Radioactive Decay Law

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Radioactive Decay Law

This lesson aligns with NGSS PS1.C

Introduction

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. This spontaneous transformation occurs in radioactive isotopes, which are variants of chemical elements with different numbers of neutrons. The goal of radioactive decay is to achieve a more stable nuclear configuration. In this educational article, we will delve into the mathematical expressions of the Radioactive Decay Law, its derivation and provide illustrative examples to enhance our understanding.

Radioactive Decay

Radioactive decay refers to the process by which elementary particles are emitted from an unstable nucleus, leading to a transformation of the unstable element into a more stable element. There exist three distinct types of radioactive decay: alpha emission, beta emission, and gamma emission.

Each type of decay involves the emission of a specific particle, resulting in a corresponding change in the type of product generated.

The Radioactive Decay Law: Mathematical Formulation

The Radioactive Decay Law provides a quantitative description of the rate at which radioactive substances decay over time. The law is expressed mathematically as:

Where:

N(t) is the quantity of the radioactive substance remaining at time t.
[math]N_0[/math] is the initial quantity of the radioactive substance.
λ is the decay constant, a characteristic property of each radioactive isotope.
t is the time elapsed

When a radioactive substance experiences α, β, or γ-decay, the rate at which nuclei undergo decay per unit time is directly proportional to the overall number of nuclei present in the given sample material.

If N = total number of nuclei in the given sample and

ΔN = number of nuclei that undergo decay in time Δt then,

where λ = radioactive decay constant or disintegration constant. Now, the change in the number of nuclei in the given sample is, dN = – ΔN in time Δt. Hence, the rate of change of N (in the limit Δt → 0) is,

Now, integrating both the sides of the above equation, we get,

Here,

[math]N_0[/math] represents the count of radioactive nuclei within the sample at a chosen time [math]t_0[/math] ,and N denotes the count of radioactive nuclei at any subsequent time t. Subsequently, by setting [math][/math]=0 and rearranging the aforementioned equation (3), we obtain:

Equation 4 is known as radioactive decay law.

The equation illustrates an exponential decay model, demonstrating that the rate of decay is proportional to the remaining quantity of the substance. The decay constant (λ) is a measure of how quickly a particular isotope undergoes decay. The larger the decay constant, the faster the decay process.

The graph illustrates the relationship between the number of radioactive nuclei at any given time, denoted as Nt, and the corresponding time, t. It is evident from the graph that the initial number of radioactive nuclei, [math]N_0[/math], observed at t = 0, experiences a rapid decline initially, followed by a more gradual decrease in a characteristic exponential fashion.

All three curves depicted in this context exhibit exponential characteristics, with the only variation being in the Decay Constant. It is noteworthy that a lower Decay Constant corresponds to a relatively gradual decrease in the curve, whereas a larger Decay Constant results in a more rapid decline.

Half-Life and its Relation to Radioactive Decay

Half-life , which is the time required for half of the radioactive substance to decay. The relationship between the half-life and the decay constant is given by:

This formula highlights the inversely proportional relationship between the half-life and the decay constant. A smaller decay constant corresponds to a longer half-life, indicating a slower rate of decay.

Sample Problems of Radioactive Decay:

1. Determine the decay constant for Carbon-14, if it has a half-life of 5730 years?

Carbon-14 is a radioactive isotope used in archaeological and geological dating. It undergoes beta decay, converting a neutron into a proton while emitting a beta particle.

By putting the values in equation 4,

The decay constant (λ) for carbon-14 is approximately 1.21×[math]10^-4[/math][math]years^-1[/math]

2. If the initial amount of carbon-14 is 100g then What is the remaining amount of carbon-14 after 5730 years?

Suppose we start with an initial quantity ([math]N_0[/math]) of 100 grams of carbon-14. After 5730 years (the half-life of carbon-14), the remaining quantity (N(t)) can be calculated using the Radioactive Decay Law.

The remaining amount of carbon-14 will decay down to 50g after 5730 years.

Summary

Radioactive decay refers to the process by which elementary particles are emitted from an unstable nucleus, transforming into a more stable element.
The Radioactive Decay Law provides a quantitative description of the rate at which radioactive substances decay over time.
There is an inverse relationship between the half-life and the decay constant.
A smaller decay constant corresponds to a longer half-life, indicating a slower rate of decay.

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