Question

For the elementary reaction \(\mathbf{M} \rightarrow \mathbf{N}\), the rate of disappearance of \(\mathbf{M}\) increases by a factor of 8 upon doubling the concentration of \(\mathbf{M}\). The order of the reaction with respect to \(\mathbf{M}\) is (A) 4 (B) 3 (C) 2 (D) 1

Short answer

The order of the reaction with respect to M is 3 (Option B).

Step-by-step solution

Understand the Relationship Between Rate and Concentration

For an elementary reaction, the rate of the reaction is directly proportional to the concentration of the reactants raised to the power of their stoichiometric coefficients in the rate law equation. The rate law for the disappearance of M can be written as rate = k[M]^n, where k is the rate constant and n is the order of the reaction with respect to M.

Analyze the Given Change

We are told that the rate of disappearance of M increases by a factor of 8 when the concentration of M is doubled. This can be mathematically stated as rate(new) = 8 * rate(original), and [M(new)] = 2 * [M(original)].

Use the Rate Law to Find the Reaction Order

Substituting the changes into the rate law equation we get: k * [2 * M(original)]^n = 8 * k * [M(original)]^n. Simplify to find the reaction order (n) that fits this relationship.

Solve for the Order of the Reaction

Divide both sides by k * [M(original)]^n to get: (2^n) = 8. Solving for n gives us n = 3, since 2^3 = 8. This indicates that the reaction is third order with respect to M.

Key concepts

Rate Law

The rate law is a mathematical equation that relates the rate of a chemical reaction to the concentration of its reactants. Through this equation, we can deduce how changes in concentration affect the reaction rate. In the context of our exercise, the rate law is expressed as rate = k[M]^n, where k is the rate constant, [M] is the concentration of reactant M, and n is the reaction order with respect to M. When the concentration of M doubles, the rate increases eightfold, indicating that the concentration of M has a significant impact on the rate.

When determining the reaction order, it is crucial to understand that it is an indicator of how dependent the rate is on the concentration of M. If the reaction order is one (n = 1), the rate changes linearly with concentration changes. However, in our case, an eightfold change in rate upon the doubling of concentration meant we were looking for a cube relation between rate and concentration, which makes the reaction third order with respect to M (n = 3). By comprehending the rate law, you grasp the very core of how chemical reactions proceed and can predict the effects of varying reactant concentrations.

Stoichiometry

Stoichiometry is the quantitative relationship between the reactants and products in a chemical reaction. It is based on the principle of the conservation of mass, where the total mass of the reactants equals the total mass of the products. Stoichiometry uses the balanced chemical equation to calculate the quantities (in moles, mass, or volume) of reactants and products.

For instance, if we consider a simplified reaction where one mole of A reacts to form one mole of B, doubling the amount of A would double the amount of B produced, provided that A is the limiting reactant. However, in kinetics, particularly when we examine reaction orders as seen in the rate law, stoichiometry provides the coefficients that may indicate the initial reaction order in an elementary reaction, although this could be modified by the reaction mechanism. Therefore, stoichiometry offers a starting point for understanding reaction proportions but must be paired with experimental data to ascertain the true kinetics of a reaction.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors affecting them. It provides insights into how fast a reaction occurs and reveals the steps involved in the transformation from reactants to products, known as the reaction mechanism. The rate of a reaction depends on various factors such as reactant concentrations, temperature, catalyst presence, and surface area of solid reactants or catalysts.

In the exercise we are discussing, the rate at which M disappears is a key focus of chemical kinetics. By examining how the rate changes when the concentration of M is altered, we can derive the rate law and, consequently, uncover part of the reaction's mechanism—the reaction order. Understanding chemical kinetics allows scientists and engineers to optimize reactions for industrial processes, control the formation of products, and ensure safety by predicting the speed of potentially dangerous reactions.

Elementary Reaction

An elementary reaction is a single-step process with a one-phase reaction mechanism. It is assumed that the reaction occurs in a single collision of reactant molecules. Unlike complex multi-step reactions, the stoichiometric coefficients in an elementary reaction can often be taken as the exponents in the rate law, which is a direct reflection of the molecularity of the reaction. For example, a bimolecular reaction would typically be second order because it involves the collision between two reactant molecules.

However, the designation of 'elementary' does not always guarantee that the stoichiometric coefficients will exactly match the reaction orders. In the given exercise, the reaction M → N is stated to be elementary, and through experimental data that showed an eightfold increase in rate from doubling M's concentration, we determined that the reaction is indeed third order with respect to M. This sheds light on the experienced reaction dynamics and provides a simple relation between concentration changes and rate changes.