Question

The reaction \(A \longrightarrow\) products is second order. The initial rate of decomposition of \(A\) when \([\mathrm{A}]_{0}=0.50 \mathrm{M}\) is \((\mathrm{a})\) the same as the initial rate for any other value of \([\mathrm{A}]_{0} ;\) (b) half as great as when \([\mathrm{A}]_{0}=1.00 \mathrm{M} ;(\mathrm{c})\) five times as great as when \([\mathrm{A}]_{0}=[\mathrm{A}]_{0}=0.25 \mathrm{M}.\)

Short answer

The initial rate of reaction for a second order reaction is not the same for all initial concentrations of the reactant. When the initial concentration is halved from 1.00 M to 0.50 M, the initial rate is halved. When the initial concentration is decreased to a quarter (0.25 M), the initial rate becomes five times greater.

Step-by-step solution

Identify the Rate Law

For a second-order reaction, the rate equation is given by \(rate = k[A]^2\). This means that the rate of reaction is directly proportional to the square of the concentration of the reactant A.

Calculate the Rate for Different Concentrations

(a) The rate of reaction will not be the same for any other value of [A], as it is dependent on the concentration of A. (b) When [A] = 1.00 M, the rate will be (1.00 M)^2 = 1, which is twice as large as when [A] = 0.50 M (rate = (0.50 M)^2 = 0.25), thus the rate is half when [A] = 0.50 M. (c) When [A] = 0.25 M, the rate is (0.25 M)^2 = 0.0625, which is five times less than when [A] = 0.50 M.

Summarize the Results

In conclusion, the initial rate of reaction is dependent on the square of the initial concentration of the reactant for a second-order reaction. The rate when [A] = 0.50 M is twice as large as when [A] = 1.00 M and five times as large as when [A] = 0.25 M.

Key concepts

Rate Law

Understanding the rate law is essential in the field of chemical kinetics. It's a mathematical equation that relates the reaction rate to the concentration of the reactants. In the context of a second-order reaction, the rate law expression takes the form \( rate = k[A]^2 \), where \(k\) is the rate constant, and \[A\] represents the concentration of reactant A. This specific form implies that the rate of the reaction is directly proportional to the square of the concentration of the reactant. This quadratic relationship indicates a stark increase in reaction rate with increasing concentration, making precise concentration measurements crucial for predicting reaction behavior.

Reaction Rates

The concept of reaction rates dives into how quickly reactants are transformed into products in a chemical reaction. It's a dynamic aspect of chemistry that can be influenced by multiple factors including temperature, catalysts, and reactant concentration. For second-order reactions, as the concentration of reactant A doubles, the reaction rate quadruples, reflecting the squared concentration dependency. This means that minor changes in the concentration can lead to significant differences in how fast the reaction proceeds, which is a fundamental concept when predicting how a chemical process will unfold over time.

Chemical Kinetics

Chemical kinetics is the study of the speeds or rates at which chemical reactions occur. It not only considers the reaction rates but also the steps that comprise the reaction mechanism and the factors that affect these rates. It's a significant branch of physical chemistry owing to its role in industrial applications, environmental science, and even in biological processes. By comprehending kinetics, chemists can design experiments to measure and control the rates of chemical reactions, ensuring the efficiency and safety of chemical processes.

Concentration Dependence

Concentration dependence is a pivotal concept in chemical kinetics that describes how the change in concentration of the reactants affects the reaction rate. For a second-order reaction, this dependence is more pronounced as the rate is proportional to the square of the concentration. This squared term implies that if you were to cut the concentration of reactant A in half, the rate of the reaction would decrease to a quarter of its original rate. This steep dependence on concentration is key to predicting how a reaction will propagate and is particularly important in scenarios where precise control of reaction speed is desired.