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:
- Identify initial quantity ($N_0$) and remaining quantity ($N(t)$) of the radioactive isotope
- Determine half-life ($t_{1/2}$) of the isotope (carbon-14 has a half-life of 5,730 years)
- 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)