Problem 30 - Physics

Question

The half-lives in two different samples, \(A\) and \(B\), of radioactive nuclei are related according to \(T_{1 / 2, \mathrm{~B}}=\frac{1}{2} T_{1 / 2, \mathrm{~A}}\) In a certain period the number of radioactive nuclei in sample A decreases to one-fourth the number present initially. In this same period the number of radioactive nuclei in sample B decreases to a fraction \(f\) of the number present initially. Find \(\bar{f}\)

Step-by-Step Solution

Answer

In the given period, the fraction \( f \) of the initial number of nuclei left in sample B is \( \frac{1}{16} \).

Step 1: Understand the Relationship between Half-lives

Given that the half-life of sample B is half of that of sample A, we can express this relationship as \( T_{1/2, B} = \frac{1}{2} T_{1/2, A} \). This means sample B decays faster than sample A.

Step 2: Determine the Decay of Sample A

Sample A's number of nuclei decreases to one-fourth of its initial value over a certain period. If \( N_0 \) is the initial quantity, then after this period, the remaining quantity is \( \frac{1}{4} N_0 \). This occurs after two half-lives because \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \).

Step 3: Calculate the Equivalent Period for Sample B

The period mentioned is equivalent to two half-lives of sample A. Because sample B's half-life is half that of sample A, sample B will go through \( \frac{T}{T_{1/2, B}} = \frac{T}{\frac{1}{2} T_{1/2, A}} = \frac{T}{1/2} \cdot \frac{1}{T_{1/2, A}} = 4 \) half-lives in the same duration.

Step 4: Determine Remaining Fraction for Sample B

Each half-life reduces a sample by half. After 4 half-lives, sample B will be reduced by \( \left( \frac{1}{2} \right)^4 \), which equals \( \frac{1}{16} \) of its initial number.

Key Concepts

Half-life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay.
It is a constant that characterizes the rate of decay of a radioactive isotope.
Understanding half-life is crucial in fields like radiometric dating, medicine, and nuclear physics. Radioactive materials decay exponentially. This means that instead of decaying at a constant rate, the rate decreases over time.
As a consequence, in every half-life period, the amount of radioactive substance is halved.
For example:

• After one half-life, you have half of the original material.
• After two half-lives, one-quarter (\( \frac{1}{4} \)) remains.
• After three half-lives, one-eighth (\( \frac{1}{8} \)) remains.

This exponential decay pattern is key to predicting changes in the amount of a radioactive substance over time.

Radioactive Nuclei

Radioactive nuclei are unstable atoms that emit radiation when they decay into more stable forms.
The process involves the transformation of a parent nucleus into a daughter nucleus plus the radiation emitted.
Common types of radiation include alpha particles, beta particles, and gamma rays. Each element has its own characteristic type and rate of decay, leading to different half-lives.
These behaviors are cataloged in nuclear decay series that help scientists understand what happens during decay events.
In the exercise, radioactive nuclei were present in two samples: A and B. When handling radioactive nuclei, understanding their decay properties helps predict the remaining nuclei over time. This prediction is crucial for applications such as:

• Medical treatments utilizing radiation therapy.
• Determining the age of archaeological finds through carbon dating.
• Managing nuclear waste by knowing how long it remains hazardous.

Correctly interpreting the decay of radioactive nuclei ensures safer and more efficient use of nuclear technologies.

Decay Rate

Decay rate refers to the speed at which radioactive nuclei transform into stable atoms.
Mathematically, it is often expressed using the decay constant (\( \lambda \)), which indicates how quickly a substance disintegrates.
The decay constant relates directly to half-life via the formula:\[T_{1/2} = \frac{\ln(2)}{\lambda}\]In the exercise, the decay rate of samples A and B varied due to different half-lives. Understanding decay rates is essential for predicting how long a specific amount of radioactive material will last.
Applications of this knowledge include:

• Ensuring correct dosages in radioactive medical therapies.
• Calculating shielding requirements for radiation protection.
• Designing nuclear reactors with accurate predictions of fuel life.

The decay rate provides a more specific understanding of how rapidly a material becomes non-radioactive, allowing for precise planning in scientific and industrial contexts.