Problem 85 - 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) [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) Decreases, (b) Increases, (c) No change, (d) No change, (e) No change, (f) No change.

Step 1: Analyze How [cyclopropane] Changes

As the reaction progresses, cyclopropane is converted into propene. Since the decomposition is first-order, the concentration of cyclopropane \([\text{cyclopropane}]\) decreases over time as it reacts to form propene.

Step 2: Analyze How [propene] Changes

Propene is the product formed from the conversion of cyclopropane. As the reaction progresses, more cyclopropane is converted to propene, so the concentration of propene \([\text{propene}]\) increases over time.

Step 3: Analyze How [catalyst] Changes

The platinum catalyst facilitates the reaction but is not consumed or altered in the process. Therefore, the concentration of the catalyst remains unchanged throughout the reaction.

Step 4: Analyze How the Rate Constant Changes

The rate constant \(k\) for a reaction is dependent only on temperature and the nature of the reaction itself. Assuming the temperature remains constant, \(k\) does not change as the reaction progresses.

Step 5: Analyze How the Reaction Order Changes

The reaction is described as first-order in cyclopropane. The order of a reaction is a fundamental property that does not change as the reaction progresses; hence, the order remains first-order.

Step 6: Analyze How the Half-life of Cyclopropane Changes

For a first-order reaction, the half-life is calculated as \( t_{1/2} = \frac{0.693}{k} \). This means the half-life depends only on the rate constant \(k\) and not on the concentration of cyclopropane, making it constant as long as the reaction temperature is constant.

Key Concepts

Catalysis in Chemistry

Catalysts are substances that speed up chemical reactions without being consumed in the process. In the reaction of cyclopropane converting to propene, platinum plays the role of a catalyst. It lowers the activation energy required for the reaction to proceed, allowing the reaction to occur at a faster rate. Importantly, the catalyst is not used up during the reaction; it remains unchanged. Therefore, the amount of catalyst does not decrease as the reaction progresses. A catalyst works at the molecular level by providing an alternative pathway for the reaction. This new pathway has a lower energy barrier, which increases the probability of collision between reactant molecules, enhancing the overall rate of reaction. Catalysts do not influence the equilibrium position of a reaction; they merely help the system reach equilibrium faster. A few vital points about catalysts include:

• They do not alter the final products of the reaction.
• Their presence in a reaction does not affect the reaction order.
• They are usually specific in their action, only catalyzing particular reactions.

Reaction Rate Constant

The reaction rate constant, denoted as \( k \), is a pivotal component in the rate equation for chemical reactions. It correlates the reaction rate to the concentrations of the reactants, such as the conversion of cyclopropane to propene, which is a first-order reaction. For this kind of reaction, the rate expression can be written as:
\[ ext{Rate} = k[ ext{Cyclopropane}] \]Here, the rate constant \( k \) remains constant throughout the reaction at a given temperature. This means that as long as the reaction occurs at a constant temperature, \( k \) does not change regardless of the concentrations of the reactants or products.The value of \( k \) provides insight into how fast a reaction proceeds. Factors affecting \( k \) include:

• Temperature: Generally, as temperature increases, the rate constant also increases.
• The presence of a catalyst: Although a catalyst does not change \( k \), it effectively allows the reaction to proceed faster by lowering the activation energy.

Consequently, for temperature-driven processes, understanding \( k \) is essential for controlling reaction rates in industrial and laboratory settings.

Chemical Reaction Order

The chemical reaction order is an integral concept, defining the power to which the concentration of a reactant is raised in the rate law expression. In this context, the reaction between cyclopropane and propene is first-order, indicating that the rate of reaction directly correlates to the concentration of cyclopropane. The expression is simplified as:
\[ ext{Rate} = k[ ext{Cyclopropane}]^1 \]In a first-order reaction, if the concentration of cyclopropane doubles, the reaction rate doubles as well. Reaction order is determined experimentally and is crucial for understanding how changes in reactant concentrations affect the reaction rate. Unlike other parameters, the order of a reaction is not influenced by the presence of catalysts or changes in concentration during the reaction. Some key points include:

• It provides insight into the reaction mechanism and steps involved.
• Not all reactions have the same order; these vary from zero to higher orders based on the reaction dynamics.
• The sum of the reaction orders of all reactants in a multi-component equation gives the overall reaction order.

Half-life in Chemical Reactions

The half-life of a reaction is the time required for half of the reactant to be consumed. For first-order reactions, like in our cyclopropane to propene example, the half-life is constant and does not depend on the initial concentration. This constancy is expressed by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]Where \( t_{1/2} \) is the half-life and \( k \) is the rate constant. Due to the exponential decay property of first-order reactions, the half-life is the same regardless of how much reactant is present at the start. For chemical processes, knowing the half-life aids in:

• Predicting how long a reactant will sustain in a system.
• Estimating the speed of reaction progress over time.
• Designing reaction conditions for appropriate reaction completion.

Thus, understanding half-life is vital in fields such as pharmaceuticals, where metabolism and breakdown rates of compounds are crucial.