Problem 91 - Chemistry and Chemical Reactivity

Question

The reaction cyclopropane \(\longrightarrow\) 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) [propene] (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 the reaction order

The reaction given is first order in cyclopropane, meaning the rate law can be expressed as \( \text{Rate} = k [\text{cyclopropane}] \). This tells us how the concentration of cyclopropane affects the rate of reaction.

Step 2: Analyze changes in concentration of cyclopropane

As the reaction progresses, the concentration of cyclopropane decreases because it is being converted into propene. This is characteristic of a reactant being consumed in a chemical reaction.

Step 3: Analyze changes in concentration of propene

The concentration of propene increases as the reaction progresses because propene is the product formed from the decomposition of cyclopropane.

Step 4: Evaluate changes in catalyst concentration

The concentration of the catalyst (platinum in this case) remains unchanged during the reaction since catalysts are not consumed in the reaction. They only facilitate the process.

Step 5: Assess the rate constant

The rate constant \( k \) for a given reaction at a fixed temperature remains constant as the reaction progresses. Since the temperature remains at \( 200^{\circ} \text{C} \), \( k \) does not change.

Step 6: Consider the order of the reaction

The order of the reaction remains constant throughout; it is first order in cyclopropane, as initially specified in the problem statement.

Step 7: Explain how half-life is affected

For first-order reactions, the half-life \( t_{1/2} \) is independent of the initial concentration and remains constant as the reaction progresses. It is given by \( t_{1/2} = \frac{0.693}{k} \).

Key Concepts

Reaction Kinetics

Reaction kinetics is the study of the speed or rate at which chemical reactions occur and the factors that affect these rates. In the case of the cyclopropane breakdown into propene, the reaction is first-order. This means the rate at which cyclopropane converts into propene depends linearly on the concentration of cyclopropane. The rate law for a first-order reaction is written as:
\( \text{Rate} = k [\text{cyclopropane}] \),
where \( k \) is the rate constant. This fundamental relationship tells us that as the concentration of cyclopropane decreases, the reaction rate decreases as well. However, the rate constant \( k \) remains unchanged, as it is specific to the reaction under a given set of conditions (i.e., temperature and catalyst presence). Knowing the reaction order is crucial since it allows for predicting how a change in concentration will affect the speed of the reaction, which is a key aspect of reaction kinetics.

Catalysis

Catalysis is a process by which the rate of a chemical reaction is increased by a substance known as a catalyst. The catalyst, often a metal like platinum as in the cyclopropane to propene reaction, provides an alternative pathway with a lower activation energy. As a result, reactions can proceed faster and more efficiently. Importantly, catalysts do not get consumed in the reaction.
- They remain unchanged after facilitating the reaction.
- Allow the reaction to take place under more manageable conditions, such as lower temperatures or milder pressures.
In our specific example, platinum serves as the catalyst, increasing the rate at which cyclopropane converts to propene without being used up in the process. This consistent presence underlines the unchanging nature of the catalyst concentration throughout the reaction.

Chemical Concentration

Chemical concentration refers to the amount of a given substance in a unit volume of solution. In reactions such as the conversion of cyclopropane to propene, the concentration of each component determines how the reaction progresses.
- The concentration of cyclopropane decreases over time as it gets converted to propene.
- Meanwhile, the concentration of propene increases as it's formed from cyclopropane.
The behavior of these concentrations is a direct reflection of the substances being consumed or produced during the reaction. Understanding these changes in concentration is essential for predicting the direction and extent of the reaction. It's important to note that while the concentrations of reactants and products change, the rate constant and catalyst remain unaffected by these shifts in concentrations.

Half-Life in Reactions

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. In first-order reactions, like the given cyclopropane to propene conversion, this half-life is uniquely constant and does not depend on the initial concentration of the reactant.
- The formula for half-life in first-order reactions is: \( t_{1/2} = \frac{0.693}{k} \)
This consistent half-life provides valuable information for understanding how long a reaction will take to significantly progress. It also allows chemists to predict durations for practical applications, such as synthesis in industrial processes. Constant half-lives are specifically characteristic of first-order reactions, setting them apart from other reaction orders where half-lives may change as concentrations decrease.