Question

The integrated rate law for the zero-order reaction \(\mathrm{A} \longrightarrow \mathrm{B}\) is \([\mathrm{A}]_{t}=[\mathrm{A}]_{0}-k t .\) (a) Sketch the follow- ing plots: (i) rate versus \([\mathrm{A}]_{t}\) and (ii) \([\mathrm{A}]_{t}\) versus \(t\). (b) Derive an expression for the half-life of the reaction. (c) Calculate the time in half-lives when the integrated rate law is no longer valid, that is, when \([\mathrm{A}]_{t}=0\)

Short answer

The rate of the reaction is constant and does not depend on the concentration of A, hence the rate versus [A]_t plot is a horizontal line. The [A]_t versus t plot is a straight line with a negative slope. The half-life of the reaction is [A]_0 / (2k), and the time when the reaction ends and the integrated rate law is no longer valid, in half-lives, is 2.

Step-by-step solution

Understanding the rate law for a zero-order reaction

The rate law for a zero-order reaction is described by the equation [A]_t = [A]_0 - kt, where [A]_t is the concentration of the reactant at time t, [A]_0 is the initial concentration of the reactant, k is the rate constant and t is the reaction time.

Plotting the graphs

(i) Plotting rate versus [A]_t: Since this is a zero-order reaction, the rate would be constant and not change with varying [A]_t. Therefore, the plot will be a horizontal straight line. (ii) Plotting [A]_t versus t: [A]_t decreases linearly with time as per the rate law [A]_t = [A]_0 - kt. Thus, the plot is a straight line with a negative slope.

Deriving the expression for the half-life

The half-life is the time it takes for half of the reactant to be consumed. It can be calculated by setting [A]_t = 1/2 [A]_0 in the rate law. This gives t_(1/2) = [A]_0 / (2k). Hence, the half-life is inversely proportional to the rate constant and directly proportional to the initial concentration.

Determining the validity of the integrated rate law

The integrated rate law is no longer valid when there is no reactant left, i.e., [A]_t equals zero. Substituting [A]_t = 0 in the rate law gives t = [A]_0 / k. This means that the time when the integrated rate law is no longer valid is the time it takes for all reactants to be consumed. To calculate this time in half-lives, we divide this time by the half-life, which gives the result as [A]_0 / (t_(1/2)) = 2.