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\(\textbf{We Are Stardust.}\) In 1952 spectral lines of the element technetium-99 (\(^9$$^9\)Tc) were discovered in a red giant star. Red giants are very old stars, often around 10 billion years old, and near the end of their lives. Technetium has no stable isotopes, and the half-life of \(^9$$^9\)Tc is 200,000 years. (a) For how many halflives has the \(^9$$^9\)Tc been in the red giant star if its age is 10 billion years? (b) What fraction of the original \(^9$$^9\)Tc would be left at the end of that time? This discovery was extremely important because it provided convincing evidence for the theory (now essentially known to be true) that most of the atoms heavier than hydrogen and helium were made inside of stars by thermonuclear fusion and other nuclear processes. If the \(^9$$^9\)Tc had been part of the star since it was born, the amount remaining after 10 billion years would have been so minute that it would not have been detectable. This knowledge is what led the late astronomer Carl Sagan to proclaim that "we are stardust".

Step by step solution

01

Calculate the number of half-lives

To find out how many half-lives have elapsed over 10 billion years, we divide the total age of the star by the half-life of technetium-99. \[\text{Number of half-lives} = \frac{10,000,000,000 \text{ years}}{200,000 \text{ years/half-life}} \] Calculate this to find the number of half-lives.

02

Calculate the number of half-lives

\[\text{Number of half-lives} = \frac{10,000,000,000}{200,000} = 50,000 \text{ half-lives}\]This indicates that the technetium-99 in the star has undergone 50,000 half-lives.

03

Calculate the remaining fraction of technetium-99

After knowing the number of half-lives, use the formula for exponential decay to determine the remaining fraction of technetium-99: \[\text{Remaining fraction} = \left(\frac{1}{2}\right)^{n} \] where \(n\) is the number of half-lives (50,000).

04

Calculate the small fraction

Insert \(n = 50,000\) into the formula to calculate: \[\text{Remaining fraction} = \left(\frac{1}{2}\right)^{50,000} \approx 0 \]Practically, this fraction is extremely small, close to zero, indicating that almost none of the original technetium-99 remains after 10 billion years.

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Key Concepts

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

Half-life

Understanding the concept of "half-life" is crucial when studying radioactive elements like technetium-99. A "half-life" is the time it takes for half of a radioactive substance to decay. This means that after every half-life, only 50% of the radioisotope's original quantity remains.

For example:

• If a radioactive substance has a half-life of 200,000 years, like technetium-99, after 200,000 years, you would have only half of the substance left.
• After 400,000 years, you would have a quarter left, as it would half again.
• This process continues over successive half-lives, reducing the substance to smaller and smaller fractions.

You can see the power of this concept when applied to the red giant star. Over 10 billion years, technetium-99 in the star would undergo 50,000 half-lives, greatly diminishing its original quantity.

Exponential decay

Exponential decay describes how the quantity of a radioactive element decreases over time. It's characterized by a rapid drop-off followed by a slower decrease, a process best illustrated through the half-life concept.

The mathematical formula for exponential decay is:

• \[N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\]

where:

• \(N(t)\) is the remaining quantity of the substance at time \(t\).
• \(N_0\) is the initial quantity of the substance.
• \(t_{1/2}\) is the half-life of the substance.

For technetium-99 in a red giant, after 50,000 half-lives, the remaining quantity becomes practically zero. This is due to how quickly and exponentially the element decays over multiple half-lives, emphasizing the significant reduction from the original amount.

Red giant star

Red giant stars are fascinating bodies in space, often marking the late stages of stellar evolution. As stars exhaust the hydrogen in their cores, they undergo expansion, cooling the outer layers to form a red giant.

Some key characteristics of red giants include:

• They are typically over 10 billion years old, representing an advanced age for stellar bodies.
• The surface appears red due to lower temperatures compared to younger, hotter stars.
• Red giants play a significant role in the synthesis of elements through nuclear processes.

Discovering elements like technetium in such stars provides evidence for ongoing elemental synthesis, important for understanding stellar life cycles.

Thermonuclear fusion

Thermonuclear fusion is a powerful process occurring in the heart of stars, including red giants. It involves fusing lighter atomic nuclei to form heavier nuclei, releasing immense amounts of energy.

Highlights of thermonuclear fusion in stars include:

• Hydrogen atoms fuse into helium, which in later stages, can fuse into heavier elements.
• This process is how stars, including our sun, produce the energy that powers them and emits light and heat.
• Elements heavier than helium, such as technetium, are formed in stars through fusion and subsequent processes.

Technetium's presence in a red giant proves that elements heavier than hydrogen and helium originate in stars, supporting theories like "we are stardust." This phrase suggests that many elements found in our solar system and planet were created long ago in the fiery cores of ancient stars.