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Another set of reactions that fuses hydrogen into helium in the Sun and especially in hotter stars is called the CNO cycle: $$\begin{aligned} &^{12} \mathrm{C}+^{1} \mathrm{H} \rightarrow^{13} \mathrm{N}+\gamma\\\ &^{13} \mathrm{N} \rightarrow^{13} \mathrm{C}+e^{+}+v_{\mathrm{e}}\\\ &^{13} \mathrm{C}+^{1} \mathrm{H} \rightarrow^{14} \mathrm{N}+\gamma\\\ &^{14} \mathrm{N}+^{1} \mathrm{H} \rightarrow^{15} \mathrm{O}+\gamma\\\ &^{15} \mathrm{O} \rightarrow^{15} \mathrm{N}+e^{+}+v_{\mathrm{e}}\\\ &^{15} \mathrm{N}+^{1} \mathrm{H} \rightarrow^{12} \mathrm{C}+^{4} \mathrm{He} \end{aligned}$$ This process is a "cycle" because \(^{12} \mathrm{C}\) appears at the beginning and end of these reactions. Write down the overall effect of this cycle (as done for the proton-proton chain in \(2 e^{-}+4^{1} \mathrm{H} \rightarrow^{4} \mathrm{He}+2 v_{\mathrm{e}}+6 \gamma\) ). Assume that the positrons annihilate electrons to form more \(\gamma\) rays.

Short Answer

Expert verified
The overall effect of the CNO cycle is given by the equation: \[^{12}C + 6^{1}H + 2e^{-} \rightarrow^{4}He + 2\nu_{e} + 9\gamma\] This means that six hydrogen nuclei, one carbon-12 nucleus, and two electrons combine to create one helium-4 nucleus, two electron neutrinos, and nine gamma rays.

Step by step solution

01

Add up all reactions

Begin by adding up all the reactions given in the CNO cycle: \(^{12}C+^{1}H \rightarrow^{13}N+\gamma\\ ^{13}N \rightarrow^{13}C+e^{+}+\nu_{e}\\ ^{13}C+^{1}H \rightarrow^{14}N+\gamma\\ ^{14}N+^{1}H \rightarrow^{15}O+\gamma\\ ^{15}O \rightarrow^{15}N+e^{+}+\nu_{e}\\ ^{15}N+^{1}H \rightarrow^{12}C+^{4}He\)
02

Combine similar terms

Combine similar elements and particles on both sides of the equations to simplify the overall effect: \(^{12}C + 6^{1}H + 2e^{-} \rightarrow^{4}He + 2\nu_{e} + 5\gamma + 2e^{+}\) Now we should account for the fact that positrons annihilate electrons to form more gamma rays.
03

Account for positron-electron annihilation

Since we have 2 positrons and 2 electrons in the simplified equation, we can assume that these will annihilate each other to form additional gamma rays. In a positron-electron annihilation, one positron and one electron form 2 gamma rays: \(2e^{+} + 2e^{-} \rightarrow 4\gamma\) With this information, we can write the overall effect of the CNO cycle: \(^{12}C + 6^{1}H + 2e^{-} \rightarrow^{4}He + 2\nu_{e} + 5\gamma + 4\gamma\)
04

Simplify the final result

Combine the gamma-ray terms in the equation and rewrite the overall effect of the CNO cycle: \(^{12}C + 6^{1}H + 2e^{-} \rightarrow^{4}He + 2\nu_{e} + 9\gamma\) This is the overall effect of the CNO cycle: six hydrogen nuclei, one carbon-12 nucleus, and two electrons combine to create one helium-4 nucleus, two electron neutrinos, and nine gamma rays.

