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The isotope Technetium \(99 \mathrm{~m}\) is used in medical imaging. It has a half-life of about 6 hours, a useful feature for radioisotopes that are injected into humans. The Technetium, having such a short half-life, is created artificially on scene by harvesting from a more stable isotope, \({ }^{99} \mathrm{Mb}\). If \(10 \mathrm{~g}\) of \(99_{m}\) Tc are "harvested" from the Molybdenum, how much of this sample remains after 9 hours?

Short Answer

Expert verified
Approximately 3.54 grams remain after 9 hours.

Step by step solution

01

Understand the Half-life

The half-life of a substance is the time required for half of the material to undergo radioactive decay. For Technetium-99m, the half-life is 6 hours.
02

Use the Decay Formula

To determine how much of a radioactive substance remains after a certain time, we use the formula: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{t/T} \] where \( N(t) \) is the remaining quantity of the substance, \( N_0 \) is the initial quantity, \( t \) is the elapsed time, and \( T \) is the half-life of the substance.
03

Identify Values and Substitute

In this problem, the initial quantity \( N_0 = 10 \text{ grams} \), the elapsed time \( t = 9 \text{ hours} \), and the half-life \( T = 6 \text{ hours} \). Substituting these values into the formula, we get:\[ N(9) = 10 \left( \frac{1}{2} \right)^{9/6} \]
04

Calculate the Exponent

Calculate \( \frac{9}{6} = 1.5 \). The decay factor is \( \left( \frac{1}{2} \right)^{1.5} \).
05

Evaluate the Decay Factor

Compute \( \left( \frac{1}{2} \right)^{1.5} = \frac{1}{\sqrt{2}} \times \frac{1}{2} = \frac{1}{2 \sqrt{2}} \approx 0.3536 \).
06

Calculate Remaining Quantity

Multiply the initial amount by the decay factor: \[ N(9) = 10 \times 0.3536 = 3.536 \] Thus, approximately 3.54 grams of Technetium-99m remain after 9 hours.

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Key Concepts

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

Half-Life
Half-life refers to the period it takes for half the atoms of a radioactive substance to decay. In the context of radioactive elements, it's a crucial metric that tells us how quickly or slowly a substance breaks down over time.
For Technetium-99m, its half-life is relatively short at just 6 hours. This short duration is beneficial in medical applications because it allows imaging procedures without prolonged radiation exposure to patients. In simple terms, after one half-life, you'll have half of the initial radioactive material left.
Understanding half-life is important because it's used to predict how long a radioactive substance will stay active in any given environment, helping in the planning and timing of its use.
  • After 6 hours, only half of the Technetium-99m remains.
  • This predictable decay makes it safe and effective for temporary medical imaging.
Decay Formula
The decay formula helps us calculate how much of a radioactive substance remains after a certain time has passed. This is extremely useful for determining the effectiveness of radioactive materials over known periods.
The formula is expressed as \[ N(t) = N_0 \left( \frac{1}{2} \right)^{t/T} \]
Here,
  • \(N(t)\) is the remaining amount of substance after time \(t\).
  • \(N_0\) is the initial quantity.
  • \(t\) is the time elapsed.
  • \(T\) is the half-life of the substance.

Applying this formula involves plugging in these values to compute how much of the substance is left after a specified period.
This systematic calculation addresses uncertainties by converting real-world information into usable numbers, showing exactly how much of a substance remains active.
Technetium-99m
Technetium-99m is especially valuable in the field of nuclear medicine. This radioisotope is used widely for imaging and functional studies of organs because of its short half-life and decay properties.
It's one of the most commonly used isotopes in diagnostics due to its ability to provide clear images of the inside of the body without exposing patients to prolonged radiation.
  • Technetium-99m emits gamma rays, which are detectable by a camera, providing detailed organ images.
  • It is used across various procedures, from bone scans to cardiac stress tests.
This isotope is derived on-site from a more stable parent compound, making it convenient and efficient for immediate use.
The rapid decay of Technetium-99m is both its strength and selling point in medical imaging settings.
Molybdenum Isotope
Molybdenum acts as the parent isotope from which Technetium-99m is harvested. Known as Molybdenum-99, this isotope provides a stable source for the production of Technetium-99m used in various diagnostic tests.
Molybdenum-99 itself has a longer half-life, approximately 66 hours, which makes it suitable for transport and use in nuclear medicine facilities.
  • The advantageous long half-life allows for the generation and "harvesting" of Technetium-99m whenever needed.
  • This synergy between Molybdenum-99 and Technetium-99m supports continual and ready-to-use medicinal radiopharmaceuticals.
Since Molybdenum-99 decays into Technetium-99m, it serves as a reliable supply chain point for hospitals, enabling streamlined medical imaging processes.
The reliance on Molybdenum-99 is essential for the healthcare industry, ensuring that modern medical procedures can be carried out efficiently and safely. This isotope cycle allows for a sustainable process in diagnostic imaging.

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