7.1 Principles of radiometric dating

Radiometric dating uses radioactive decay to determine the age of rocks and minerals. It's based on the principle that radioactive isotopes decay at a constant rate, measured by half-life. The ratio of parent to daughter isotopes in a sample reveals its age.

Key concepts include half-life, decay constant, and the radioactive decay equation. Techniques like isochron dating and understanding secular equilibrium help scientists accurately date geological samples. These methods are crucial for unraveling Earth's history and studying geological processes.

Radioactive Decay Principles

Half-life and Decay Constant

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• Half-life represents the time required for half of the original amount of a radioactive isotope to decay into its daughter product
• Varies widely among different radioactive isotopes, ranging from fractions of a second to billions of years
• Decay constant ($\lambda$) is the probability that a single atom will decay per unit time
  • Related to half-life ($t_{1/2}$) by the equation: $t_{1/2} = \frac{\ln(2)}{\lambda}$
  • Larger decay constants indicate more rapid radioactive decay and shorter half-lives

Parent and Daughter Isotopes

• Parent isotope is the original radioactive isotope that undergoes decay (uranium-238, potassium-40)
• Daughter isotope is the stable product formed by the radioactive decay of the parent isotope (lead-206, argon-40)
• Daughter isotopes accumulate over time as the parent isotope decays
  • Ratio of parent to daughter isotopes can be used to determine the age of a sample

Radioactive Decay Equation

• Describes the exponential decay of a radioactive isotope over time
• General form: $N(t) = N_0 e^{-\lambda t}$
  • $N(t)$ is the number of parent atoms remaining at time $t$
  • $N_0$ is the initial number of parent atoms
  • $\lambda$ is the decay constant
• Can be rearranged to solve for the age of a sample: $t = \frac{1}{\lambda} \ln(\frac{N_0}{N(t)})$

Radiometric Dating Techniques

Radiometric Clock and Closed System

• Radiometric clock starts when a mineral or rock forms and incorporates parent and daughter isotopes
  • Ratio of parent to daughter isotopes at formation is known or can be estimated
• Closed system assumes no loss or gain of parent or daughter isotopes after formation
  • Essential for accurate radiometric dating
  • Processes like weathering, metamorphism, or recrystallization can disrupt the closed system

Isochron Dating

• Technique used to determine the age of a rock or mineral using multiple samples with different parent-daughter ratios
• Plots the ratio of daughter isotope to a stable reference isotope (y-axis) against the ratio of parent isotope to the same reference isotope (x-axis)
  • Samples that formed at the same time will fall on a straight line called an isochron
• Slope of the isochron represents the age of the samples
• Accounts for potential initial daughter isotope present at formation

Advanced Concepts

Secular Equilibrium

• Occurs when the rate of decay of the parent isotope equals the rate of decay of the daughter isotope
• Achieved after approximately 6-10 half-lives of the daughter isotope
• Ratio of parent to daughter isotopes remains constant over time
  • Can be used to determine the age of very old samples (>1 million years) using long-lived isotopes (uranium-238, thorium-232)
• Example: In the uranium-238 decay series, secular equilibrium is reached between uranium-234 and thorium-230 after approximately 500,000 years