Honors Physics
Table of Contents
Unit 22 Overview: The Atom
22.1 The Structure of the Atom
22.2 Nuclear Forces and Radioactivity
22.3 Half Life and Radiometric Dating
22.4 Nuclear Fission and Fusion
22.5 Medical Applications of Radioactivity: Diagnostic Imaging and Radiation

⚾️honors physics review

22.3 Half Life and Radiometric Dating

Citation:

Radioactive decay is a fascinating process where unstable atoms lose energy by emitting radiation. The half-life, a key concept, measures how long it takes for half of a radioactive substance to decay. This predictable rate allows scientists to date ancient materials.

Calculations help us understand decay rates and solve real-world problems. We use equations to determine how much of a substance remains after a certain time. This knowledge is crucial for applications like carbon dating and nuclear medicine.

Radioactive Decay and Half-Life

Concept of radioactive half-life

  • Radioactive decay process where an unstable atomic nucleus loses energy by emitting radiation
  • Occurs at a characteristic rate for each radioactive isotope measured by its half-life (uranium-238, carbon-14)
  • Half-life time required for a quantity of a radioactive substance to reduce to half of its initial value
    • Remains constant for each specific isotope (potassium-40 has a half-life of 1.3 billion years)
    • Independent of external factors (temperature, pressure, chemical environment)
  • Radiometric dating utilizes predictable decay rates of radioactive isotopes to determine the age of materials (fossils, rocks)
    • Compares ratio of a radioactive isotope to its decay products in a sample (uranium-lead dating)
    • Calculates time elapsed since the sample formed based on the isotope's known half-life (rubidium-strontium dating)
    • Involves measuring the ratio of parent isotope to daughter isotope in the sample

Calculations for decay rates

  • Decay rate of a radioactive substance calculated using the equation $N(t) = N_0 e^{-\lambda t}$
    • $N(t)$ quantity of the radioactive isotope at time $t$
    • $N_0$ initial quantity of the radioactive isotope
    • $\lambda$ decay constant related to the half-life by $\lambda = \frac{\ln 2}{t_{1/2}}$
    • $t$ elapsed time
  • Solving problems related to half-life:
    1. Identify initial quantity ($N_0$) and remaining quantity ($N(t)$) of the radioactive isotope
    2. Determine half-life ($t_{1/2}$) of the isotope (carbon-14 has a half-life of 5,730 years)
    3. Use decay equation or concept of half-life to calculate elapsed time or remaining quantity (if half the original amount of a radioactive isotope remains after 1,000 years, the half-life is 1,000 years)

Decay Chains and Equilibrium

  • Some radioactive isotopes decay through a series of steps called a decay chain
  • Each step in the chain involves the decay of one isotope into another
  • Radioactive equilibrium occurs when the rate of production of a daughter isotope equals the rate of its decay
  • Radiogenic isotopes are the stable end products of radioactive decay chains

Carbon-14 Dating

Carbon-14 dating process and limitations

  • Carbon-14 ($^{14}C$) radioactive isotope of carbon produced in the upper atmosphere by cosmic ray bombardment
  • Incorporated into living organisms through the carbon cycle (plants absorb $^{14}C$ during photosynthesis, animals consume plants)
  • Ratio of $^{14}C$ to stable carbon isotopes remains constant in living organisms (1 part per trillion)
  • When an organism dies, it stops exchanging carbon with the environment
    • Amount of $^{14}C$ in the organism begins to decrease through radioactive decay (half-life of 5,730 years)
  • Carbon-14 dating measures remaining $^{14}C$ in an organic sample to determine time elapsed since the organism died (wood, charcoal, bone)
  • Limitations of carbon-14 dating:
    • Effective dating range limited to about 50,000-60,000 years due to relatively short half-life of $^{14}C$ (cannot date dinosaur fossils)
    • Accuracy affected by contamination or isotopic fractionation (older carbon introduced, different rates of $^{14}C$ uptake)
    • Assumes constant atmospheric $^{14}C$ to $^{12}C$ ratio, which may vary due to factors like solar activity or fossil fuel emissions (industrial revolution, nuclear weapons testing)

