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Chapter 18: Problem 39

Starting with \(1 \mathrm{~g}\) of a radioactive isotope whose half-life is 10 days, sketch a graph showing the pattern of decay for that material. On the \(x\)-axis, plot time (you may want to simply show multiples of the half-life), and on the \(y\)-axis, plot mass of material remaining. Then after completing the graph, explain why a sample never really gets to the point where all of its radioactivity is considered to be gone.

Short Answer

Expert verified
The isotope's mass halves every 10 days, generating points for a decaying graph. Complete decay never occurs because the mass continually halves, approaching but never reaching zero.

Step by step solution

01

- Understand the Decay Process

Radioactive decay means that the quantity of the isotope decreases by half every half-life period. Here, the half-life is 10 days.
02

- Define the Initial Conditions

Begin with 1 gram of the radioactive isotope.
03

- Write the Decay Formula

The mass of the remaining isotope can be expressed as a function of time using the formula: \[ M(t) = M_0 \times \frac{1}{2}^{(t/T_{1/2})} \] where \( M_0 \) is the initial mass, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life.
04

- Calculate Remaining Mass for Multiple Half-Lives

Calculate the remaining mass at each multiple of the half-life (10 days, 20 days, 30 days, etc.). For example: \( t = 10 \) days: \( 1 \text{ g} \times \frac{1}{2}^{1} = 0.5 \text{ g} \)\( t = 20 \) days: \( 1 \text{ g} \times \frac{1}{2}^{2} = 0.25 \text{ g} \)
05

- Sketch the Graph

On the x-axis, plot time in multiples of 10 days (0, 10, 20, 30, 40, etc.). On the y-axis, plot the remaining mass using the calculated values.
06

- Explain Why Complete Decay Never Occurs

As time progresses, the mass of the radioactive sample continuously halves, theoretically never reaching zero. Because the mass is always halving, it gets increasingly closer to zero but never fully disappears.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

half-life
The concept of 'half-life' is crucial in understanding radioactive decay. A half-life is the time required for a quantity to reduce to half its initial value. For example, if you start with 1 gram of a radioactive isotope with a half-life of 10 days, after 10 days, only 0.5 grams will remain. After another 10 days, it will halve again to 0.25 grams. This process continues indefinitely.
exponential decay
Radioactive decay follows an 'exponential decay' pattern. This means that the quantity of the radioactive isotope decreases exponentially over time. This can be mathematically represented using the formula:
\(M(t) = M_0 \times \frac{1}{2}^{(t/T_{1/2})}\)
where
\(M_0\) is the initial mass,
\(t\) is the time elapsed, and
\(T_{1/2}\) is the half-life.
One of the features of exponential decay is that the remaining amount never actually reaches zero. It keeps reducing by half, becoming an increasingly smaller fraction over time.
radioisotopes
Radioisotopes are isotopes of elements that exhibit radioactivity. They spontaneously emit radiation to reach a more stable state. The decay process of radioisotopes can be used in various applications such as medicine, archaeology (carbon dating), and energy production. Understanding how radioisotopes decay over time is crucial for using them effectively and safely.
mass calculation
Performing 'mass calculation' for radioactive decay involves using the exponential decay formula to determine how much of a substance remains after a given time period. For example, if you start with 1 gram of a radioactive substance with a half-life of 10 days:
  • After 10 days, 0.5 grams remain.
  • After 20 days, 0.25 grams remain.
  • After 30 days, 0.125 grams remain.
This calculation is essential for sketching the decay graph and understanding the long-term behavior of the radioactive substance. By plotting these values, you'll see the exponential decay curve that approaches zero but never actually reaches it.

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