Problem 63 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

As the initial concentration increases from \(0.75\) to \(1.55 \mathrm{M}\) in a reaction, \(t_{1 / 2}\) decreases from 60 to \(29 \mathrm{~s}\). The order of the reaction is (a) zero (b) first (c) second (d) third

Step-by-Step Solution

Answer

The reaction is of second-order (c).

Step 1: Analyze the Relation Between Concentration and Half-life

To determine the order of the reaction, examine how the half-life (\(t_{1/2}\)) changes with the concentration of the reactant. In a zero-order reaction, the half-life increases with the concentration. In a first-order reaction, the half-life remains constant regardless of the concentration changes. In a second-order reaction, the half-life decreases with an increase in concentration. Third-order reactions are less common and typically show more complex dependencies.

Step 2: Compare Given Data Points

We are given that as the concentration increases from 0.75 to 1.55 M, the half-life decreases from 60 s to 29 s. This is a sign of a second-order reaction because the half-life is decreasing while the concentration is increasing.

Step 3: Eliminate Other Options

Since the half-life is not constant, we can rule out a first-order reaction. It does not increase with an increase in concentration, so we can rule out a zero-order reaction. Third-order reactions generally have more complex behaviors not described by this scenario, so it is also unlikely.

Key Concepts

Chemical Kinetics

Chemical kinetics, a subfield of physical chemistry, is a study of the speed or rate at which chemical reactions occur and the factors that affect these rates. In the context of chemical kinetics, the rate of a reaction often depends on the concentration of reactants, the temperature, and the presence of a catalyst. The rate law expresses the relationship of the reaction rate with the concentration of reactants. It is usually given by an equation of the form \( rate = k[Reactant]^n \) where \( k \) is the rate constant, \( [Reactant] \) is the concentration of the reactant, and \( n \) reflects the reaction order with respect to that reactant. Understanding how these factors influence the speed of a reaction is pivotal in various fields such as pharmaceuticals, environmental science, and materials engineering, where controlling the reaction rate is essential.

Half-life of Reaction

The half-life of a chemical reaction, denoted as \( t_{1/2} \), is the time required for the concentration of a reactant to decrease to half its initial value. The concept of half-life is not only useful in the field of nuclear chemistry but also in chemical kinetics, where it helps understand reaction rates. For a zero-order reaction, half-life is directly proportional to the initial concentration of the reactant. In contrast, for a first-order reaction, the half-life remains unchanged regardless of the initial concentration. A second-order reaction shows an inverse relationship between half-life and initial concentration: as the starting amount of reactant increases, the half-life decreases, as demonstrated by the given exercise where \( t_{1/2} \) decreases from 60 to 29 seconds as the concentration rises from 0.75 to 1.55 M. This characteristic allows chemists to predict how long it will take for a reaction to reach a certain stage and plan accordingly.

Concentration-Dependence

The dependence of reaction rate and half-life on reactant concentrations is a key concept in chemical kinetics. As seen in the exercise, understanding this relationship can lead to identifying the order of a reaction. Different reaction orders reflect varying degrees of sensitivity to changes in concentration. A zero-order reaction rate is independent of the concentration of the reactant, meaning that the rate does not change as the reactant is consumed. For a first-order reaction, the rate is directly proportional to the concentration, meaning that if the concentration doubles, so does the rate. Lastly, second-order reactions have rates that are proportional to the square of the reactant concentration, indicating a more significant impact of concentration changes on the rate. This concentration-dependence plays an integral role in predicting how varying conditions will affect the progress of a reaction, which is crucial for chemical manufacturing and process control.