Key Terms to Review (31)

Curie: The curie (symbol Ci) is a unit of radioactivity, named after the renowned physicist and chemist Marie Curie. It is used to measure the intensity of radioactive sources and is particularly relevant in the contexts of half-life and radiometric dating.
Alpha Decay: Alpha decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle, consisting of two protons and two neutrons. This process results in the transformation of the parent nucleus into a daughter nucleus with a lower atomic number.
Ernest Rutherford: Ernest Rutherford was a pioneering New Zealand-born physicist who made significant contributions to the understanding of radioactivity and atomic structure. His groundbreaking work laid the foundation for modern nuclear physics and had a profound impact on the fields of half-life and radiometric dating.
Parent Isotope: A parent isotope is the original, unstable form of a radioactive element that undergoes radioactive decay to produce a more stable daughter isotope. The parent isotope is the starting point for the radioactive decay process.
Radiogenic Isotope: A radiogenic isotope is a stable or unstable isotope of an element that is produced through the radioactive decay of a parent isotope. These isotopes are commonly used in radiometric dating techniques to determine the age of geological and archaeological samples.
Radiometric Dating: Radiometric dating is a technique used to determine the age of rocks, minerals, and other geological materials by measuring the radioactive decay of their constituent elements. It is a fundamental tool in the study of the Earth's history and the evolution of life on our planet.
Potassium-Argon Dating: Potassium-Argon dating is a radiometric dating method used to determine the age of rocks and minerals by measuring the decay of radioactive potassium-40 (40K) into argon-40 (40Ar). This technique is particularly useful for dating igneous and metamorphic rocks that are millions to billions of years old.
Uranium-238: Uranium-238 is a naturally occurring isotope of the radioactive element uranium. It is the most abundant isotope of uranium found in the Earth's crust and is the primary isotope used in nuclear power generation and nuclear weapons.
Decay Chain: A decay chain, also known as a radioactive decay series, is a sequence of radioactive decays in which an unstable atomic nucleus undergoes a series of transformations, emitting radiation in the form of alpha or beta particles, until it reaches a stable configuration. This process is crucial in the understanding of half-life and radiometric dating.
Potassium-40: Potassium-40 is a radioactive isotope of the element potassium that is found naturally in the environment. It is an important radioisotope used in the context of half-life and radiometric dating, two key topics in understanding the age and composition of the Earth and the universe.
Isotopic Fractionation: Isotopic fractionation refers to the process by which the relative abundance of different isotopes of an element is altered during physical, chemical, or biological processes. This phenomenon is particularly relevant in the context of radioactive decay and radiometric dating, as it can influence the measured isotopic ratios used to determine the age of geological and archaeological samples.
Radiocarbon Dating: Radiocarbon dating is a scientific method used to determine the age of organic materials by measuring the amount of radioactive carbon-14 they contain. It is a widely used technique in archaeology, geology, and other scientific fields to date materials and events from the past.
Radioactive Isotope: A radioactive isotope, also known as a radioisotope, is an unstable variant of a chemical element that undergoes radioactive decay, emitting excess energy in the form of radiation. These isotopes are of great importance in the study of nuclear physics, nuclear chemistry, and various applications, including medical imaging, cancer treatment, and radiometric dating.
Mass Spectrometer: A mass spectrometer is an analytical instrument that measures the mass-to-charge ratio of charged particles. It is widely used in various fields, including physics, chemistry, and biology, to identify and quantify the composition of samples by determining the masses of their constituent atoms or molecules.
Rubidium-Strontium Dating: Rubidium-strontium dating is a radiometric dating method that uses the radioactive decay of rubidium-87 (87Rb) to strontium-87 (87Sr) to determine the age of rocks and minerals. This technique is particularly useful for dating igneous and metamorphic rocks, as well as some sedimentary rocks.
Daughter Isotope: A daughter isotope is a radioactive isotope that is produced as the result of the radioactive decay of a parent isotope. This term is particularly relevant in the context of understanding half-life and radiometric dating, as the formation of daughter isotopes is a key aspect of these concepts.
