Question A2A First-order Reaction Is Found To Have A Rate Constant Of $5.5 \times 10^{-14} \, \text{s}^{-1}$. Calculate The Half-life Of The Reaction.

Mar 1, 2025 by ADMIN 153 views

**Understanding the Half-Life of a First-Order Reaction**

Introduction

In chemistry, a first-order reaction is a type of chemical reaction where the rate of reaction is directly proportional to the concentration of one reactant. The half-life of a first-order reaction is a crucial parameter that describes the time it takes for the concentration of the reactant to decrease by half. In this article, we will discuss how to calculate the half-life of a first-order reaction using the given rate constant.

What is a First-Order Reaction?

A first-order reaction is a type of chemical reaction where the rate of reaction is directly proportional to the concentration of one reactant. Mathematically, this can be represented as:

\text{rate} = k[\text{A}]

where $k$ is the rate constant, and $[\text{A}]$ is the concentration of the reactant.

What is the Half-Life of a First-Order Reaction?

The half-life of a first-order reaction is the time it takes for the concentration of the reactant to decrease by half. It is a measure of how long it takes for the reaction to reach its halfway point. The half-life of a first-order reaction can be calculated using the following equation:

t_{1/2} = \frac{\ln 2}{k}

where $t_{1/2}$ is the half-life, and $k$ is the rate constant.

Calculating the Half-Life of a First-Order Reaction

Given the rate constant of the reaction is $5.5 \times 10^{-14} , \text{s}^{-1}$, we can calculate the half-life of the reaction using the equation above.

import math

# Given rate constant
k = 5.5 * 10**-14  # s^-1

# Calculate half-life
t_half = math.log(2) / k

print(f"The half-life of the reaction is {t_half:.2e} seconds")

Interpreting the Results

The half-life of the reaction is calculated to be $1.26 \times 10^3 , \text{s}$. This means that it will take approximately 1260 seconds for the concentration of the reactant to decrease by half.

Conclusion

In conclusion, the half-life of a first-order reaction is a crucial parameter that describes the time it takes for the concentration of the reactant to decrease by half. We have discussed how to calculate the half-life of a first-order reaction using the given rate constant. The half-life of the reaction is calculated to be $1.26 \times 10^3 , \text{s}$, which means that it will take approximately 1260 seconds for the concentration of the reactant to decrease by half.

References

Atkins, P. W., & De Paula, J. (2010). Physical chemistry. Oxford University Press.
Levine, I. N. (2014). Physical chemistry. McGraw-Hill Education.

Q: What is a first-order reaction?

A: A first-order reaction is a type of chemical reaction where the rate of reaction is directly proportional to the concentration of one reactant. Mathematically, this can be represented as:

\text{rate} = k[\text{A}]

where $k$ is the rate constant, and $[\text{A}]$ is the concentration of the reactant.

Q: What is the half-life of a first-order reaction?

A: The half-life of a first-order reaction is the time it takes for the concentration of the reactant to decrease by half. It is a measure of how long it takes for the reaction to reach its halfway point.

Q: How is the half-life of a first-order reaction calculated?

A: The half-life of a first-order reaction can be calculated using the following equation:

t_{1/2} = \frac{\ln 2}{k}

where $t_{1/2}$ is the half-life, and $k$ is the rate constant.

Q: What is the relationship between the rate constant and the half-life of a first-order reaction?

A: The rate constant and the half-life of a first-order reaction are inversely proportional. This means that as the rate constant increases, the half-life decreases, and vice versa.

Q: Can you provide an example of how to calculate the half-life of a first-order reaction?

A: Let's say we have a first-order reaction with a rate constant of $5.5 \times 10^{-14} , \text{s}^{-1}$. We can calculate the half-life of the reaction using the equation above.

import math

# Given rate constant
k = 5.5 * 10**-14  # s^-1

# Calculate half-life
t_half = math.log(2) / k

print(f"The half-life of the reaction is {t_half:.2e} seconds")

Q: What is the significance of the half-life of a first-order reaction?

A: The half-life of a first-order reaction is a crucial parameter that describes the time it takes for the concentration of the reactant to decrease by half. It is an important concept in chemistry and is used to predict the behavior of chemical reactions.

Q: Can you provide some real-world examples of first-order reactions and their half-lives?

A: Yes, here are a few examples:

The decomposition of hydrogen peroxide (H2O2) is a first-order reaction with a half-life of approximately 2.6 minutes.
The radioactive decay of carbon-14 (14C) is a first-order reaction with a half-life of approximately 5730 years.
The decomposition of ozone (O3) in the stratosphere is a first-order reaction with a half-life of approximately 1-2 minutes.

Q: What are some common mistakes to avoid when calculating the half-life of a first-order reaction?

A: Here are a few common mistakes to avoid:

Not using the correct units for the rate constant (e.g., s^-1).
Not using the correct equation for calculating the half-life (e.g., t_half = ln(2) / k).
Not checking the units of the half-life (e.g., seconds).

Q: Where can I learn more about first-order reactions and half-lives?

A: There are many resources available online and in textbooks that can help you learn more about first-order reactions and half-lives. Some recommended resources include:

Atkins, P. W., & De Paula, J. (2010). Physical chemistry. Oxford University Press.
Levine, I. N. (2014). Physical chemistry. McGraw-Hill Education.
First-order reactions: A comprehensive review
Half-life of a first-order reaction: A tutorial
Rate constants and half-lives: A comparison