Radioisotopes and Nuclear Reactions

Radioisotope Decay and Half-Life

• Radioisotopes undergo decay, a process where their radioactivity decreases.
• Each radioisotope has a unique half-life (symbol: $T$) which is the time required for half of the radioactive atoms in a sample to decay, or for its radioactivity to decrease by half.
• Half-lives vary significantly, from milliseconds to billions of years.

Half-Life Calculation

• After a certain number of half-lives ($n$), the remaining activity ($A<em>{rem}$) or mass is a fraction of the original activity ($A</em>{orig}$) or mass.
• The general equation for remaining activity is: $A<em>{rem} = A</em>{orig} \cdot (\frac{1}{2})^n$
  • $n$ represents the number of half-lives that have passed ($n = \frac{\text{total time}}{\text{half-life}}$).
• This equation can be used to solve for $A<em>{rem}$, $A</em>{orig}$, or $n$ if the other two values are known.
• The mass of the radioisotope in a sample also decreases proportionally to its radioactivity over time.

Carbon-14 Dating

• Principle: The percentage of carbon-14 ($^{\text{14}}\text{C}$) compared to total carbon in the atmosphere (and thus in living organisms) is assumed to be constant.
• When an organism dies, it stops exchanging carbon with the environment, and its $^{\text{14}}\text{C}$ begins to decay.
• By measuring the remaining $^{\text{14}}\text{C}$ in a fossil or organic sample and comparing it to the environmental level, scientists can estimate how many half-lives have passed and thus determine the sample's age.
• The half-life of $^{\text{14}}\text{C}$ is approximately $5730$ years.
• Limitation: Only applicable to organic (carbon-containing) samples.
• Potential Error: If environmental $^{\text{14}}\text{C}$ levels were unusually high at the time of an organism's death, carbon dating could underestimate its age.

Medical Applications of Radioactivity

• Food Preservation: Radiation kills viruses, bacteria, and pests in fruits and vegetables, extending their shelf life.
• Radiotherapy: Uses concentrated radiation (e.g., from Cobalt-60 gamma sources) to kill tumor cells in cancer treatment.
• MRI (Magnetic Resonance Imaging): Uses ingested positron-emitting radioisotopes to create detailed images of internal body structures and functions (though MRI itself primarily uses strong magnetic fields and radio waves, the transcript describes a method using radioisotopes).
• Urea Breath Test: Uses $^{\text{14}}\text{C}$-labeled urea to detect Helicobacter pylori bacteria in the stomach, which metabolize urea into $^{\text{14}}\text{C}$-labeled carbon dioxide.
• Radioactive Capsules: Small capsules containing radioisotopes (e.g., Iodine-131) can be implanted near localized tumors to deliver targeted radiation.

Balancing Nuclear Reactions

• In a nuclear reaction, both the total atomic number (sum of protons, Z) and the total mass number (sum of protons and neutrons, A) must be conserved before and after the reaction.
• Gamma emission ($^{0}_{0}\gamma$): No change in Z or A, pure energy release.
• Beta decay ($^{0}_{-1}e$): The atomic number (Z) of the parent nucleus increases by one, while the mass number (A) remains the same.