Question

For the zero order reaction \(A \rightarrow B+C ;\) initial concentration of \(A\) is \(0.1 M\). If \(A=0.08 M\) after 10 minutes, then it's half-life and completion time are respectively: (a) \(10 \mathrm{~min} ; 20 \mathrm{~min}\) (b) \(2 \times 10^{-3} \min ; 4 \times 10^{-3} \min\) (c) \(25 \mathrm{~min}, 50 \mathrm{~min}\) (d) \(250 \mathrm{~min}, 500 \mathrm{~min}\)

Short answer

The half-life and completion time are respectively 25 minutes and 50 minutes.

Step-by-step solution

Understanding zero order kinetics

For a zero-order reaction, the rate is independent of the concentration of the reactant. The rate law is given by \[\frac{d[A]}{dt} = -k\] where \[k\] is the rate constant and \[A\] is the concentration of reactant.

Calculate the rate constant

We can rearrange the rate law to solve for the rate constant \[k\]: \[k = \frac{-\Delta[A]}{\Delta t}\]. Given \[A=0.08 M\] after 10 minutes, \[\Delta[A] = 0.1 M - 0.08 M = 0.02 M\] and \[\Delta t = 10\ min\]. Substituting the values, we get \[k = \frac{0.02 M}{10 min} = 0.002 M/min\].

Calculate the half-life

For a zero order reaction, the half-life \(t_{1/2}\) is given by \[t_{1/2} = \frac{[A]_0}{2k}\] where \[ [A]_0\] is the initial concentration. Substituting the known values, \[t_{1/2} = \frac{0.1 M}{2 \times 0.002 M/min} = 25 min\].

Calculate the completion time

The completion time, \(t_{complete}\), is when the concentration of \(A\) reaches zero. This can be found by \[t_{complete} = \frac{[A]_0}{k}\]. Substituting the known values, \[t_{complete} = \frac{0.1 M}{0.002 M/min} = 50 min\].

Match with the given options

The calculated half-life and completion time are 25 minutes and 50 minutes, respectively. Therefore, the correct answer is option (c).

Key concepts

Rate Law

Understanding the rate law is fundamental in chemical kinetics, as it connects the velocity of a chemical reaction with the concentrations of reactants. For zero-order reactions, the rate law has a deceptively simple form:

The rate of reaction is constant and does not depend on the concentration of the reactant. Mathematically, this is represented as \br\br>\(\frac{d[A]}{dt} = -k\), where \([A]\) denotes the reactant's concentration and \(k\) is the rate constant. This equation implies that the reaction proceeds at a steady pace until one of the reactants is depleted. For students, it's crucial to understand that in zero-order kinetics, only the disappearance of reactant \(A\) over time \br\br>\(t\) is directly proportional to the rate constant \(k\), without the influence of how much \(A\) is initially present.

Half-life of Reaction

The half-life of a chemical reaction is a key concept describing the time it takes for half of the reactant to be consumed in a reaction. In zero order kinetics, the half-life is directly proportional to the initial concentration of the reactant and inversely proportional to the rate constant. Given by \br\br>\(t_{1/2} = \frac{[A]_0}{2k}\),

where \([A]_0\) is the initial concentration. This relationship shows that, as opposed to first-order reactions, the half-life for zero-order reactions will change as the concentration changes. This can be somewhat counterintuitive to students who must remember that for zero-order reactions, half-life depends on the starting conditions of the reactant.

Rate Constant Calculation

Calculating the rate constant, \(k\), is an integral step in solving kinetic problems for zero order reactions. It is determined from the observed change in reactant concentration over time, according to the relationship

\(k = \frac{-\triangle[A]}{\triangle t}\).

For students, calculating \(k\) involves identifying the concentration change, \(\triangle[A]\), and the corresponding time interval, \(\triangle t\). In the exercise provided, when the concentration decreases by \(0.02\text{ M}\) over \(10\text{ minutes}\), the rate constant is computed to be \(0.002\text{ M/min}\). This straightforward formula highlights the importance of measuring changes precisely to ensure accurate calculation of the rate constant, a key factor governing the reaction's progress.

Chemical Kinetics Problems

Problem solving in chemical kinetics requires understanding of concepts such as rate laws, reaction order, and the calculation of various parameters including half-life and rate constants. Key to tackling these problems is a methodical approach: starting with interpreting the given reaction order, using the appropriate formulas, and substituting the known quantities to solve for the unknown variables.

The step-by-step solution for the problem given demonstrates this approach by computing the half-life and completion time of a zero-order reaction. It emphasizes the need to methodically follow through each calculation phase, ensuring students grasp the sequential nature of solving kinetics problems.