The concept of half-life is fundamental in nuclear physics, particularly in understanding radioactive decay. It defines the time required for half of the nuclei in a radioactive sample to undergo decay. This topic is essential for students preparing for the Cambridge IGCSE Physics - 0625 - Core examination, providing insights into nuclear stability, decay processes, and practical applications in various scientific fields.

Key Concepts

Understanding Radioactive Decay

Radioactive decay is a spontaneous process by which unstable atomic nuclei lose energy by emitting radiation. This process transforms the original nucleus into a different atomic configuration, which may be a different element or a different isotope of the same element. The rate at which this decay occurs is characterized by the half-life of the radioactive substance.

Definition of Half-Life

The half-life ($T_{1/2}$) of a radioactive isotope is the time required for half of the radioactive nuclei in a sample to decay. It is a constant property for each isotope and does not depend on the initial quantity of the material or the environmental conditions, such as temperature and pressure.

Mathematical Representation of Half-Life

The decay of a radioactive substance can be modeled using the exponential decay law. The number of undecayed nuclei $N(t)$ at a time $t$ is given by:

$$ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} $$

where:

$N_0$ is the initial number of radioactive nuclei
$T_{1/2}$ is the half-life
$t$ is the elapsed time

Decay Constant

The decay constant ($\lambda$) is another parameter that characterizes the rate of radioactive decay. It is related to the half-life by the equation:

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

This constant represents the probability per unit time that a nucleus will decay.

Exponential Decay Law

The exponential decay law combines the concepts of half-life and decay constant to describe the decay process mathematically. It provides a continuous function to predict the quantity of a radioactive substance remaining after a given time.

$$ N(t) = N_0 e^{-\lambda t} $$

Example Calculation

Consider a sample containing $800$ radioactive atoms with a half-life of $2$ hours. The number of atoms remaining after $6$ hours can be calculated as follows:

$$ N(t) = 800 \times \left(\frac{1}{2}\right)^{\frac{6}{2}} = 800 \times \left(\frac{1}{2}\right)^3 = 800 \times \frac{1}{8} = 100 $$

Thus, after $6$ hours, $100$ atoms remain undecayed.

Graphical Representation

The decay of a radioactive substance over time can be graphically represented by a decreasing exponential curve. The y-axis represents the number of undecayed nuclei or the activity, while the x-axis represents time. The curve asymptotically approaches zero, indicating that theoretically, an infinite time would be required for complete decay.

Activity and Half-Life

Activity ($A$) of a radioactive sample is the rate at which decays occur, measured in becquerels (Bq). It is directly proportional to the number of undecayed nuclei:

$$ A(t) = \lambda N(t) = \lambda N_0 e^{-\lambda t} $$

As the number of undecayed nuclei decreases, so does the activity of the sample.

Applications of Half-Life

Understanding half-life is crucial in various fields, including:

Radiometric Dating: Determining the age of archaeological finds and geological samples.
Medicine: Utilizing radioactive isotopes in diagnostics and treatment, such as in PET scans and cancer radiotherapy.
Nuclear Power: Managing radioactive waste and understanding the longevity of nuclear fuel.

Factors Affecting Half-Life

While half-life remains constant under specific conditions, certain factors can influence it:

Electron Capture Isotopes: Changes in electron configurations can slightly affect half-life.
Extreme Environmental Conditions: Theoretically, extreme pressures or chemical states might influence decay rates, though such effects are negligible for most isotopes.

Practical Considerations

In practical applications, half-life helps in predicting the behavior of radioactive materials over time. For instance, in nuclear waste management, knowing the half-life of isotopes informs storage duration and safety protocols.

Half-Life and Stability of Nuclei

The half-life of an isotope is indicative of its nuclear stability. Isotopes with short half-lives are typically less stable and decay rapidly, while those with longer half-lives are more stable. This stability is influenced by the balance between protons and neutrons within the nucleus.

