.

Reaction kinetics offers insights not only into how a reaction progresses but also into the detailed mechanisms through which products are formed. It serves as the backbone for designing chemical products and processes that are efficient and cost-effective.

First-order Reactions

First-order reactions are a specific type of chemical reaction where the rate depends directly on the concentration of one reactant. In simple terms, if you double the concentration of the reactant, the rate of the reaction also doubles.

The reaction of cyclopropane converting into propene is a classic example of a first-order reaction. The rate equation is expressed as:

• \( \text{Rate} = k \cdot [\text{cyclopropane}] \)

This means that the concentration of cyclopropane directly affects the speed at which it turns into propene. As the reaction proceeds, the amount of cyclopropane decreases, subsequently affecting the reaction rate.

First-order reactions are common in both nature and industry. Understanding them can help us comprehend the dynamics of numerous biological processes, including enzyme-driven reactions and drug metabolism.

Rate Constant

The rate constant, often denoted by the symbol \(k\), is a vital component in the study of reaction kinetics. It provides a quantitative measure of the rate at which a reaction occurs. For first-order reactions, the rate constant can be determined by observing how the concentration of reactants decreases over time.

The rate constant is influenced by temperature but remains constant when the temperature is fixed. In our example reaction, the rate constant for cyclopropane turning into propene remains the same throughout the process, given the temperature does not change.

This characteristic allows scientists to predict how long a reaction will take under controlled conditions, aiding in the formulation of reactions and the synthesis of products on larger scales. It's important to note that the rate constant is specific to every individual reaction and reflects the unique nature of its chemical pathway.

Half-life of Reactions

The half-life of a reaction is the time it takes for the concentration of a reactant to reduce to half its initial amount. In the context of first-order reactions, the half-life is unique because it remains constant throughout the reaction.

For a first-order reaction, the formula for calculating the half-life \( t_{1/2} \) is:

• \( t_{1/2} = \frac{\ln(2)}{k} \)

This equation shows that the half-life is independent of the initial concentration of the reactant, which is quite different from other reaction orders.

This consistency makes half-life a practical measure in various fields, including pharmacology, where it helps determine how long a drug will remain active in the body. Understanding the half-life concept is also critical in environmental sciences for assessing the decay of pollutants over time.