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Key Concepts

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

Nuclear Fusion
Nuclear fusion is a powerful natural process that fuels stars, including our Sun. It occurs when atomic nuclei come together to form a heavier nucleus. In stars, this usually involves hydrogen nuclei fusing to form helium.
Unlike nuclear fission, which splits atoms, fusion joins them together. This process releases a tremendous amount of energy. Essentially, the sun shines because of the energy generated by nuclear fusion in its core.
In the context of stars, fusion begins when the internal temperatures and pressures are high enough to overcome the electrostatic repulsion between positively charged protons. This is why nuclear fusion occurs at the heart of stars where conditions are extreme.
There are different types of fusion processes. Apart from the CNO cycle, another important one is the proton-proton chain that occurs in stars like our Sun. These processes help scientists understand not just how stars generate energy, but also the creation of elements.
Stellar Processes
Stellar processes are a collection of activities and phenomena occurring in stars. These processes are crucial for the life and evolution of stars. At the heart of these activities is nuclear fusion, which is responsible for converting lighter elements into heavier ones, releasing energy.
The CNO Cycle is a part of these stellar processes. It describes a series of reactions in which carbon, nitrogen, and oxygen nuclei act as catalysts to fuse hydrogen into helium. This cycle predominates in stars that are hotter and more massive than the Sun. While it is less significant for energy production in the Sun, it's crucial for understanding the life cycles of larger stars.
These processes also determine how stars evolve over time. Depending on their mass, stars can end their lives in different ways, such as becoming white dwarfs, neutron stars, or black holes. Each stage of a star’s life cycle involves different stellar processes which play a role in element formation and energy output.
  • Energy production involves converting hydrogen to helium via various fusion cycles.
  • Element creation involves synthesis of heavier elements from lighter ones.
  • Life cycle developments include star formation, main sequence activities, and end-of-life stages.
Gamma Rays
Gamma rays are a form of electromagnetic radiation, similar to visible light but with much higher energy and shorter wavelengths.
They are a critical aspect of the nuclear reactions occurring in stars. During processes like the CNO Cycle, when atomic nuclei combine or decay, gamma rays are emitted.
These rays carry away energy produced in the fusion reactions within a star’s core. They eventually contribute to the light and heat brought to us from the star, after undergoing various interactions in the star's layers.
One of the interesting phenomena is that when positrons (anti-electrons) meet electrons, they annihilate each other, producing additional gamma rays, as seen in the CNO cycle.
Gamma rays are important because they help scientists understand stellar processes and conditions within stars. Observing gamma-ray emissions allows astronomers to uncover hidden aspects of star behavior that aren’t visible with other forms of radiation.
  • Gamma rays originate from nuclear fusion and decay processes.
  • They are produced both intrinsically in reactions and through pairs like positron-electron annihilation.
  • Understanding these rays provides insights into the energetics of stellar interiors.

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Most popular questions from this chapter

A sample of radioactive material is obtained from a very old rock. A plot \(\ln A\) verses \(t\) yields a slope value of \(-10^{-9} \mathrm{s}^{-1}\) (see Figure \(10.10(\mathrm{b})\) ). What is the half-life of this material?

The Galileo space probe was launched on its long journey past Venus and Earth in \(1989,\) with an ultimate goal of Jupiter. Its power source is \(11.0 \mathrm{kg}\) of \(^{238} \mathrm{Pu}\) a by-product of nuclear weapons plutonium production. Electrical energy is generated thermoelectrically from the heat produced when the \(5.59-\mathrm{MeV} \quad \alpha\) particles emitted in each decay crash to a halt inside the plutonium and its shielding. The half-life of \(^{238} \mathrm{Pu}\) is 87.7 years. (a) What was the original activity of the \(^{238} \mathrm{Pu}\) in becquerels? (b) What power was emitted in kilowatts? (c) What power was emitted 12.0 y after launch? You may neglect any extra energy from daughter nuclides and any losses from escaping \(\gamma\) rays.

According to your lab partner, a 2.00-cm-thick sodium-iodide crystal absorbs all but \(10 \%\) of rays from a radioactive source and a 4.00-cm piece of the same material absorbs all but \(5 \%\) ? Is this result reasonable?

A radioactive sample initially contains \(2.40 \times 10^{-2}\) mol of a radioactive material whose half-life is 6.00 h. How many moles of the radioactive material remain after \(6.00 \mathrm{h}\) ? After 12.0 h? After 36.0 h?

One half the \(\gamma\) rays from \(^{99 \mathrm{m}} \mathrm{Tc}\) are absorbed by a 0.170-mm-thick lead shielding. Half of the \(\gamma\) rays that pass through the first layer of lead are absorbed in a second layer of equal thickness. What thickness of lead will absorb all but one in 1000 of these \(\gamma\) rays?

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