Question

$$ \begin{aligned} &\text { Use the table below to answer questions } 18-20 .\\\ &\begin{array}{|c|c|} \hline \text { Number of Half-lives } & \text { Parent Isotope Remaining (\%) } \\ \hline 1 & 100 \\ \hline 2 & X \\ \hline 3 & 25 \\ \hline 4 & 12.5 \\ \hline 5 & \mathrm{Y} \\ \hline \end{array} \end{aligned} $$ Explain the relationship between the number of half-lives that have elapsed and the amount of parent isotope remaining.

Short answer

Each half-life results in the parent isotope being reduced by 50%.

Step-by-step solution

Understanding Half-Lives

A half-life is the time required for half of the parent isotopes in a sample to decay. Each half-life results in a 50% reduction of the remaining parent isotopes.

Analyzing Initial Conditions

Initially, at 0 half-lives, 100% of the parent isotope is present. Each subsequent half-life reduces the remaining percentage by half.

Calculating Remaining Isotopes for Two Half-Lives

For one half-life, 50% of the isotope remains (half of the initial 100%). For two half-lives: \[ ext{Remaining Isotope} = 50\% imes 0.5 = 25\% \]This means X must be 25%.

Confirming Remaining Isotopes for Subsequent Half-Lives

For three half-lives, we are given the remaining amount is 25%, which is consistent with the calculated result in Step 3. For four half-lives, 25% reduces by half to:\[ 25\% imes 0.5 = 12.5\% \]This matches the given table value.

Calculating and Validating for Five Half-Lives

For five half-lives:\[ 12.5\% imes 0.5 = 6.25\% \]Therefore, Y must be 6.25%, which is the consistent reduction by half from the previous amount.

Key concepts

Radioactive decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation. During this process, a parent isotope transform into a more stable form, known as the daughter isotope. The rate at which this transformation takes place is measured in half-lives, each characterized by a 50% reduction in the amount of the parent isotope. The decay process is random but governed by probabilities, which makes it predictable over a large number of atoms.

• A single decay event is unpredictable, but collective behaviors are governed by statistical rules.
• The energy released can take several forms like particles (alpha, beta) or electromagnetic radiation (gamma rays).
• Each decay type has different implications for the remaining isotopes.

Understanding this process is vital to fields like archeology, medicine, and nuclear physics as it allows for dating artifacts, diagnosing illnesses, or generating energy.

Parent isotope

The parent isotope is the original unstable isotope that undergoes a transformation during radioactive decay. It is this isotope that begins the decay process, starting the sequence of energy release and transformation.
The amount of this parent isotope decreases over time as it decays into a more stable form. This decrease follows an exponential pattern, as observed in half-life calculations.

Key aspects of the parent isotope include:

• Initial Composition: 100% at the beginning of the observation.
• Transformation: Converts into a daughter isotope over time.
• Measurement: The amount remaining is often expressed as a percentage of the original quantity.

Tracking the decay of the parent isotope allows scientists to date samples and understand the age of geological and archaeological samples.

Isotope remaining

Isotope remaining refers to the portion of the parent isotope that still exists after a certain number of half-lives have elapsed. The remaining amount of isotope is crucial for determining how far a sample has progressed through its decay process.

Each half-life reduces the remaining isotope by 50%, leading to a predictable pattern:

• After 1 half-life: 50% of the parent isotope remains.
• After 2 half-lives: 25% remains.
• After 3 half-lives: 12.5% remains, and so forth.

Understanding the amount of isotope remaining is important for dating methods, medical decay treatments, and nuclear reactions as it indicates the usable quantity of a substance at any given time.

Decay process

The decay process is a central aspect of radioactive decay, which describes the transformation of a parent isotope into its daughter isotopes over time. It provides a roadmap of changes as isotopes lose stability and convert into new forms. Each cycle of decay contributes to an exponentially decreasing amount of the original isotope.
Here’s what happens during the decay process:

• Energy Emission: Radiation is emitted, causing changes in the atomic structure.
• Transformation: The isotope moves toward a more stable state.
• Elapsed Time: The number of half-lives indicates the stage of the decay process.

This understanding helps in calculating the half-life accurately, predicting the composition of materials in the future, and managing resources effectively in applications such as nuclear power and waste management.