Radioactive Equilibrium: Radioactive equilibrium is a state in which the rate of radioactive decay of a radioactive isotope is balanced by the rate of formation of that isotope, resulting in a constant activity level over time. This concept is crucial in understanding the principles of half-life and radiometric dating.
Precambrian: The Precambrian is the earliest geological eon, spanning from the formation of the Earth around 4.6 billion years ago to the beginning of the Cambrian period approximately 541 million years ago. This vast period of time encompasses the origins of the planet, the emergence of the first lifeforms, and the gradual development of more complex organisms.
Willard Libby: Willard Libby was an American physical chemist who developed the radiocarbon dating technique, which revolutionized the field of archaeological and geological dating. His groundbreaking work on the application of radioactive carbon-14 to determine the age of organic materials has had a profound impact on our understanding of the Earth's history and the evolution of life.
Uranium-235: Uranium-235 is a fissile isotope of the element uranium that is the primary fuel used in nuclear reactors and nuclear weapons. It is characterized by its ability to undergo nuclear fission, a process in which the nucleus of the atom splits, releasing a large amount of energy in the form of heat and radiation.
Becquerel: The becquerel (Bq) is the International System of Units (SI) derived unit of radioactivity, representing one nuclear decay or transformation per second. It is used to quantify the amount of radioactive material present in a sample or environment, providing a measure of the rate of radioactive decay.
Sievert: The sievert (Sv) is the unit used to measure the effective dose of ionizing radiation received by an individual. It takes into account the type of radiation and the sensitivity of the exposed tissues, providing a measure of the biological impact of radiation exposure on the human body.
Isochron Dating: Isochron dating is a radiometric dating method used to determine the age of geological samples by analyzing the relative abundances of specific isotopes within the sample. It is a powerful technique that can provide accurate age estimates for a wide range of materials, including rocks, minerals, and even meteorites.
Beta Decay: Beta decay is a type of radioactive decay where an atomic nucleus emits an electron (or a positron) and an antineutrino (or a neutrino) to transform one type of nucleon into another, changing the atomic number while keeping the mass number constant. This process is a fundamental part of understanding radioactive dating and half-life.
Gamma Decay: Gamma decay is a type of radioactive decay in which an atomic nucleus emits a high-energy electromagnetic radiation called a gamma ray. This process occurs when an excited nucleus transitions to a lower energy state, releasing the excess energy in the form of a gamma photon.
Carbon-14: Carbon-14 is a radioactive isotope of carbon with a relatively long half-life, making it a valuable tool for radiometric dating of organic materials. It is constantly being formed in the upper atmosphere and is incorporated into living organisms through photosynthesis and the food chain, allowing it to be used to determine the age of once-living materials.
Paleozoic: The Paleozoic Era is the earliest of the three major geologic eras, spanning from approximately 541 to 252 million years ago. This era is characterized by the emergence and diversification of complex life forms, including the first vertebrates, arthropods, and plants with vascular tissues.
Half-Life: Half-life is the time it takes for a radioactive substance to decay to half of its original amount. It is a fundamental concept in the study of radioactivity and is crucial for understanding radiometric dating and medical applications of radioactivity.
Exponential Decay: Exponential decay is a mathematical model that describes the gradual reduction of a quantity over time. It is a fundamental concept in various scientific fields, including physics, chemistry, and biology, and is particularly relevant in the context of half-life and radiometric dating.
Decay Constant: The decay constant, denoted by the Greek letter lambda (λ), is a fundamental parameter that describes the rate of radioactive decay. It represents the probability of a radioactive nucleus undergoing decay in a given time interval and is a crucial concept in understanding both half-life and radiometric dating.
Uranium-Lead Dating: Uranium-lead dating is a radiometric dating method that uses the radioactive decay of uranium isotopes to determine the age of rocks and minerals. It is a widely used technique in geochronology to estimate the absolute age of various geological formations and events.