Question

Calculate how many half-lives have passed in a rock containing one-eighth the original radioactive material and seven-eighths of the daughter product.

Short answer

Three half-lives have passed.

Step-by-step solution

Understand the half-life concept

A half-life is the time required for half of a radioactive substance to decay into its daughter product. After one half-life, you will have half of the original material remaining. After two half-lives, you will have a quarter of the original material remaining (since half of a half is a quarter). This pattern continues as follows: one-eighth after three half-lives, one-sixteenth after four half-lives, and so on.

Set up the initial condition

Recognize that the problem states that one-eighth of the original radioactive material remains. This means that we started with a full amount (1) and now have 1/8 of that original material left. This matches the amount expected after a certain number of half-lives.

Match the condition to the half-life pattern

As mentioned in Step 1, one-eighth of the original material remains after 3 half-lives. Let's confirm this by following the decay pattern: - After 1 half-life: 1/2 of the original remains. - After 2 half-lives: 1/4 of the original remains. - After 3 half-lives: 1/8 of the original remains.

Key concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This leads to the transformation of the original element into a different one. During decay, the nucleus of an atom emits particles or electromagnetic waves, resulting in the loss of atomic number or mass. This process can happen in different ways:

• Alpha decay: The nucleus sheds 2 protons and 2 neutrons, forming an alpha particle.
• Beta decay: A neutron changes into a proton, or vice versa, which results in the emission of a beta particle and a neutrino.
• Gamma decay: The nucleus loses energy by releasing gamma radiation, without changing the number of protons or neutrons.

The time it takes for half the atoms in a radioactive sample to decay is called the half-life. Over successive half-lives, the remaining amount of radioactive material reduces by half, ultimately leading to the formation of a stable element.

Daughter Product

A daughter product is what remains after a radioactive parent isotope undergoes decay. This resulting substance can further decay until a stable isotope is formed. The transition from the parent isotope to the daughter product happens at a constant rate. This reliability makes it possible to calculate timeframes for geologic events by measuring the ratios of parent and daughter isotopes in a sample.

The term 'daughter' refers to the direct result of a decay event. If a parent isotope undergoes multiple transformations, each successive element is a step closer to stability until reaching an isotope that no longer alters further through decay.

• Parent Isotope: The original unstable isotope before decay.
• Daughter Isotope: The new isotope formed after decay.
• Decay Chain: Series of decay events that transform the parent isotope into stable daughters.

Understanding the relationship between parent and daughter isotopes enables scientists to determine the age of materials and geological formations.

Material Decay

Material decay refers to the overall loss of radioactive material over time. This decay is not uniform, as it depends on the half-life of the radioactive isotope involved. After each half-life cycle, the percentage of the remaining material decreases exponentially, creating a predictable decay trend.

To comprehend material decay effectively, it's essential to grasp the exponential nature of the process. Initially, you start with a full amount. After one half-life, the quantity will reduce by half; after two half-lives, it reduces to a quarter; then one-eighth after three half-lives, and so forth. By recognizing these patterns, scientists can estimate the age of objects or samples by observing how much material remains.

• Use of exponential decay formula: Allows precise calculations of remaining amounts.
• Several half-lives can pass before significant reduction of material.
• Critical for dating ancient samples using radioactive isotopes.

Overall, the concept of material decay is pivotal in studying both archaeological finds and natural formations.

Rock Dating

Rock dating involves determining the age of a rock sample by measuring the decay of radioactive isotopes within it. This scientific process is known as radiometric dating. By examining the ratio of parent isotopes to daughter products, geologists can accurately determine the age of rocks.

Several isotopes are used in rock dating, each suitable for dating different types of rocks and timescales.

• Uranium-Lead dating: Effective for rocks over a million years old.
• Carbon-14 dating: Ideal for dating recent remains up to 50,000 years old.
• Potassium-Argon dating: Useful for volcanic rocks.

Calculating the number of half-lives that have passed provides valuable age information. For instance, if a rock contains one-eighth of its original radioactive material, it has gone through three half-lives. Thus, understanding the half-life and how it applies to different isotopes is essential for accurate rock dating and piecing together Earth's history.