Calculating Number of Half-Lives

Determining the number of half-lives that have occurred over a given time period is useful for solving decay problems. It is calculated as:

$$ n = \frac{t}{T_{1/2}} $$

Where $n$ is the number of half-lives, $t$ is the elapsed time, and $T_{1/2}$ is the half-life. The remaining quantity after $n$ half-lives is:

$$ N(t) = N_0 \times \left(\frac{1}{2}\right)^n $$

Integrated Form of Decay Law

Integrating the differential form of the decay law provides another perspective on the decay process. Starting from:

$$ \frac{dN}{dt} = -\lambda N $$

Integrating both sides yields:

$$ N(t) = N_0 e^{-\lambda t} $$

This form underscores the continuous nature of radioactive decay.

Half-Life vs. Mean Lifetime

While half-life is the time required for half the nuclei to decay, mean lifetime ($\tau$) is the average time a nucleus exists before decaying. They are related by:

$$ \tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln(2)} $$>

Advanced Concepts

Derivation of Decay Equations

Starting with the differential equation governing radioactive decay:

$$ \frac{dN}{dt} = -\lambda N $$>

We separate variables and integrate:

$$ \int \frac{dN}{N} = -\lambda \int dt $$> $$ \ln(N) = -\lambda t + C $$>

Exponentiating both sides:

$$ N(t) = e^{C} e^{-\lambda t} $$>

Letting $e^{C} = N_0$ (the initial quantity):

$$ N(t) = N_0 e^{-\lambda t} $$>

This derivation demonstrates the exponential nature of radioactive decay and the intrinsic link between the decay constant and half-life.

Solving Complex Decay Problems

Consider a scenario where a $^{137}$Cs sample with a half-life of $30.17$ years is used in medical treatments. If a patient is administered $2$ grams initially, how much remains after $75.51$ years?

First, calculate the number of half-lives:

$$ n = \frac{75.51}{30.17} = 2.5 $$>

Using the decay formula:

$$ N(t) = 2 \times \left(\frac{1}{2}\right)^{2.5} = 2 \times \frac{1}{\sqrt{32}} \approx 2 \times 0.17678 \approx 0.3536 \text{ grams} $$>

Thus, approximately $0.354$ grams of $^{137}$Cs remain after $75.51$ years.

Interdisciplinary Connections

The concept of half-life extends beyond nuclear physics, influencing various disciplines:

Medicine: Radioactive tracers in diagnostic imaging rely on isotopes with known half-lives to provide accurate imaging timelines.
Archaeology: Radiocarbon dating utilizes the half-life of $^{14}C$ to estimate the age of organic materials.
Environmental Science: Tracking the decay of isotopes helps in understanding pollutant dispersion and longevity in ecosystems.
Chemistry: Nuclear magnetic resonance (NMR) spectroscopy depends on the understanding of nuclear properties, including half-life.

Decay Chains and Half-Life

In nature, many radioactive isotopes undergo decay chains, where a parent nucleus decays into a series of daughter nuclei until a stable isotope is formed. Each step in the chain has its own half-life, affecting the overall behavior and stability of the resulting elements.

Stable vs. Unstable Isotopes

Isotopes with exceptionally long half-lives are considered practically stable, as their decay over observable timescales is negligible. Conversely, isotopes with short half-lives are highly unstable and decay rapidly, making them useful in applications requiring quick decay, such as medical diagnostics.

Implications in Nuclear Energy

Half-life plays a critical role in nuclear energy, particularly in the management of nuclear reactors and waste. Understanding the half-life of fuel materials ensures efficient energy production and informs strategies for long-term waste storage.

Impact on Radiometric Dating Accuracy

The precision of radiometric dating hinges on accurately knowing the half-life of isotopes used. Any uncertainty in half-life values directly affects the reliability of age estimates for geological and archaeological samples.

