Problem 19 - Chemistry Matter and Change

Question

Write the rate law for the reaction \(a A \rightarrow b B\) if the reaction is third order in \(A\) . \([B]\) is not part of the rate law.

Step-by-Step Solution

Answer

The rate law is rate = \(k[A]^3\).

Step 1: Identify the Order

The exercise specifies that the reaction is third order with respect to reactant \(A\). This means that the order of reactant \(A\) in the reaction is 3.

Step 2: Understand Rate Law Formulation

The rate law for a reaction is generally given by the expression, rate \( = k[A]^m[B]^n\), where \(k\) represents the rate constant, and \(m\) and \(n\) are the orders of reactants \(A\) and \(B\) respectively. In this exercise, since \([B]\) is not part of the rate law, we need not consider \(B\).

Step 3: Formulate the Rate Law

Since the order of reaction with respect to \(A\) is 3, the rate law will only involve \(A\). Therefore, the rate law can be expressed as: rate \( = k[A]^3\). \(k\) is the rate constant for this specific reaction.

Key Concepts

Understanding Reaction Order

Reaction order is crucial in determining how the concentration of reactants affects the rate of a chemical reaction. It gives us insight into the mechanics of the reaction and reveals the relationship between the concentration of reactants and the reaction rate.
In the provided exercise, the reaction is described as being third order with respect to reactant \(A\). This means that any changes in the concentration of \([A]\) will have a significant impact on the reaction rate. Specifically, a threefold change. For instance:

• If the concentration of \([A]\) is doubled, the reaction rate increases by a factor of eight \((2^3)\)
• If the concentration of \([A]\) is halved, the reaction rate decreases by a factor of eight \((\frac{1}{2})^3\)

This exponential relationship indicates the power in which the concentration of \(A\) is raised in the rate law which, in this case is 3. Thus, illustrating its third-order reaction nature.

Concept of the Rate Constant

The rate constant, represented by \(k\), is an essential component of the rate law equation and provides insights into the speed of a chemical reaction at a given temperature. While the reaction order tells us about the dependency on reactant concentrations, \(k\) serves as a proportional factor that aligns with the specific condition of temperature and medium.
The value of the rate constant can be influenced by:

• Temperature, as most reactions tend to speed up with an increase in temperature leading to a higher \(k\)
• The presence of a catalyst, which can lower the activation energy and alter the rate constant

It is important to note that \(k\) is unique for every reaction under specific conditions and must be determined experimentally. In our specific exercise scenario, the rate law is represented as rate \( = k[A]^3\). Here, \(k\) is customized for the reaction where \(A\) is the only concentrate considered.

Exploration of Chemical Kinetics

Chemical kinetics delves into the rates of chemical processes and allows us to explore the dynamic nature of reactions. It includes the study of how different conditions alter the speed and mechanism of reactions.
Through chemical kinetics, we can

• Predict how changes in concentration, temperature, or a catalyst will affect the rate of a reaction
• Understand the step-by-step approach of reaction pathways and intermediates
• Gain insight into reaction mechanisms which helps in the development of new reactions

In our exercise, the third-order rate law reflects a specific case of chemical kinetics where only the concentration of \([A]\) affects the reaction's progress. Thus, emphasizes the role of chemical kinetics in simplifying complex reactions into formula-based rates that predict behaviors of the reacting chemicals effectively.