Half-life of reactions Knowledge-Based Study

Study notes

1. The half-life of a chemical reaction, denoted as $t_{1/2}$, is the time required for the concentration of a reactant to decrease to exactly half of its initial value.

   Mathematically, when time $t = t_{1/2}$, the reactant concentration $[A]$ reaches:

   $$[A] = \frac{[A]_0}{2}$$

   where $[A]_0$ is the initial concentration of the reactant at $t = 0$. Half-life serves as a valuable metric in chemical kinetics to characterize how rapidly a reactant is consumed and to determine the underlying reaction order.

   For a first-order reaction with the rate law $\text{Rate} = k[A]$, the integrated rate law is expressed as:

   $$\ln\left(\frac{[A]_0}{[A]}\right) = kt$$

   Substituting $[A] = \frac{[A]_0}{2}$ at $t = t_{1/2}$ gives:

   $$\ln\left(\frac{[A]_0}{\frac{[A]_0}{2}}\right) = \ln(2) = k t_{1/2}$$

   Solving for $t_{1/2}$ yields:

   $$t_{1/2} = \frac{\ln(2)}{k} \approx \frac{0.693}{k}$$

   Key Insight: In a first-order process, the half-life is completely independent of the initial reactant concentration. Each successive half-life takes the exact same duration regardless of starting amount.

   Unlike first-order processes, the half-life of zero-order and second-order reactions depends directly on the initial concentration of the reactant.

   Zero-Order Reactions

   For a zero-order reaction ($\text{Rate} = k$), the integrated rate equation is $[A] = [A]_0 - kt$. Substituting $[A] = \frac{[A]_0}{2}$ yields:

   $$t_{1/2} = \frac{[A]_0}{2k}$$

   Here, $t_{1/2}$ is directly proportional to $[A]_0$. As the reaction proceeds and reactant is depleted, successive half-lives become shorter.

   Second-Order Reactions

   For a second-order reaction involving a single reactant ($\text{Rate} = k[A]^2$), the integrated rate equation is $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$. Substituting $[A] = \frac{[A]_0}{2}$ yields:

   $$t_{1/2} = \frac{1}{k[A]_0}$$

   Here, $t_{1/2}$ is inversely proportional to $[A]_0$. As the reactant concentration decreases, successive half-lives double in duration.

   A concise summary of how half-life relates to reactant concentration across common reaction orders:

   • Zero-Order: $t_{1/2} = \frac{[A]_0}{2k}$ — Half-life decreases as concentration decreases.
   • First-Order: $t_{1/2} = \frac{0.693}{k}$ — Half-life remains constant throughout the reaction.
   • Second-Order: $t_{1/2} = \frac{1}{k[A]_0}$ — Half-life increases as concentration decreases.

   Monitoring whether successive half-lives remain constant, decrease, or increase over time provides a straightforward experimental diagnostic for deducing the reaction order.