Radioactive Decay Types and Radiocarbon Dating

Radioactive Decay Types

Alpha Decay

  • An element loses the equivalent of a helium atom (He^{2+}), which consists of two protons and two neutrons.
  • Equation Balancing: The numbers on the right side of the equation must add up to the numbers on the left side.
    • The bottom number represents the number of protons and must be the same on both sides.
    • The top number represents the total number of protons and neutrons, which must also balance on both sides.

Beta Decay

  • A neutron is converted into a proton and an electron.

Neutron Conversion

  • A neutron (n) deteriorates into a proton (p) and an electron (e^-).
  • Neutron: 0 protons, 1 neutron.
  • Proton: 1 proton.
  • Electron: Represented as -1 (to balance the charge).
  • Equation: n \rightarrow p + e^-

Carbon-14 Dating

  • Beta decay is crucial for carbon-14 dating, an important isotropic dating process.
  • Carbon-14 is present in almost all living organisms and many other materials.

Carbon-14 Decay

  • Carbon-14 (^{14}C) decays by giving off an electron (beta - decay).
  • Carbon has 6 protons. Isotopes include carbon-12 (6 neutrons), carbon-13 (7 neutrons), and carbon-14 (8 neutrons).
  • Equation: ^{14}C \rightarrow ^{14}N + e^-
    • Carbon-14 (6 protons, 8 neutrons) becomes Nitrogen-14 (7 protons, 7 neutrons) plus an electron.

Positron Emission (Beta Positive)

  • A relatively rare decomposition where a proton deteriorates into a neutron plus a positive electron (positron).
  • Equation: p \rightarrow n + e^+
    • A proton becomes a neutron plus a positron.

Example: Copper-64 Decay

  • Copper (Cu) with 29 protons decays into Nickel (Ni) after emitting a positron.
  • Equation: ^{64}Cu \rightarrow ^{64}Ni + e^+
    • Copper-64 (29 protons) becomes Nickel (28 protons) plus a positron.

Electron Capture

  • An electron interacts with a proton to produce a neutron.
  • Equation: e^- + p \rightarrow n

Example: Potassium-40 Decay

  • Potassium-40 (^{40}K) captures an electron and transforms into Argon (^{40}Ar).
  • Potassium is important in rocks, fossils, and human bodies.
  • Equation: ^{40}K + e^- \rightarrow ^{40}Ar
    • Potassium-40 (19 protons) captures an electron to become Argon (18 protons).

Decay Probabilities

  • Many compounds, especially larger ones, undergo multiple decay processes.
  • Decay is generally not influenced by the environment; it is a probabilistic process.
  • Example: Potassium-40 decays via beta emission and positive emission with specific percentages (e.g., 80% beta, 20% positive).

Gamma Emission

  • An unstable compound releases gamma radiation to stabilize.

Decomposition

  • Highly unstable elements decompose into other metals and gases while emitting neutrons.
  • Example: A heavy, complex element deteriorates into Xenon gas and Ruthenium metal.

Example: Rubidium-87 Beta Decay

  • Rubidium-87 decays by emitting a beta particle.
  • Equation: ^{87}Rb \rightarrow ^{87}Sr + e^-
    • Rubidium-87 (37 protons, 50 neutrons) becomes Strontium-87 (38 protons) plus an electron.

Radiocarbon Dating

Carbon Isotopes

  • Carbon has three naturally occurring isotopes:
    • Carbon-12 (^{12}C): 99% (6 protons, 6 neutrons).
    • Carbon-13 (^{13}C): ~1% (6 protons, 7 neutrons).
    • Carbon-14 (^{14}C): Trace amounts (6 protons, 8 neutrons).

Production of Carbon-14

Natural Process

  • Cosmic rays interact with atoms in the atmosphere, generating neutrons.
  • Neutrons interact with Nitrogen-14 (^{14}N).
  • Equation: ^{14}N + n \rightarrow ^{14}C + H
    • Nitrogen-14 captures a neutron to produce Carbon-14 and a hydrogen atom.

Artificial Process

  • Nuclear reactions from nuclear bombs and power stations also produce Carbon-14.
  • Nuclear weapon testing in the 1960s significantly increased Carbon-14 levels.

Environmental Incorporation

  • Carbon-14 oxidizes into carbon monoxide (CO), then carbon dioxide (CO_2).
  • Carbon dioxide is absorbed by plants and animals, entering the ecosystem.

Radiocarbon Dating

  • Carbon-14 has a half-life of approximately 5,730 years.
  • Willard Libby won the Nobel Prize for pioneering radiocarbon dating.
  • The dating limit is around 55,000 years (approximately 10 half-lives).

Decay Process

  • Living organisms continuously absorb Carbon-14.
  • Upon death, the absorption stops, and Carbon-14 decays back to Nitrogen-14 via beta decay.

Half-Life

Definition

  • The time required for half of the radioactive nuclei to decay.

Carbon-14 Half-Life

  • After 5,730 years, half of the Carbon-14 in a sample will have decayed.
  • After 11,460 years, a quarter of the original Carbon-14 will remain.

Exponential Decay

  • The decay of Carbon-14 follows an exponential curve.
  • The number of Carbon-14 nuclei (N) at time (t) is given by: N(t) = N0 e^{-\lambda t}, where N0 is the initial amount and \lambda is the decay constant.

Logarithmic Scale

  • Using a logarithmic scale transforms the exponential decay curve into a linear curve, simplifying analysis.