Problem 21 - College Physics

Question

We are stardust. In \(1952,\) spectral lines of the element technetium-99 \(\left(^{99} \mathrm{Tc}\right)\) 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 \(n o\) stable isotopes, and the half life of \(^{99} \mathrm{Tc}\) is \(200,000\) years. (a) For how many half-lives has the \(^{99} \mathrm{Tc}\) been in the red-giant star if its age is 10 billion years? (b) What fraction of the original \(^{99} \mathrm{Tc}\) 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 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

Answer

(a) 50,000 half-lives. (b) The remaining fraction is effectively zero.

Step 1: Calculate the Number of Half-Lives

To find how many half-lives have passed for technetium-99 (^{99}Tc) in a red giant star, divide the age of the star by the half-life of ^{99}Tc. The age of the star is 10 billion years, and the half-life is 200,000 years, so you calculate: \[ \text{number of half-lives} = \frac{10 \text{ billion years}}{200,000 \text{ years/half-life}} = \frac{10 \times 10^9}{2 \times 10^5} = 5 \times 10^4 \text{ half-lives}. \]

Step 2: Determine the Remaining Fraction of Technetium-99

To determine the fraction of the original ^{99}Tc still present after these half-lives, use the exponential decay formula: \[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^{\text{number of half-lives}}. \] Substituting the number of half-lives found in the previous step gives: \[ \left( \frac{1}{2} \right)^{5 \times 10^4}. \] This fractional value will be extremely small, effectively zero, indicating no detectable ^{99}Tc would remain.

Key Concepts

Technetium-99 Decay

Technetium-99 ( ^{99}Tc) is a radioactive element with no stable isotopes. It is of considerable interest in nuclear physics and astronomy due to its unstable nature. When technetium-99 decays, it transforms into other elements over time, releasing particles and energy. This decay is significant in understanding the processes in stars because technetium-99 does not occur naturally on Earth due to its instability and relatively short half-life. The existence of technetium-99 in stars, especially older red giants, is evidence of nucleosynthesis. This is because technetium cannot survive for billions of years outside of a star without being replenished. Thus, its presence in a star confirms the creation and decay cycles driven by nuclear processes within stars.

Half-Life Calculation

The concept of half-life is pivotal in understanding radioactive decay. It refers to the time required for half of a given isotope to decay. For technetium-99, this time is 200,000 years. Understanding half-life helps in calculating how much of a radioactive isotope remains after a certain period.To find the number of half-lives that fit into a given timeframe, you simply divide the total time by the half-life. For example, for a star 10 billion years old containing technetium-99, you calculate this as: \[ \text{number of half-lives} = \frac{10 \text{ billion years}}{200,000 \text{ years/half-life}} = 5 \times 10^4\text{ half-lives.} \] This calculation provides a framework for predicting how much of the isotope might remain under the decay process.

Stellar Nucleosynthesis

Stellar nucleosynthesis is the process by which stars generate new elements. It occurs through nuclear reactions transforming existing elements in the star's core and layers. This process is responsible for the formation of elements heavier than hydrogen and helium. In stars like red giants, these reactions occur primarily through fusion processes. As stars age and evolve, they fuse lighter elements into heavier ones, such as carbon, oxygen, and eventually, heavier elements like technetium. This ongoing element fusion enriches the universe with complex chemical elements necessary for life and planets. Technetium-99's presence in stars provides evidence of this process, showcasing that elements are continuously created and destroyed within these cosmic forges.

Exponential Decay Formula

Radioactive decay is a probabilistic process described by the exponential decay formula. It quantifies how the quantity of a radioactive isotope decreases over time.The decay formula is given by:\[\text{Remaining fraction} = \left( \frac{1}{2} \right)^{\text{number of half-lives}}\]This formula illustrates that after each half-life, half of the existing radioactive isotope remains. Applying this to technetium-99 in a star over 10 billion years indicates very little of the original isotope would remain, as shown by: \[ \left( \frac{1}{2} \right)^{5 \times 10^4} \]This exponential decrease demonstrates why any original technetium-99 in an ancient star would have diminished to undetectable levels, supporting the hypothesis that such isotopes result from ongoing stellar processes.