Problem 89 - Chemistry The Central Science In Si Units

Question

Consider the reaction \(2 \mathrm{~A} \longrightarrow \mathrm{B}\). Is each of the following statements true or false? (a) The rate law for the reaction must be, Rate \(=k[A]^{2} \cdot(\mathbf{b})\) If the reaction is an elementary reaction, the rate law is second order. \((\mathbf{c})\) If the reaction is an elementary reaction, the rate law of the reverse reaction is first order. (d) The activation energy for the reverse reaction must be smaller than that for the forward reaction.

Step-by-Step Solution

Answer

(a) True, (b) True, (c) True, (d) False.

Step 1: Analyze Statement (a)

For the reaction \(2A \longrightarrow B\) to have a rate law expression \(\text{Rate} = k[A]^2\), the reaction must be elementary. Typically, an elementary reaction's rate law corresponds directly to its stoichiometry. Here, the stoichiometry is 2 for \(A\), so the statement would be true if we assume the reaction is elementary.

Step 2: Evaluate Statement (b)

If the reaction is elementary, the rate law is derived from the stoichiometry of the reaction. Given the stoichiometry of \(2A\), the rate law would indeed be second order, meaning this statement is true.

Step 3: Assess Statement (c)

The reverse reaction for \(2A \longrightarrow B\) would be \(B \longrightarrow 2A\). For it to be elementary, the rate law would be based on the stoichiometry of the reverse reaction, which shows first order in \(B\). Therefore, statement (c) is true.

Step 4: Evaluate Statement (d)

There is no direct rule that the activation energy for the reverse reaction is smaller than that for the forward reaction. These energies depend on the potential energy surface of the reaction and specific details of the energy barrier, thus making this statement false unless further specific details are provided.

Key Concepts

Rate Law

When we talk about rate laws in chemistry, we refer to the mathematical expression that describes the speed of a chemical reaction. A rate law shows how the rate depends on the concentration of the reactants. For the reaction given, if it is an elementary reaction, the stoichiometry directly influences the rate law. This means the power of each concentration term in the rate law is equal to the coefficient of the reactant in the balanced equation.
In the example of the reaction \(2A \longrightarrow B\), the rate law is expressed as \( \text{Rate} = k[A]^2 \). Here, \([A]^2\) shows that the reaction rate depends quadratically on the concentration of \(A\). The reaction must be elementary for this direct correlation to hold true, highlighting the importance of knowing whether a reaction is elementary when determining the rate law.

For more complex reactions, where the sequence of steps isn't clear, the rate law can't be simply deduced from stoichiometry and requires experimental data.

Elementary Reactions

Elementary reactions are single-step processes that cannot be broken down into simpler steps. They occur exactly as written with no intermediates. In such reactions, the rate law can be directly derived from the coefficients of the reactants in the balanced equation.
For instance, with the simple reaction \(2A \longrightarrow B\), if it is considered an elementary reaction, the rate law \( \text{Rate} = k[A]^2 \) directly mirrors the reactants’ stoichiometry.

Elementary reactions provide insights into the mechanism of a reaction, the sequence of steps that lead from reactants to products. Each step represents an elementary reaction with its unique rate law that, when combined, gives the overall reaction rate law. Understanding whether a reaction is elementary helps in predicting its behavior and calculating its rate effectively.

Activation Energy

Activation energy is the minimum energy required for a chemical reaction to occur. It acts as a barrier that reactants must overcome for a product to form. Both the forward and reverse reactions have their own activation energy barriers.
In the context of the exercise, whether the activation energy for the forward reaction \(2A \longrightarrow B\) is larger or smaller than that of the reverse \(B \longrightarrow 2A\) isn't straightforward. These energies depend on the specific details of the energy landscape of the reaction, which are not always available without experimental data.

Activation energy is a crucial concept in determining the rate of reaction. A higher activation energy means a slower reaction, assuming temperatures are constant. This is often depicted in an energy profile diagram where the height of the peak represents the activation energy.

Chemical Stoichiometry

Stoichiometry involves the quantitative relationships between the amounts of reactants and products in a chemical reaction. It gives you the theoretical yield of a product and is crucial for calculating concentrations, volumes, and comparing reaction outcomes.
In chemical equations like \(2A \longrightarrow B\), stoichiometry shows that two moles of \(A\) are needed to produce one mole of \(B\). This ratio helps in determining the rate law if the reaction is elementary and assists in predicting the amount of product formed from given reactants.

Stoichiometry is foundational for understanding reaction mechanisms as it lays out a blueprint for how reactants convert to products. It also helps in balancing chemical equations, ensuring mass conservation according to the law of conservation of mass.