Question

The rate equation for the reaction \(2 \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}\) is found to be rate \(=k[\mathrm{A}][\mathrm{B}] .\) For this reaction, we can conclude that (a) the unit of \(k=s^{-1} ;\) (b) \(t_{1 / 2}\) is constant; (c) the value of \(k\) is independent of the values of \([\mathrm{A}]\) and \([\mathrm{B}] ;\) (d) the rate of formation of \(\mathrm{C}\) is twice the rate of disappearance of A.

Short answer

(a) Incorrect, the unit should be \(M^{-1} s^{-1}\); (b) Incorrect, \(t_{1/2}\) is not constant for a second-order reaction; (c) Correct, \(k\) is independent of the reactant concentrations; (d) Incorrect, the rate of formation of C is half the rate of disappearance of A.

Step-by-step solution

Analyze Statement (a)

The statement provides a given unit for the rate constant \(k\). However, the unit of the rate constant depends on the overall order of the reaction. Here, the rate equation is given as \(rate = k[\mathrm{A}][\mathrm{B}]\) which indicates a second order reaction (first order with respect to both A and B). So, the units of \(k\) for a second order reaction are \(M^{-1} s^{-1}\), not \(s^{-1}\). Hence, statement (a) is incorrect.

Analyze Statement (b)

The half-life of a reaction (\(t_{1/2}\)) indicates the time it takes for the concentration of a reactant to decrease to half its initial concentration. For a second order reaction, \(t_{1/2}\) is not constant but depends on the initial concentration of the reactants. Hence, statement (b) is not correct.

Analyze Statement (c)

The rate constant (\(k\)) is indeed independent of the concentrations of the reactants. The value of \(k\) is a constant for a given reaction at a fixed temperature. Therefore, statement (c) is correct.

Analyze Statement (d)

Inspecting the stoichiometric equation (i.e. \(2A + B → C\)), it is clear that two molecules of A react to form one molecule of C. Hence, the rate at which C forms (rate of formation of C) is half the rate at which A disappears (rate of disappearance of A) and not the other way around. Therefore, statement (d) is incorrect.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors influencing those rates. It is a subfield of physical chemistry that deals with understanding how and why chemical reactions occur as quickly or as slowly as they do.

In chemical kinetics, the reaction rate signifies how fast a reaction proceeds. It's often expressed as the change in concentration of a reactant or product per unit of time. Factors such as temperature, concentration of reactants, surface area, catalysts, and the presence of light can affect reaction rates.

Understanding kinetics can help us control reactions in industrial processes, biological systems, and even food preparation. The better we comprehend the factors affecting reaction rates, the more effectively we can manipulate them for desired outcomes.

Reaction Order

Reaction order is a concept in kinetics that describes the power to which the concentration of a reactant is raised in the rate equation. The order gives insight into the relationship between the concentration of reactants and the reaction rate.

A reaction can be zero, first, second, or of higher order, often determined by experiment. For example, a first-order reaction implies that the rate is directly proportional to the concentration of one reactant.

Understanding the Reaction Order

In the reaction from the exercise, the rate equation is proportional to the concentrations of both A and B, meaning both have a reaction order of one, making the overall reaction second order.

Knowing the reaction order is crucial because it influences how we predict the change in reaction rate with changing reactant concentrations and has implications for calculating the half-life of a reaction.

Rate Constant

The rate constant, denoted by the symbol 'k,' is a crucial parameter in the rate equation that shows the speed of a chemical reaction. It relates the rate of the reaction to the concentrations of the reactants for a given reaction at a specific temperature.

What's unique about the rate constant is that it does not change with the concentrations of the reactants but can vary with temperature. Essentially, it's the intrinsic 'speed' at which the reaction can proceed under certain conditions.

Contemplating the practical significance, the rate constant is used to compare the rates of different reactions and to measure the effect of temperature on reaction rates through the Arrhenius equation. Understanding the rate constant is vital as it helps in the design and control of chemical processes.

Half-Life of a Reaction

The half-life of a reaction, often represented as \(t_{1/2}\), is the time required for the concentration of a reactant to reduce to half its initial value. This concept is widely used in radiochemistry, pharmacokinetics, and chemical kinetics.

The half-life varies depending on the order of the reaction. For instance, in a first-order reaction, the half-life is constant and independent of initial concentration. However, in second-order and higher-order reactions, the half-life depends on the initial concentration of the reactants, as shown in our exercise where statement (b) suggested a constant half-life which is incorrect for the given second-order reaction.

Knowing the half-life helps chemists understand the duration over which reactants will be depleted or the time frame within which products will be formed, which is essential in processes like drug delivery and environmental pollution control.