Understanding Radioactive Decay and First-order Reactions: A Practical Guide

Radioactive decay is a fundamental process in physics and chemistry, underpinning our understanding of nuclear reactions and the behavior of atomic isotopes. This phenomenon is best described through the concept of a first-order reaction, where the rate of decay is directly proportional to the amount of the substance present. In this guide, we will delve into the practical application of radioactive decay and first-order reactions using a specific example involving a radioactive substance with a half-life of 3.8 hours.

An Introduction to Half-life

In nuclear physics, the half-life of a radioactive substance is the time required for half of the substance to decay. It is an intrinsic property of the substance and is constant for a given isotope. For the substance in question, the half-life is 3.8 hours. This means that every 3.8 hours, the amount of the radioactive substance is halved.

First-order Reaction and Decay Rate

A first-order reaction is characterized by the rate law: rate k[A], where A is the concentration of the reactant, and k is the rate constant. For radioactive decay, the decay follows a first-order rate law:

rate  -k[A]

The negative sign indicates that the concentration of the substance is decreasing over time. The decay constant k can be calculated using the half-life t_{1/2} through the relation:

Formula: k 0.693/t_{1/2}

Substituting the given half-life of 3.8 hours, we calculate:

k  0.693 / 3.8 h ≈ 0.18 h-1

Practical Application: Decay of a Radioactive Substance

Let's consider the scenario where, at a certain time, the amount of the radioactive substance is 30% of its original value. We need to determine how many hours it will take for this decay to occur. The decay equation for a first-order reaction is given by:

ln(A/A0)  -kt

Where:

A is the amount of the substance at time t A0 is the original amount of the substance k is the decay constant t is the time in hours

In our case, A is 0.30 times the original amount A0. Substituting the values, we get:

ln(0.30/1)  -0.18t

Solving for t:

t  -ln(0.30) / 0.18 h ≈ 6.6 h

Conclusion

Through this exploration, we have gained a deeper understanding of the relationship between half-life and decay rate, as well as the mathematical principles governing first-order reactions. By applying the decay constant and the decay equation, we can predict the time required for a specific amount of a radioactive substance to decay to a given percentage of its original value. This knowledge is crucial in various fields, including nuclear medicine, geology, and environmental science.

Further Reading

For more detailed information on radioactive decay and first-order reactions, you may refer to the following resources:

NIST Radioactive Decay Constants LibreTexts - Radioactive Decomposition and Radioactivity U.S. EPA - Radioactive Decay Partitions and Half-lives: Relation to Radiation Decay