Radioactive decay is a fascinating process where unstable atoms break down over time. Half-life, the time it takes for half of a substance to decay, is key to understanding this phenomenon. It's constant for each radioisotope and helps us calculate how much remains after a given period.

Radioactive dating uses these principles to determine the age of materials. By comparing the ratio of a radioactive isotope to its decay product, scientists can estimate how old something is. This technique is crucial for dating fossils, artifacts, and even rocks.

Radioactive Decay and Half-Life

Half-life in radioactive decay

• Half-life represents the time required for half of a given quantity of a radioactive substance to decay into more stable elements or isotopes
  • Remains constant for each specific radioisotope (carbon-14, uranium-238)
  • Independent of the initial amount of the substance present at the start of the decay process
• Significance in radioactive decay
  • Determines the rate at which a radioactive substance decays over time
  • Allows for the calculation of the amount of substance remaining after a given time period has elapsed
  • Enables radioactive dating techniques to estimate the age of organic (carbon-14 dating) and inorganic (uranium-lead dating) materials

Calculations with half-lives

• Exponential decay formula: $N(t) = N_0 \cdot (\frac{1}{2})^{t/t_{1/2}}$
  • $N(t)$ represents the amount of substance remaining at time $t$
  • $N_0$ represents the initial amount of substance present at the start of the decay process
  • $t$ represents the elapsed time since the start of the decay process
  • $t_{1/2}$ represents the half-life of the substance undergoing decay
• Alternative formula for calculating remaining substance: $N(t) = N_0 \cdot (0.5)^n$
  • $n$ represents the number of half-lives that have elapsed since the start of the decay process
• Amount of substance remaining is halved with each passing half-life, resulting in an exponential decrease over time

Radioactive dating techniques

• Radioactive dating relies on comparing the ratio of a radioactive isotope to its decay product within a sample
  • Carbon-14 dating commonly used for organic materials (fossils, artifacts)
    • Half-life of carbon-14 is approximately 5,730 years, allowing for dating of relatively recent materials
  • Uranium-lead dating commonly used for inorganic materials (rocks, minerals)
    • Half-life of uranium-238 is approximately 4.5 billion years, allowing for dating of ancient geological samples
• Steps involved in radioactive dating
  1. Measure the amount of radioactive isotope and its decay product present in the sample
  2. Calculate the ratio of the radioactive isotope to its decay product based on the measured amounts
  3. Compare the calculated ratio to a standard ratio to determine the age of the sample, taking into account the known half-life of the radioactive isotope

Half-life vs decay rate

• Rate of radioactive decay is inversely proportional to the half-life of the substance
  • Shorter half-life indicates a faster rate of decay (radon-222, half-life of 3.8 days)
  • Longer half-life indicates a slower rate of decay (potassium-40, half-life of 1.3 billion years)
• Rate of decay is expressed as the number of disintegrations per unit time
  • Measured in becquerels (Bq), with 1 Bq representing one disintegration per second
  • Alternatively measured in curies (Ci), with 1 Ci representing 3.7 × 10^10 disintegrations per second
• Relationship between rate of decay and half-life expressed mathematically: $\lambda = \frac{\ln 2}{t_{1/2}}$
  • $\lambda$ represents the decay constant, which is a measure of the rate of decay
  • $t_{1/2}$ represents the half-life of the substance undergoing decay
• Understanding the relationship between half-life and rate of decay is crucial for predicting the behavior of radioactive substances over time and assessing their potential hazards or applications