Radioactive Dating

Definition:
Radioactive dating, also known as radiometric dating, is a technique used to determine the age of
materials such as rocks, fossils, and archaeological artifacts by measuring the relative
proportions of specific radioactive isotopes and their decay products.

Principle of Radioactive Dating:

Radioactive dating is based on the concept of radioactive decay, where unstable isotopes (parent
isotopes) transform into stable isotopes (daughter isotopes) over time at a predictable rate. This
rate is characterized by the isotope's half-life, which is the time required for half of the parent
isotopes to decay.

Key Components:
1. Radioactive Isotope (Parent): The unstable isotope that undergoes decay.
2. Stable Isotope (Daughter): The product of the decay process.
3. Half-Life: The time it takes for half of the parent isotope to decay.
4. Decay Constant (λ\lambdaλ): A value that represents the probability of decay per unit
time.
The age of a sample can be calculated using the formula:
t=ln⁡(N0Nt)λt = \frac{\ln\left(\frac{N_0}{N_t}\right)}{\lambda}t=λln(NtN0)
Where:
• ttt = age of the sample
• N0N_0N0 = initial amount of parent isotope
• NtN_tNt = remaining amount of parent isotope
• λ=ln⁡(2)Half-Life\lambda = \frac{\ln(2)}{\text{Half-Life}}λ=Half-Lifeln(2)

Common Isotopes Used in Radioactive Dating:

Isotope Half-Life Application

Carbon-14 5730 years Dating organic materials (e.g., bones, wood).

Uranium-238 4.5 billion years Dating rocks and the Earth's crust.

Potassium-40 1.3 billion years Dating volcanic rocks.

Isotope Half-Life Application

Rubidium-87 49 billion years Dating ancient minerals.

Steps in Radioactive Dating:

1. Sample Collection: Obtain a sample of material to be dated.
2. Isotope Analysis: Measure the quantities of the parent and daughter isotopes using
advanced instruments like mass spectrometers.
3. Age Calculation: Apply the radioactive decay formula to determine the time elapsed since
the sample formed.

Applications of Radioactive Dating:

1. Archaeology:
o Dating ancient artifacts and remains (e.g., using carbon-14 dating for bones and
wooden tools).
2. Geology:
o Determining the age of Earth's oldest rocks.
o Studying geological events such as volcanic eruptions.
3. Paleontology:
o Estimating the age of fossils to understand the evolution of life.
4. Astronomy:
o Dating meteorites to study the age and formation of the solar system.
Radioactive dating is based on the concept of radioactive decay, where unstable isotopes (parent
isotopes) transform into stable isotopes (daughter isotopes) over time at a predictable rate. This rate is
characterized by the isotope's half-life, which is the time required for half of the parent isotopes to decay.

Example Problem:
A fossil contains 25% of its original carbon-14 content. Calculate the age of the fossil. (Half-life
of carbon-14 is 5730 years.)
Solution:
The fossil is approximately 11,460 years old. t=n⋅Half-Life=2⋅5730=11,460 years.t = n \cdot
\text{Half-Life} = 2 \cdot 5730 = 11,460 \text{ years.}t=n⋅Half-Life=2⋅5730=11,460 years.
Advantages of Radioactive Dating:
• Provides precise and reliable age estimates.
• Applicable to a wide range of materials and time scales.

Limitations:
• Requires accurate measurement of isotope concentrations.
• Assumes no contamination of the sample since formation.
• Some methods are only applicable to certain types of materials (e.g., carbon-14 is only
effective for samples up to ~50,000 years old).
 HALF LIFE
Introduction:
Half-life is a key concept in science, describing the time it takes for half the quantity of a
radioactive substance, unstable atom, or certain chemical to decay or transform into a stable
form. This principle, rooted in exponential decay, is critical in fields like nuclear physics,
chemistry, archaeology, and medicine. Understanding half-life helps scientists estimate the age of
fossils, determine drug dosages, and handle nuclear waste.

Mathematical Formulation:
The decay of a substance over time is governed by an exponential decay law, expressed as:
Nt=N0e−λt
Where:
• Nt: Amount of the substance at time ttt.
• N0: Initial amount of the substance.
• λ: Decay constant, representing the probability of decay per unit time.
• t: Time elapsed.
The relationship between the decay constant (λ) and the half-life (t1/2) is given by:
This equation shows that the half-life is inversely proportional to the decay constant.
t1/2= ln(2)/ λ ≈ 0.693/ λ

Practical Problems:
Problem 1:
A radioactive isotope has a half-life of 5 years. If 160 grams of the substance are initially present,
how much will remain after 15 years?
Solution:
• Number of half-lives: n=Half-Life/Total Time=15/5=3
• Remaining quantity:
Nt=N0×(1/2)n=160×(1/3)3=160×1/8=20grams

Problem 2:
The activity of a sample decreases to 25% of its original value in 12 hours. What is its half-life?
Solution:
• 25% remaining indicates n=2 half-lives.
• Total time for 2 half-lives = 12 hours.
• Half-life (t1/2) = Total Time/Number of Half-Lives=12/2=6 hours

Applications of Half-Life:
1. Radioactive Dating:
o Carbon-14 Dating: Measures the age of organic materials by assessing the decay
of carbon-14.
o Uranium-238 Dating: Determines the age of rocks and Earth's crust, crucial for
geology.
2. Medical Science:
o Used in radiopharmaceuticals for imaging (e.g., PET scans) and treatments (e.g.,
iodine-131 for thyroid disorders).
3. Nuclear Energy:
o Understanding the decay of nuclear fuel and managing radioactive waste safely.
4. Drug Metabolism:
o Half-life helps pharmacologists design proper dosage schedules to ensure
effective drug levels in the bloodstream.

Visualization:
A graph of exponential decay shows the amount of substance decreasing with time. The curve is
steep initially and levels off as it approaches zero.
Steps to Create a Graph:
• X-axis: Time (e.g., hours, years).
• Y-axis: Remaining quantity (percentage or mass).
• Plot points for each half-life (e.g., 100%, 50%, 25%, 12.5%, etc.).
• Connect points to form a smooth curve.
The shape highlights that the decay rate slows over time, though the half-life remains constant.
Limitations:
• Assumes no contamination of the sample.
• Requires accurate initial measurements and conditions (e.g., no external interference).
• Some isotopes have extremely short or long half-lives, limiting their practical
applications.
Research Component:
Case Study: Carbon-14 in Archaeology
Carbon-14, with a half-life of 5730 years, is widely used in dating artifacts up to ~50,000 years
old. Researchers measure the remaining 14C in organic material and calculate its age using:
t=t1/2×log2(N0/Nt)
For instance, the dating of ancient tools and fossils has revolutionized our understanding of
human history.
Conclusion:
Half-life is a pivotal concept that bridges multiple disciplines, from understanding the age of the
Earth to designing life-saving medical treatments. Its mathematical simplicity belies its profound
implications in research and technology. By mastering half-life, we unlock a deeper appreciation
for the processes that shape our universe over time.