Problem 4 - Calculus Early Transcendentals

Question

Explain the meaning of half-life.

Step-by-Step Solution

Answer

Answer: Half-life is the time it takes for half of the radioactive atoms in a sample to undergo radioactive decay, transforming into different elements called decay products or daughter isotopes. This concept is crucial for various scientific fields, such as nuclear energy, geology, and carbon dating. In nuclear energy, knowledge of half-life helps manage nuclear waste and protect people from radiation exposure. In geology, the half-life of isotopes is used to determine the age of rocks and the Earth. In carbon dating, the half-life of Carbon-14 is used to estimate the age of artifacts or fossils containing organic materials, providing valuable information about the past.

Step 1: Definition of Half-Life

Half-life is the time required for a quantity of a radioactive substance to decay and reduce to half of its initial amount. In other words, it is the time it takes for half of the radioactive atoms in a sample to undergo radioactive decay.

Step 2: Radioactive Decay

Radioactive decay is a random process, and it happens when an unstable atomic nucleus loses energy by emitting ionizing radiation. This can be in the form of alpha particles, beta particles, or gamma photons. As the radioactive atoms decay, they change into atoms of a different element, called decay products or daughter isotopes.

Step 3: Examples of Half-life

Let's look at a well-known example, the radioactive isotope Carbon-14. Carbon-14 has a half-life of approximately 5,730 years. This means that after 5,730 years, half of the initial amount of Carbon-14 in a sample will have decayed into other elements, and half will still be Carbon-14. After another 5,730 years (a total of 11,460 years), half of the remaining Carbon-14 will decay, leaving only 25% of the initial amount. Another example is the radioactive element Uranium-238, which has a half-life of about 4.5 billion years. It means that if you have a sample containing 100 grams of Uranium-238 today, after 4.5 billion years there will be 50 grams of Uranium-238 remaining, and the other 50 grams will have decayed into its daughter isotopes.

Step 4: Application of Half-Life Knowledge

Understanding half-life is crucial for various scientific fields, such as nuclear energy, geology, and carbon dating. In nuclear energy, knowing the half-life of radioactive materials is essential to safely manage nuclear waste and protect people from radiation exposure. In geology, the half-life of isotopes is used to determine the age of rocks and the Earth itself. In carbon dating, the half-life of Carbon-14 is utilized to estimate the age of artifacts or fossils containing organic materials, such as wooden objects or mummified remains, providing valuable information about the past.