Question

Reaction \(A \rightarrow B\) follows second order kinetics. Doubling the concentration of \(A\) will increase the rate of formation of \(B\) by a factor of : (a) \(1 / 4\) (b) \(1 / 2\) (c) 2 (d) 4

Short answer

Doubling the concentration of A will increase the rate of formation of B by a factor of 4 (option (d)).

Step-by-step solution

Understanding Second Order Kinetics

For a reaction that follows second order kinetics, the rate of the reaction is proportional to the square of the concentration of the reactant. Mathematically, this can be represented as rate = k[A]^2, where k is the rate constant and [A] is the concentration of reactant A.

Assessing the Effect of Doubling the Concentration

If the initial concentration of A is [A], upon doubling the concentration it becomes 2[A]. The rate of formation of B with this new concentration is then k(2[A])^2.

Comparing the Rates

To find the factor by which the rate increases, compare the new rate to the old rate: (k(2[A])^2) / (k[A]^2) = (2^2[A]^2) / [A]^2 = 4. Thus, the rate of formation of B increases by a factor of 4.

Key concepts

Chemical Kinetics

Chemical kinetics is a branch of physical chemistry that deals with understanding how chemical reactions occur and quantifying the rates at which they proceed. The main focus of chemical kinetics is to measure the speed of a reaction and identify the steps through which it occurs. There are various factors that affect the speed or rate of a chemical reaction, including temperature, concentration of reactants, surface area, catalysts, and the presence of light.

In the study of chemical kinetics, reactions are categorized by their order, which refers to the power to which the concentration of reactant is raised in the rate law. The rate law is an equation that relates the rate of a reaction to the concentrations of the reactants. For a second order reaction, the rate is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. This dependence on concentrations is a pivotal aspect of second order kinetics and plays a critical role when predicting how the rate will change when concentrations vary.

Reaction Rate

The reaction rate is a measure of how quickly the concentration of a substance in a chemical reaction changes over time. It is usually expressed in terms of the change in concentration per unit time, for example, mol/L/s. In the context of second order kinetics, the reaction rate increases with the square of the concentration of the reactant.

Let's dive deeper into the exercise to see this concept in action. You learned that the reaction rate of second order reactions is proportional to the concentration of reactant squared, represented by the mathematical expression: rate = k[A]^2. This equation tells us that if you were to double the concentration of reactant A, as the exercise suggests, the reaction rate would increase not just by double, but by a factor of 2 squared, which is 4. Hence, in second order kinetics, the effect of changing the reactant concentration is more dramatic compared to a first order reaction where the rate would simply double with the doubling of the reactant.

Concentration Dependence

Concentration dependence in chemical kinetics refers to how the rate of a reaction changes as the concentrations of the reactants change. For a reaction where the rate is dependent on the concentration of a single reactant to the first power (first order), a change in concentration results in a directly proportional change in the rate.

However, with second order kinetics, as illustrated in the textbook exercise, the rate law indicates that the rate is proportional to the concentration of the reactant squared. Doubling the concentration of reactant A, therefore, does not simply double the reaction rate. Instead, according to the second order rate law (rate = k[A]^2), it results in the rate increasing by the square of the factor by which the concentration increases - which is fourfold in this scenario (since 2 squared equals 4).

This strong dependence on concentration means that small changes in the concentration of reactants can have large effects on the rate of reaction. Understanding this concept is crucial for chemists when they design experiments and analyze reaction mechanisms, as well as for industries where controlled reaction rates are essential, such as pharmaceuticals and materials manufacturing.