Problem 41 - Physics for Scientists and Engineers with Modern Physics

Question

(I) What fraction of a sample of \(\frac{68}{32} \mathrm{Ge},\) whose half-life is about 9 months, will remain after 2.0 yr?

Step-by-Step Solution

Answer

After 2 years, approximately 15.6% of the original sample remains.

Step 1: Understand the Concept of Half-Life

The half-life of a substance is the time it takes for half of the sample to decay. For this problem, the half-life of Germanium-68 is 9 months. We need to find out what fraction of the original sample remains after 2 years.

Step 2: Convert Years to Months

Since the half-life is given in months, we first convert the 2 years into months. There are 12 months in a year, so 2 years is equal to 2 × 12 = 24 months.

Step 3: Determine Number of Half-Lives

A single half-life is 9 months. To find out how many half-lives have passed in 24 months, divide 24 by 9. Number of half-lives = \( \frac{24}{9} \approx 2.67 \) half-lives.

Step 4: Calculate Remaining Fraction

The fraction of the sample remaining after a certain number of half-lives is given by the formula \[ \left(\frac{1}{2}\right)^n \]where \( n \) is the number of half-lives. Substitute 2.67 for \( n \).Remaining fraction = \( \left(\frac{1}{2}\right)^{2.67} \approx 0.156 \).

Key Concepts

Radioactive Decay

Radioactive decay is a natural process in which an unstable atomic nucleus loses energy by emitting radiation. This decay can result in the formation of a different element, as the number of protons or neutrons in the nucleus changes. Radioactive decay is random and spontaneous, which means we can't predict when a specific atom will decay, but we can determine the rate at which a sample of atoms will decay over time.

The process of decay involves the transformation of one element into another through

• the emission of alpha particles (two protons and two neutrons),
• beta particles (electrons or positrons),
• or gamma rays (high-energy photons).

The rate of decay is often described in terms of the element's half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding half-life is crucial for calculating how much of a substance will remain after a given period, as demonstrated with the Germanium-68 problem.

Germanium-68

Germanium-68 is a radioactive isotope of the element germanium. It is denoted as \(^{68} \text{Ge}\), where the number 68 represents its atomic mass number, comprised of protons and neutrons.

Germanium-68 is used in various scientific applications, including in medical imaging and tracer studies in nuclear medicine. The isotope decays through electron capture to form the stable isotope, gallium-68, \(^{68} \text{Ga}\), which is useful in positron emission tomography (PET) scans.

For Germanium-68, the half-life is about 9 months. This relatively long half-life allows for the decay process to be measured more easily over extended periods, making it ideal for research studies. Knowing its half-life helps scientists calculate how much of a sample remains after a specific time, which is essential for applications in radiation physics.

Radiation Physics

Radiation physics explores the phenomena, properties, and effects of radiation. It deals with understanding how radiation interacts with matter, especially in the context of radioactive decay, like that observed in Germanium-68.

Radiation comes in various forms, including alpha, beta, and gamma radiation, each with unique properties and interactions with matter:

• Alpha particles have a strong ionizing ability but are not deeply penetrating.
• Beta particles are less ionizing than alpha particles and can penetrate further into materials.
• Gamma rays are highly penetrating and can pass through most materials, requiring dense materials like lead for shielding.

In radiation physics, understanding these interactions is vital to ensuring the safe use and management of radioactive materials in various fields such as medicine, industry, and research. Proper calculations, like determining remaining substances through half-life, are essential to safely harness and apply the benefits of radioactive isotopes like Germanium-68.