The half-life for the radioactive decay of C-14 is 5730 years and is independent of the initial concentration. How long does it take for 25% of the C-14 atoms in a sample of C-14 to decay? If a sample of C-14 initially contains 1.5 mmol of C-14, how many millimoles are left after 2255 years?

Understanding the half-life of a radioactive isotope like Carbon-14 (C-14) is crucial in determining how its concentration changes over time. The half-life is the time required for exactly half of the entities to decay, reducing the original quantity to 50%. It's a constant figure for each radioactive isotope and for C-14, it stands at 5730 years.

For example, if we start with a 100% concentration of C-14, after one half-life (5730 years), only 50% remains. This concept allows us to predict how much of a substance will remain after any given period. If you were asked how long it would take for 25% of a C-14 sample to decay, which means you're left with 75% of the original, you could use this concept to deduce that it would take exactly one half-life. That's because going from 100% to 50% takes one half-life, and as radioactive decay is a logarithmic process, the next 50% to decay (leaving 25% of the original) also takes one half-life.

To express the decay of radioactive isotopes mathematically, we use the radioactive decay formula: \[ N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \]Where:

• \( N \) represents the remaining quantity of the isotope after time \( t \).
• \( N_0 \) is the initial quantity of the isotope.
• \( t \) is the time elapsed.
• \( T \) is the half-life of the isotope.

This exponential formula reflects the logarithmic nature of radioactive decay. For instance, to determine how much C-14 remains after 2255 years, you would plug the values into the formula and solve:\[ N = 1.5 \times \left(\frac{1}{2}\right)^{\frac{2255}{5730}} \]First, calculate the exponent \( \frac{2255}{5730} \), then use this result to determine the power of \( \frac{1}{2} \) that represents the proportion of C-14 remaining after 2255 years. After evaluating the power, you'd multiply it by the initial amount to find the final quantity of C-14 left.

Carbon-14 dating, also known as radiocarbon dating, is a method used to determine the age of an object containing organic material by measuring the levels of C-14 it contains. Because living organisms continuously exchange carbon with their environment, including a fixed ratio of C-14, the proportion of C-14 in a sample reflects the time elapsed since the organism ceased exchanging carbon (usually its death).

For instance, archaeologists implement carbon-14 dating to determine the age of ancient artifacts or fossils. By measuring how much C-14 has decayed in the sample and using the known half-life of C-14, scientists can calculate how many half-lives have passed since the death of the organism. This leads to an age estimate for the sample. The half-life is the cornerstone in these calculations, making the understanding of its calculation, along with the decay formula, essential for accurate carbon dating.