Problem 16 - General Chemistry: Principles and Modern Applications

Question

One of the following statements is true and the other is false regarding the first-order reaction \(2 \overrightarrow{\mathrm{A}} \longrightarrow \mathrm{B}+\mathrm{C} .\) Identify the true statement and the false one, and explain your reasoning. (a) A graph of [A] versus time is a straight line. (b) The rate of the reaction is one-half the rate of disappearance of A.

Step-by-Step Solution

Answer

Statement (a) is false: A graph of [A] vs. time for a first-order reaction is an exponential decay curve, not a straight line. Statement (b) is true: The rate of the reaction is one-half the rate of disappearance of A, due to the stoichiometry of the reaction.

Step 1: Analyzing Statement (a)

A first-order reaction is characterized by a constant half-life, and its rate depends only on the concentration of one reactant, in this case A. The equation for a first-order reaction is \([A] = [A]_0 \cdot \exp^{-kt}\), where \([A]\) is the concentration of A at time \(t\), \([A]_0\) is the initial concentration of A, \(k\) is the rate constant, and \(t\) is time. If you graph \([A]\) vs. \(t\), you would get an exponential decay curve, not a straight line. So, statement (a) is false.

Step 2: Analyzing Statement (b)

In the given reaction \(2A \rightarrow B + C\), the rate of disappearance of A is \(−\frac{1}{2} \cdot \frac{d[A]}{dt}\) while the rate of the reaction is normally defined as \(+\frac{1}{d} \cdot \frac{d[B]}{dt} = +\frac{1}{d} \cdot \frac{d[C]}{dt}\). Due to the stoichiometry of the equation (for every 2 particles of A that react, 1 particle of B and 1 particle of C is produced), the rate of the reaction is indeed half the rate of disappearance of A. So, statement (b) is true.

Key Concepts

Reaction Rate in First-Order Reactions

In a first-order reaction, the rate at which the reaction proceeds is directly proportional to the concentration of one reactant. This means if you double the concentration of the reactant, the reaction rate also doubles. The general expression for the rate of a reaction can be written as:- Rate = \( k[A] \)Here, \( k \) is the rate constant that is specific to the reaction at a certain temperature, and \([A]\) is the concentration of the reactant A.One of the key features of a first-order reaction is its constant half-life, which is the time required for half of the reactant to be consumed. The half-life does not depend on the initial concentration, making it a unique characteristic of first-order kinetics.It's also important to note that for a first-order reaction, the rate of reaction and the rate of disappearance of reactants can differ due to stoichiometry, which we'll explore next.

Understanding Stoichiometry in Reaction Rates

Stoichiometry is a concept that involves the quantitative relationships between reactants and products in a chemical reaction. For the reaction given: \( 2A \rightarrow B + C \), stoichiometry tells us that for every 2 units of A that react, 1 unit of B and 1 unit of C are formed.In terms of rates, this relationship affects how we describe the disappearance of A versus the formation of B and C. The rate of disappearance of A is \( -\frac{d[A]}{dt} \), and for the given stoichiometric coefficient, it becomes \( -\frac{1}{2} \) \( \frac{d[A]}{dt} \) when expressing it in terms of the rate of the reaction.Thus, the actual reaction rate can be represented as half the rate of disappearance of A due to this 1:2 stoichiometry in the balanced equation. Understanding these stoichiometric coefficients is crucial to interpreting experimental data and developing accurate rate expressions.

Exponential Decay in First-Order Reactions

First-order reactions are characterized by an exponential decay in the concentration of the reactant over time. The formula representing this is:- \([A] = [A]_0 \cdot e^{-kt}\)where \([A]_0\) is the initial concentration of A, \(k\) is the rate constant, and \(t\) is time.Exponential decay implies that even though the concentration of A decreases over time, it never truly reaches zero but approaches it infinitely. This creates a curve when plotting \([A]\) versus time, which contrasts with a straight line for zero-order reactions where the rate of reaction is constant and independent of the concentration.Understanding exponential decay helps clarify why first-order reactions don't produce a linear graph of concentration over time. Instead, they showcase the characteristic curve that reflects the nature of the process and the diminishing concentration of reactants as time progresses.