Statistical Interpretation of Half-Life

Half-life can also be viewed from a statistical standpoint, where it represents the median decay time in a large population of identical nuclei. This interpretation underscores the probabilistic nature of radioactive decay.

Half-Life in Cosmology

Half-life measurements of isotopes found in meteorites and cosmic dust provide valuable information about the processes and timescales involved in the formation of the solar system and other celestial phenomena.

Role in Forensic Science

Radioactive isotopes with known half-lives are used in forensic science to estimate the time of death in criminal investigations, based on the decay of isotopes in the body.

Environmental Half-Life Versus Nuclear Half-Life

While nuclear half-life refers to the decay of nuclei, environmental half-life accounts for the time it takes for a substance to reduce to half its concentration in the environment through all processes, including decay, dilution, and chemical reactions.

Comparison Table

Aspect	Nuclear Half-Life	Environmental Half-Life
Definition	Time for half of the radioactive nuclei to decay.	Time for half of a substance to be removed from the environment.
Process Involved	Radioactive decay of nuclei.	Processes like dilution, chemical reactions, and decay.
Dependency	Intrinsic property of the isotope.	Depends on environmental conditions and processes.
Application	Nuclear medicine, radiometric dating.	Environmental pollution studies, pollutant monitoring.

Summary and Key Takeaways

Half-life is the time required for half of a radioactive sample to decay.
It is a constant property specific to each radioactive isotope.
The exponential decay law models the decrease in undecayed nuclei over time.
Half-life has diverse applications across fields like medicine, archaeology, and environmental science.
Understanding half-life is crucial for managing nuclear materials and interpreting radiometric data.

Examiner Tip

Tips

Use the mnemonic "HALF" to remember Half-life: H for Half, A for Always constant, L for Logarithmic decay, and F for Formula application. Additionally, practice converting between the decay constant and half-life using the relationship $\lambda = \frac{\ln(2)}{T_{1/2}}$ to reinforce understanding for exam success.

Did You Know

Some isotopes, like Technetium-99, have no stable forms and are solely produced artificially for medical diagnostics. Additionally, the concept of half-life isn't limited to radioactive substances; it also applies to the decay of particles in pharmacology, influencing how long drugs remain active in the body.

Common Mistakes

One frequent error is confusing half-life with mean lifetime, leading to incorrect decay calculations. For example, using the half-life formula when the mean lifetime is required can skew results. Another common mistake is forgetting to account for the number of half-lives elapsed, which affects the remaining quantity of the substance.

FAQ

What is half-life?

Half-life is the time required for half of the radioactive nuclei in a sample to decay.

Does half-life depend on the initial amount of the substance?

No, half-life is an intrinsic property of the isotope and does not depend on the initial quantity of the substance.

How is the decay constant related to half-life?

The decay constant ($\lambda$) is inversely related to half-life by the equation $\lambda = \frac{\ln(2)}{T_{1/2}}$.

Can environmental factors affect the half-life of an isotope?

Generally, half-life is unaffected by environmental factors like temperature and pressure, as it is a nuclear property. However, in specific cases like electron capture, slight variations can occur under extreme conditions.

What applications rely on the concept of half-life?

Applications include radiometric dating in archaeology, medical diagnostics and treatment, nuclear power management, and environmental pollutant tracking.

How do you calculate the remaining quantity after multiple half-lives?

Use the formula $N(t) = N_0 \times \left(\frac{1}{2}\right)^n$, where $n$ is the number of half-lives elapsed.

1. Motion, Forces, and Energy

1.1 Energy Resources

1.1.1 Sources of energy (fossil fuels, biofuels, hydro, geothermal, nuclear, solar)

1.1.2 Advantages and disadvantages of different energy sources

1.1.3 Concept of efficiency

1.2 Power

1.2.1 Definition and calculation of power

1.2.2 Power as energy transferred per unit time

1.3 Pressure

1.3.1 Definition and calculation of pressure