Problem 30 - Fundamentals of Physics

Question

The half-life of a particular radioactive isotope is \(6.5 \mathrm{~h}\). If there are initially \(48 \times 10^{19}\) atoms of this isotope, how many remain at the end of \(26 \mathrm{~h}\) ?

Step-by-Step Solution

Answer

At the end of 26 hours, \(3 \times 10^{19}\) atoms remain.

Step 1: Understanding Half-life Concept

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms present to decay. In this problem, the isotope has a half-life of 6.5 hours. This means every 6.5 hours, the quantity of the isotope is reduced to half.

Step 2: Determine Number of Half-lives

To find out how many half-lives have passed in 26 hours, divide the total time duration by the half-life. So, we calculate \( \frac{26}{6.5} = 4 \). Thus, 4 half-lives have elapsed.

Step 3: Calculate Remaining Atoms After 4 Half-lives

After each half-life, the number of remaining atoms is halved. Starting with an initial amount of \(48 \times 10^{19}\) atoms, after the first half-life, \(24 \times 10^{19}\) remain; after the second half-life, \(12 \times 10^{19}\); after the third half-life, \(6 \times 10^{19}\); and after the fourth half-life, \(3 \times 10^{19}\) atoms remain.

Key Concepts

Half-life

The concept of half-life is central to understanding radioactive decay. Half-life refers to the amount of time it takes for half of a sample of a radioactive isotope to lose its radioactivity through decay. For example, if a radioactive isotope has a half-life of 6.5 hours, this means that in every 6.5-hour interval, half of the original radioactive atoms will have decayed into a more stable form.

Biological organisms and environmental processes also face effects from the half-life of isotopes, as it influences how long these substances remain active or detectable.

• The half-life is a constant for any given isotope and does not change over time.
• This concept helps predict how quickly a radioactive material will decay over time.
• Understanding half-life is crucial in fields like nuclear medicine, dating archaeological finds, and assessing nuclear waste management.

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are variants of chemical elements that have unstable nuclei. These nuclei can decay over time, resulting in the emission of radiation.

• Radioisotopes have different half-lives, ranging from fractions of a second to thousands of years.
• They are employed in various applications due to their radioactive nature, which can be harnessed for medical diagnostics, treatments, and scientific research.
• The natural occurrence of these isotopes also helps geologists understand geological and planetary processes through methods like radioactive dating.

It is important to handle radioisotopes with care due to the potential hazards posed by radiation. Specialized equipment and protocols are essential to contain and manage these materials safely. With this understanding, scientists and industries use radioisotopes beneficially and efficiently.

Decay Calculation

To calculate the decay of a radioactive isotope, you must determine how many half-lives have elapsed over a given period and then apply this information to find the remaining quantity of the isotope.

• First, divide the total time period by the isotope's half-life to find the number of half-lives that have passed. For example, over 26 hours with a 6.5-hour half-life, 4 half-lives will occur.
• With each half-life period, the amount of isotope reduces to half of its existing quantity. If you start with 48 \times 10^{19} atoms, after 4 half-lives only \(3 \times 10^{19}\) atoms will remain.
• This calculation helps predict how much of a radioactive isotope will remain un-decayed after a certain time, which is vital for safety assessments and applications.

By using these decay calculations, scientists can estimate the longevity and effectiveness of isotopes for their intended purposes, from medical treatments to carbon dating.