Question

Another type of autocatalytic reaction is referred to as cubic autocatalytic corresponding to the following elementary process: \\[ A+2 B \rightarrow 3 B \\] Write the rate law expression for this elementary process. What would you expect the corresponding differential rate expression in terms of \(\xi(\) the coefficient of reaction advancement) to be?

Short answer

Question: Determine the rate law expression and the differential rate expression in terms of the reaction advancement coefficient (ξ) for the given cubic autocatalytic reaction: A + 2B → 3B. Answer: The rate law expression for the given reaction is Rate = k[A][B]^2. The differential rate expression in terms of ξ is given by: dξ/dt = k(A₀-ξ)(B₀+2ξ)².

Step-by-step solution

Write the elementary process

The elementary process for the cubic autocatalytic reaction is given as follows: \\[ A+2 B \rightarrow 3 B \\]

Determine the rate law expression

To write the rate law expression, we will use the stoichiometry of the reaction, considering that the reaction rate depends on the concentrations of the reactants. For an elementary process, the reaction order for each reactant directly corresponds to its stoichiometric coefficient. The rate law expression for this reaction can be written as: \\[ \text{Rate} = k[A]^\alpha [B]^\beta \\] where k is the rate constant, α and β are the reaction orders for A and B, respectively. In this case, α = 1 and β = 2. So the rate law expression will be: \\[ \text{Rate} = k[A][B]^2 \\]

Write the differential rate expression in terms of ξ

The coefficient of reaction advancement (ξ) indicates how far the reaction has progressed. To write the differential rate expression in terms of ξ, we will express the concentrations of the reactants in terms of ξ. Let the initial concentrations of A and B be represented by [A]0 and [B]0, respectively. We can write the concentration expressions as follows: \\[ [A]=(A_0 - \xi) \\] and \\[ [B]=(B_0 + 2\xi) \\] Substituting these expressions into the rate law expression, we get: \\[ \dfrac{d\xi}{dt} = k[(A_0-\xi)][(B_0+2\xi)]^2 \\] The corresponding differential rate expression in terms of ξ is: \\[ \dfrac{d\xi}{dt} = k(A_0-\xi)(B_0+2\xi)^2 \\]

Key concepts

Elementary Process

Elementary processes are fundamental steps that make up a reaction mechanism. In the context of chemical kinetics, it refers to a single step reaction where reactants are directly converted to products. Each of these processes involves certain molecules colliding effectively, leading to a chemical change.
For a cubic autocatalytic reaction, the elementary process is given by: \[A + 2B \rightarrow 3B\] In this particular reaction, one molecule of A reacts with two molecules of B to form three molecules of B. This transformation highlights the autocatalytic nature of the reaction, where one of the products (B) acts as a catalyst, speeding up its own production. Elementary processes are crucial as they allow us to directly relate the rate of reaction to the concentrations of the reactants involved.

Rate Law Expression

A rate law expression connects the reaction rate to the concentration of its reactants. For elementary reactions, the rate law is directly derived from the stoichiometry of the reaction.
The general form is:

• \[ \text{Rate} = k[A]^\alpha [B]^\beta \]

Where \(k\) is the rate constant, and \([A]\) and \([B]\) are the concentrations of the reactants raised to the power of their stoichiometric coefficients, \(\alpha\) and \(\beta\).
For the reaction \( A + 2B \rightarrow 3B \), the stoichiometry suggests that \(\alpha = 1\) and \(\beta = 2\). Consequently, the rate law becomes:

• \[ \text{Rate} = k[A][B]^2 \]

This expression emphasizes that the rate depends linearly on the concentration of A and quadratically on B. Understanding the rate law is crucial to manipulating and predicting how different conditions will affect the speed of the reaction.

Differential Rate Expression

The differential rate expression describes how the concentration of reactants and products change over time during a reaction. By using the coefficient of reaction advancement, \(\xi\), we can translate this change into measurable terms.
For an expression based on \(\xi\), we need to relate the concentration changes of reactants in terms of their initial concentrations and \(\xi\). For the cubic autocatalytic reaction where:

• \([A] = (A_0 - \xi)\)
• \([B] = (B_0 + 2\xi)\)

We replace these expressions into the rate law to find: \[\dfrac{d\xi}{dt} = k[(A_0-\xi)][(B_0+2\xi)]^2\]This equation reveals how the reaction progress is governed by the changing concentrations of A and B as represented by \(\xi\) over time. Mastering differential rate expressions assists in understanding the dynamic behavior of reactions as they proceed, allowing for better control and optimization in practical applications.