Problem 20 - College Physics

Question

The half-life of a radioactive isotope is known to be exactly \(1 \mathrm{~h}\). (a) What fraction of a sample would be left after exactly \(3 \mathrm{~h}:(1)\) one-third, (2) one-eighth, or (3) oneninth? (b) What fraction of a sample would be left after exactly 1 day?

Step-by-Step Solution

Answer

(a) One-eighth; (b) Nearly zero.

Step 1: Understanding Half-life Concept

The half-life of a substance is the time required for half of the initial amount of the substance to decay. For this isotope, the half-life is 1 hour, meaning every hour, the amount of the isotope is reduced by half.

Step 2: Calculating Fraction Left After 3 Hours

Since the half-life is 1 hour, after the first hour, half of the sample remains. After the second hour, half of the remaining half decays, leaving one-quarter. After the third hour, half of the remaining one-quarter decays, leaving one-eighth of the original sample.

Step 3: Fraction Calculation Formula

We can use the formula \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/h} \), where \( N(t) \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the time elapsed, and \( h \) is the half-life. For part (a), \( t = 3 \) hours, \( h = 1 \) hour, hence \( N(t) = N_0 \times \left( \frac{1}{2} \right)^3 = N_0 \times \frac{1}{8} \). This confirms that one-eighth of the sample remains, matching option (2).

Step 4: Calculating Fraction Left After 1 Day

One day is 24 hours. Using the same formula \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/h} \), with \( t = 24 \) hours, \( h = 1 \) hour, we have \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{24} \). Evaluating \( \left( \frac{1}{2} \right)^{24} \) results in a very small fraction, approximately \( 5.96 \times 10^{-8} \), which is practically zero.

Key Concepts

Half-Life

Half-life is one of the central concepts in understanding radioactive decay. It refers to the time it takes for half of the atoms in a radioactive substance to decay into another substance. This is a constant value for a given isotope, which means that no matter how much of the substance you have, half of it will decay over the length of one half-life.

This idea is crucial in the calculation of how much of a sample of a radioactive isotope remains after a certain period of time. For the isotope in the exercise, the half-life is specifically 1 hour. Every hour, half of the current amount of the substance decreases. This continues until the substance is essentially decayed to nothing. Understanding half-life allows us to predict how long a radioactive element will remain active and how quickly it will decline in mass.

Exponential Decay

Radioactive decay is a natural exponential decay process, which is a type of decrease where the rate of decay is directly proportional to the amount present. In simpler terms, as time passes, decay happens rapidly at first and slows down over time because there’s less of the substance left to decay.

Exponential decay can be visually represented as a curve that steeply falls and gradually levels out on a graph over time. This behavior is due to the nature of radioactive particles decaying at a rate proportional to their number. Thus, the rate decreases as time progresses, leading to lesser quantities of the isotope at later stages. The half-life is an integral part of understanding this process since it defines the time interval during which the substance's amount halves.

Isotope

An isotope is a variant of a particular chemical element that has the same number of protons but a different number of neutrons in its nucleus. This difference in neutrons means isotopes of an element have different mass numbers, and some isotopes may be unstable or radioactive.

The stability of an isotope determines its radioactivity. In the context of the exercise, the radioactive isotope has a precise half-life of 1 hour, indicating it’s quite unstable and decays rapidly. Understanding isotopes is essential not only in nuclear chemistry and physics but also in applications such as nuclear medicine, radiocarbon dating, and nuclear power generation.

• Isotopes have the same atomic number (protons) but different mass numbers (isotopes).
• Radioactive isotopes decay over time, releasing particles in the process.
• They can be used in various scientific and industrial applications.

Decay Formula

The decay formula is a mathematical expression used to calculate the remaining amount of a radioactive substance after a certain period of time. It is derived from the concept of exponential decay and incorporates the half-life of the substance into its calculation.

The decay formula is expressed as:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/h} \]Here:

• \(N(t)\) represents the remaining quantity of the substance after time \(t\).
• \(N_0\) is the initial quantity at time zero.
• \(t\) is the elapsed time.
• \(h\) is the half-life of the isotope.

Applying this formula allows us to understand and predict the decay of radioactive materials over time. In the exercise, this formula helps calculate that only one-eighth of the original sample remains after 3 hours, as \(\left( \frac{1}{2} \right)^3 = \frac{1}{8}\), and an almost negligible amount remains after one full day due to the exponential nature of decay.