Radioisotopes and Nuclear Reactions

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.

After a certain number of half-lives (n), the remaining activity (A{rem}) or mass is a fraction of the original activity (A{orig}) or mass.
The general equation for remaining activity is: A{rem} = A{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{rem}, A{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.

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.

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.

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.