The Law of Radioactive Decay Revision Notes for Leaving Cert Physics

The Law of Radioactive Decay

What is radioactive decay?

Radioactive decay is a random process where unstable atomic nuclei spontaneously break down, releasing energy and particles. While we cannot predict exactly when any individual nucleus will decay, we can make accurate predictions about large numbers of nuclei using statistical laws.

The key characteristic of radioactive decay is that it follows a predictable pattern despite being random at the individual level - similar to how we can predict roughly how many people will be in a shop during lunch time, even though we can't predict exactly who will come.

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The random nature of radioactive decay means that while individual nuclear decay events are unpredictable, the behaviour of large collections of nuclei follows precise mathematical laws. This is similar to other statistical phenomena in nature, where individual events are random but collective behaviour is predictable.

Activity and its measurement

Activity measures how quickly a radioactive sample is decaying. It represents the number of nuclei that decay per second in a given sample.

The SI unit for activity is the becquerel (Bq), where:

1 Bq = 1 radioactive disintegration per second
1 Bq = 1 s⁻¹

For example, if a sample has an activity of 500 Bq, this means that 500 nuclei are decaying every second in that sample.

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Key Properties of Radioactive Activity:

You cannot predict which specific nuclei will decay next
You cannot predict exactly when any particular nucleus will decay
The decay process is entirely random for individual nuclei
However, at any given moment, the number of nuclei decaying per second is directly proportional to the total number of undecayed nuclei present

The mathematical law of radioactive decay

The law of radioactive decay is one of the fundamental principles in nuclear physics. It states that the activity of a radioactive sample is directly proportional to the number of undecayed nuclei present:

$A \propto N$

This can be written as an equation:

$A = \lambda N$

Where:

$A$ = activity (Bq)
$\lambda$ = radioactive decay constant (s⁻¹)
$N$ = number of undecayed nuclei

The radioactive decay constant (λ) is a characteristic property of each radioactive isotope. Different isotopes have different values of λ, which determines how quickly they decay.

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In calculus notation, this relationship can also be expressed as:

$A = -\frac{dN}{dt}$

The minus sign indicates that $N$ (the number of undecayed nuclei) decreases with time.

Since $A = \lambda N$ , we can find that $\lambda = A/N$ , giving λ units of s⁻¹.

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Worked Example: Calculating Undecayed Nuclei

Question: An isotope has a decay constant of $8 \times 10^{-5}$ s⁻¹. If a sample emits $2.4 \times 10^4$ beta particles per second, how many undecayed nuclei are present?

Solution:

Given: $\lambda = 8 \times 10^{-5}$ s⁻¹, $A = 2.4 \times 10^4$ Bq
Using $A = \lambda N$ , we get: $N = \frac{A}{\lambda}$
$N = \frac{2.4 \times 10^4}{8 \times 10^{-5}} = 3 \times 10^8$ nuclei

Half-life concept

The half-life $(T_{1/2})$ of a radioactive isotope is the time taken for half of the undecayed nuclei to decay.

This is a particularly useful concept because:

It gives us an intuitive way to understand decay rates
It remains constant for any given isotope regardless of sample size
It allows easy calculation of remaining material after any number of half-lives

The graphs show how both the number of undecayed nuclei and the activity decrease exponentially over time, following the same pattern when measured in half-life intervals.

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Understanding Half-Life Patterns

After each half-life period:

Half of the remaining nuclei decay
Half of the remaining nuclei survive to the next period
The activity is also halved

This creates an exponential decay curve that is characteristic of all radioactive processes.

Number of half-lives	Time (years)	Fraction remaining	Percentage remaining
1	5	1/2	50%
2	10	1/4	25%
3	15	1/8	12.5%
4	20	1/16	6.25%
5	25	1/32	3.125%

Relationship between half-life and decay constant

The half-life and decay constant are related by the fundamental formula:

$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$

This relationship allows us to convert between these two ways of describing decay rates:

Short half-life = large decay constant = fast decay
Long half-life = small decay constant = slow decay

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Worked Example: Finding Decay Constant

Question: A radioactive isotope has a half-life of 5 years. What is its decay constant?

Solution:

Given: $T_{1/2} = 5$ years
First convert to seconds: $5 \text{ years} = 5 \times 365 \times 24 \times 3600 = 1.58 \times 10^8$ s
Using $T_{1/2} = \frac{0.693}{\lambda}$
$\lambda = \frac{0.693}{T_{1/2}} = \frac{0.693}{1.58 \times 10^8} = 4.39 \times 10^{-9}$ s⁻¹

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Worked Example: Half-Life Calculations

Question: The half-life of a radioactive isotope is 30 minutes. What fraction remains undecayed after 1.5 hours?

Solution:

Half-life = 30 minutes = 0.5 hours
Time elapsed = 1.5 hours
Number of half-lives = 1.5 ÷ 0.5 = 3 half-lives

After 3 half-lives, the fraction remaining = $(1/2)^3 = 1/8$

Therefore, 1/8 of the original sample remains undecayed.

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Key Formulas Reference

Formula	Description	Units
$A = \lambda N$	Activity equals decay constant times undecayed nuclei	A in Bq, λ in s⁻¹, N is number
$T_{1/2} = \frac{0.693}{\lambda}$	Half-life related to decay constant	$T_{1/2}$ in s, λ in s⁻¹
$A = -\frac{dN}{dt}$	Activity as rate of change of nuclei	A in Bq

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Key Points to Remember:

Activity measures the rate of radioactive decay and is measured in becquerels (Bq)
The law of radioactive decay
states that activity is proportional to the number of undecayed nuclei: $A = \lambda N$
Half-life is the time for half the nuclei to decay and is related to the decay constant by $T_{1/2} = \frac{0.693}{\lambda}$
Radioactive decay is random for individual nuclei but follows predictable statistical patterns for large numbers
Each isotope has its own characteristic decay constant and half-life that remain constant regardless of sample size

The Law of Radioactive Decay (Leaving Cert Physics): Revision Notes