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43.65. We Are Stardust. In 1952 spectral lines of the element technetium- 99 ( \(^{99} \mathrm{Tc} )\) 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 \(^{99} \mathrm{Te}\) is \(200,000\) years. (a) For how many half-lives has the \(^{99}\) Te been in the red-giant star if its age is 10 billion years? (b) What fraction of the original "Te 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 \(^{99} \mathrm{Tchadbeen}\) 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

Determine the Half-Life Duration in Number of Years

Since the element technetium-99 has a half-life of 200,000 years, we use this value as a basis for calculating how many half-lives fit into 10 billion years.

02

Calculate the Number of Half-Lives over 10 Billion Years

Divide the total time of 10 billion years by the half-life of technetium-99 to find the number of half-lives: \[ \text{Number of half-lives} = \frac{10 \text{ billion years}}{200,000 \text{ years/half-life}} = \frac{10^{{10}}}{2 \times 10^5} = 5 \times 10^4 \text{ half-lives} \]

03

Determine the Remaining Fraction of Original Technetium-99

For each half-life, the remaining fraction of a substance is halved, so after \( n \) half-lives, the fraction remaining is given by \( \left(\frac{1}{2}\right)^n \). \[ \text{Remaining fraction} = \left(\frac{1}{2}\right)^{5 \times 10^4} \] This result is an extremely small number, essentially zero, indicating that after 50,000 half-lives, there would be virtually none left.

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Key Concepts

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

Nuclear Fusion in Stars

Stars shine and produce energy through a process known as nuclear fusion. In the intense heat and pressure at the core of a star, lighter elements like hydrogen are fused together to form heavier elements such as helium. This process releases an enormous amount of energy.

Nuclear fusion in stars is fundamental because it's the source of the light and heat emitted by stars, including our Sun. These energy-producing reactions push back against the force of gravity pulling the star inward, maintaining the star's stability over billions of years.

One key aspect of nuclear fusion is that it operates like a chain reaction. As heavier elements are produced, further fusion reactions can take place, forming elements up to iron. Elements heavier than iron require different processes, like those occurring in supernovae explosions, to form.

• Fusion mainly occurs in stars' cores, where temperatures reach millions of degrees.
• It creates elements crucial for life, including carbon, oxygen, and others.
• The energy created balances the gravitational forces in the star.

Half-Life Calculations

Half-life is a concept used to understand the time it takes for half of a radioactive substance to decay. When calculating how much of a radioactive element remains over time, the concept of half-life allows us to determine the decay process.

For technetium-99, with a half-life of 200,000 years, the half-life formula is essential. The formula for the remaining quantity of a substance after a certain time is:\[\text{Remaining fraction} = \left(\frac{1}{2}\right)^{n}\]
Here, \(n\) is the number of half-lives elapsed, found by dividing the total time by the half-life duration.

• The amount of technetium-99 left is minuscule after 10 billion years, showing why it was a significant discovery.
• Knowing how to perform half-life calculations helps in understanding decay processes in various scientific fields.

Stellar Nucleosynthesis

Stellar nucleosynthesis refers to the process by which stars generate new elements in their cores. This process begins with nuclear fusion, but it doesn't stop there. As stars evolve, they create heavier and heavier elements.

In young stars, light elements like hydrogen and helium fuse to form elements up to iron. However, in more massive stars, or in their explosive ends as supernovae, even heavier elements are formed.

The discovery of technetium in a star was a breakthrough that confirmed much of our scientific understanding of stellar nucleosynthesis. Since technetium has no stable isotopes and decays relatively quickly on a cosmic timescale, finding it in a star showed it must have been produced there—not through ancient processes, but in the contemporary life of the star.

• Processes like the s-process and r-process describe how stars can create these elements.
• Most elements in the universe were formed inside stars, supporting the idea that "we are stardust."
• This process is central to our understanding of the chemical composition of the universe.