### Understanding Half-Life in Chemical ReactionsThe **half-life** of a reaction, denoted as $ t_{1/2} $, is the time required for the concentration of a reactant $[A]$ to reduce by half. This concept is crucial in understanding the kinetics of reactions. For example:- After one half-life, the concentration decreases from the initial concentration $[A]_0$ to $[A]_0/2$.- After a second half-life, it reduces to $[A]_0/4$.- After a third half-life, it becomes $[A]_0/8$, and so on.#### First-Order ReactionsFor first-order reactions, the half-life is constant, relying only on the rate constant $ k $, and is independent of the reactant concentration. It is expressed by the formula:\[t_{1/2} = \frac{\ln(2)}{k}\]#### Second-Order ReactionsIn contrast, the half-life for second-order reactions depends on both the rate constant and the initial concentration of the reactant. It is calculated using:\[t_{1/2} = \frac{1}{k[A]_0}\]### Exercise Examples#### Part AA certain first-order reaction ($A \rightarrow \text{products}$) has a rate constant of $6.30 \times 10^{-3} \, \text{s}^{-1}$ at 45°C. Determine the time it takes for the concentration of reactant $[A]$ to decrease to 6.25% of its original level. Provide your answer with the appropriate units.**Input Box for Answer:**- Value- Units- Submit button#### Part BIn a second-order reaction ($B \rightarrow \text{products}$), with a rate constant of $1.35 \times 10^{-3} \, \text{M}^{-1} \, \text{s}^{-1}$ at 27°C and an initial half-life of 240 seconds, calculate the concentration of reactant $B$ after one half-life.**Input Box for Answer:**- Value- Units- Submit buttonThese exercises help to apply the theoretical understanding of reaction kinetics in practical scenarios.---This content is structured to aid in the conceptual and practical comprehension of half

Answered: The half-life of a reaction, f/2, is…

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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### Understanding Half-Life in Chemical Reactions

The **half-life** of a reaction, denoted as $ t_{1/2} $, is the time required for the concentration of a reactant $[A]$ to reduce by half. This concept is crucial in understanding the kinetics of reactions. For example:
- After one half-life, the concentration decreases from the initial concentration $[A]_0$ to $[A]_0/2$.
- After a second half-life, it reduces to $[A]_0/4$.
- After a third half-life, it becomes $[A]_0/8$, and so on.

#### First-Order Reactions
For first-order reactions, the half-life is constant, relying only on the rate constant $ k $, and is independent of the reactant concentration. It is expressed by the formula:
\[
t_{1/2} = \frac{\ln(2)}{k}
\]

#### Second-Order Reactions
In contrast, the half-life for second-order reactions depends on both the rate constant and the initial concentration of the reactant. It is calculated using:
\[
t_{1/2} = \frac{1}{k[A]_0}
\]

### Exercise Examples

#### Part A
A certain first-order reaction ($A \rightarrow \text{products}$) has a rate constant of $6.30 \times 10^{-3} \, \text{s}^{-1}$ at 45°C. Determine the time it takes for the concentration of reactant $[A]$ to decrease to 6.25% of its original level. Provide your answer with the appropriate units.

**Input Box for Answer:**
- Value
- Units
- Submit button

#### Part B
In a second-order reaction ($B \rightarrow \text{products}$), with a rate constant of $1.35 \times 10^{-3} \, \text{M}^{-1} \, \text{s}^{-1}$ at 27°C and an initial half-life of 240 seconds, calculate the concentration of reactant $B$ after one half-life.

**Input Box for Answer:**
- Value
- Units
- Submit button

These exercises help to apply the theoretical understanding of reaction kinetics in practical scenarios.

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This content is structured to aid in the conceptual and practical comprehension of half

Expert Solution

Step 1: Defining half life of the reaction

Answer:

Time required to take the concentration of reactant to half of its initial value is called as its half life.

For first order reaction:

$straight t subscript 1 divided by 2 end subscript equals fraction numerator ln 2 over denominator straight k end fraction$

For second order reaction:

$straight t subscript 1 divided by 2 end subscript equals fraction numerator 1 over denominator straight k left square bracket straight A right square bracket subscript 0 end fraction$

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