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Show that the net result of the proton-proton fusion chain that occurs inside our sun can be summarized as $$ 6 \mathrm{p}^{+} \rightarrow{ }_{2}^{4} \mathrm{He}+2 \mathrm{p}^{+}+2 \beta^{+}+2 \gamma+2 \nu_{\mathrm{e}} $$

Short Answer

Expert verified
6 protons yield 1 helium-4, 2 protons, 2 positrons, 2 gammas, and 2 neutrinos.

Step by step solution

01

Understanding the proton-proton cycle

In the sun, the proton-proton fusion chain is a series of nuclear reactions that convert hydrogen into helium. This process primarily consists of three continuous reactions: (1) A pair of protons fuses to form a deuterium nucleus, a positron, and a neutrino; (2) The deuterium nucleus fuses with another proton to form helium-3 and release a gamma photon; (3) Two helium-3 nuclei merge to create helium-4, releasing two protons.
02

First Reaction Equation

The first reaction involves two protons (\(2 ext{p}^+\)) forming deuterium (\(_{1}^{2} ext{H}\)), a positron (\(\beta^+\)), and a neutrino (\(u_e\)). The equation is:\[2 ext{p}^+ ightarrow {}_{1}^{2} ext{H} + \beta^+ + u_e\].
03

Second Reaction Equation

In the second reaction, a deuterium nucleus (created in the first step) interacts with another proton (\(\text{p}^+\)) to form helium-3 (\(_{2}^{3} ext{He}\)) and release a gamma photon (\(\gamma\)). The equation is:\[{}_{1}^{2} ext{H} + ext{p}^+ ightarrow {}_{2}^{3} ext{He} + \gamma\].
04

Third Reaction Equation

In the third reaction, two helium-3 nuclei (\(2 imes {}_{2}^{3} ext{He}\)) react to form a helium-4 nucleus (\(_{2}^{4} ext{He}\)) along with two protons (\(2 ext{p}^+\)). The equation is:\[2 imes {}_{2}^{3} ext{He} ightarrow {}_{2}^{4} ext{He} + 2 ext{p}^+\].
05

Combining All Reactions

To derive the overall reaction, we sum all individual reactions. Start by writing all steps together:1. \(4 ext{p}^+ \rightarrow 2 {}_{1}^{2} ext{H} + 2 \beta^+ + 2 u_e\).2. \(2 {}_{1}^{2} ext{H} + 2 ext{p}^+ \rightarrow 2 {}_{2}^{3} ext{He} + 2 \gamma\).3. \(2 {}_{2}^{3} ext{He} \rightarrow {}_{2}^{4} ext{He} + 2 ext{p}^+\).By adding these equations, we simplify the terms to obtain:\[6 ext{p}^+ \rightarrow {}_{2}^{4} ext{He} + 2 ext{p}^+ + 2 \beta^+ + 2 \gamma + 2 u_e\].
06

Conclusion

The net balance shows that 6 initial protons are involved in producing a single helium-4 nucleus, with 2 protons, 2 positrons, 2 gamma rays, and 2 neutrinos as by-products. This confirms the original equation: \(6 ext{p}^+ \rightarrow {}_{2}^{4} ext{He} + 2 ext{p}^+ + 2 \beta^+ + 2 \gamma + 2 u_e\).

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Key Concepts

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

Nuclear Reactions
Nuclear reactions are processes that involve changes in the nucleus of an atom. Unlike chemical reactions, where electrons are traded or shared between atoms, nuclear reactions impact the core - the nucleus. This results in the transformation of elements. The changes typically involve the conversion of one element into another or the emission of energy in various forms. In the context of star physics, nuclear reactions are crucial as they underpin the energy generation in stars. For instance, in the sun, the prevalent fusion process is the proton-proton fusion chain, which generates energy by converting hydrogen nuclei into helium. This sequence of reactions is the powerhouse that sustains our sun's glow.

Key characteristics of nuclear reactions include:
  • They can release or absorb massive amounts of energy.
  • Involvement of forces within the atom, such as the strong nuclear force.
  • Potential production of different isotopes or elements through fusion or fission.
  • Residual particles like neutrinos or positrons might be emitted.
Understanding nuclear reactions gives insight into the fundamental processes keeping stars alive and contributing to element formation throughout the universe.
Helium Production
Helium production in stars is primarily the result of nuclear fusion reactions. Within the sun and other similar stars, hydrogen nuclei, or protons, combine under extreme pressure and temperature to form helium. This process is known as the proton-proton fusion chain, which transforms hydrogen into helium over several stages.

This conversion starts with two protons melding to create a deuterium nucleus, emitting a positron and neutrino. This initial reaction sets off subsequent fusions, eventually forming helium-3. Finally, two helium-3 nuclei collide to generate a stable helium-4 atom, releasing two protons, leaving behind the fused helium, and emitting energy in the process.

Highlights of helium production include:
  • Conversion of four hydrogen nuclei to one helium nucleus.
  • The release of energy essential for stellar radiation.
  • Production of by-products like neutrinos and gamma rays.
Helium production is not only a crucial component of star life cycles but also contributes to the synthesis of heavier elements as stars evolve.
Solar Nuclear Processes
Solar nuclear processes refer to the series of reactions occurring in the sun's core, producing energy through fusion. These processes ensure the conversion of hydrogen into heavier elements, fundamentally powering the sun's life cycle. The proton-proton chain reaction is the predominant pathway in the sun, facilitating energy flow within its interior.

This chain starts with hydrogen nuclei colliding to form deuterium, eventually leading to the synthesis of helium-3 and helium-4. Each step in this chain releases particles such as positrons, neutrinos, and gamma photons, contributing to the sun's immense energy output. The efficiency of these processes ensures the sun maintains its glow, supporting life on Earth by providing warmth and light.

Major points to grasp about solar nuclear processes:
  • The primary reaction is the fusion of hydrogen to form helium.
  • Energy is released as part of the sun's radiation and light.
  • The process involves multiple layers of nuclear reactions and particle emissions.
Understanding these processes sheds light on why stars, like the sun, have prolonged lifespans and continuous energy output.
Energy Generation in Stars
Energy generation in stars is a fundamental process that powers the universe. Stars, acting as cosmic furnaces, create energy through the fusion of lighter elements into heavier ones. In stars like our sun, energy generation occurs predominantly through the proton-proton chain. This series of fusion reactions begins with hydrogen nuclei and culminates in the production of helium-4, releasing significant energy.

This energetic output is tremendous because the mass of the fused nucleus is slightly less than the sum of its components. The lost mass is converted into energy, as explained by Einstein's mass-energy equivalence formula, \( E=mc^2 \). This conversion fuels the star, contributing to its brightness and heat.

Key aspects of energy generation in stars include:
  • Fusion reactions, combining light elements like hydrogen into heavier ones like helium.
  • Release of energy that supports the star's luminance and thermal energy.
  • Continuity of fusion processes which maintain equilibrium within the star.
Understanding how stars generate energy provides valuable insights into their internal mechanisms and life cycles, offering explanations for phenomena such as their immense brightness and longevity.

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Most popular questions from this chapter

Given that each particle contains only combinations of \(u, d, s,\) \(\bar{u}, \bar{d},\) and \(\bar{s},\) deduce the quark content of (a) a particle with charge \(+e,\) baryon number \(0,\) and strangeness \(+1 ;\) (b) a particle with charge \(+e,\) baryon number \(-1,\) and strangeness \(+1 ;\) (c) a particle with charge \(0,\) baryon number \(+1,\) and strangeness -2.

Which type of radioactive decay produces \({ }^{131} \mathrm{I}\) from \({ }^{131} \mathrm{Te} ?\) A. Alpha decay B. Beta-minus decay C. Beta-plus decay D. Gamma decay

The measured energy width of the \(\phi\) meson is \(4.0 \mathrm{MeV},\) and its mass is \(1020 \mathrm{MeV} / c^{2}\). Using the uncertainty principle (in the form \(\Delta E \Delta t \geq h / 2 \pi),\) estimate the lifetime of the \(\phi\) meson.

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 no 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?

Nuclear weapons tests in the \(1950 \mathrm{~s}\) and \(1960 \mathrm{~s}\) released significant amounts of radioactive tritium \(\left({ }_{1}^{3} \mathrm{H},\right.\) half-life 12.3 years \()\) into the atmosphere. The tritium atoms were quickly bound into water molecules and rained out of the air, most of them ending up in the ocean. For any of this tritium-tagged water that sinks below the surface, the amount of time during which it has been isolated from the surface can be calculated by measuring the ratio of the decay product, \({ }_{2}^{3} \mathrm{He},\) to the remaining tritium in the water. For example, if the ratio of \({ }_{2}^{3} \mathrm{He}\) to \({ }_{1}^{3} \mathrm{H}\) in a sample of water is \(1: 1,\) the water has been below the surface for one half-life, or approximately 12 years. This method has provided oceanographers with a convenient way to trace the movements of subsurface currents in parts of the ocean. Suppose that in a particular sample of water, the ratio of \({ }_{2}^{3} \mathrm{He}\) to \({ }_{1}^{3} \mathrm{H}\) is \(4.3: 1.0 .\) How many years ago did this water sink below the surface?

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