Question

In addition to the process described in the text, a second process called the carbon-nitrogen cycle occurs in the sun: a. What is the catalyst in this process? b. What nucleons are intermediates? c. How much energy is released per mole of hydrogen nuclei in the overall reaction? (The atomic masses of \(_{1}^{1} \mathrm{H}\) and \(\frac{4}{2} \mathrm{He}\) are 1.00782 \(\mathrm{u}\) and \(4.00260 \mathrm{u},\) respectively.)

Short answer

a. The catalyst in the carbon-nitrogen cycle is carbon. b. The intermediate nucleons in the carbon-nitrogen cycle are Nitrogen-13, Carbon-12, Carbon-13, Nitrogen-14, and Oxygen-15. c. The energy released per mole of hydrogen nuclei in the overall carbon-nitrogen cycle reaction is approximately 2.95 x 10^12 Joules/mol.

Step-by-step solution

Understanding the carbon-nitrogen cycle

The carbon-nitrogen (C-N) cycle is a set of fusion reactions by which stars convert hydrogen into helium, the primary alternative being the proton-proton chain. The C-N cycle involves heavier catalyst elements like carbon and nitrogen as intermediates for it to occur.

Identifying the catalyst in the process

The catalyst in the carbon-nitrogen cycle is carbon. Carbon plays an essential role in transforming hydrogen nuclei into helium through a series of reactions involving nitrogen and other intermediate elements.

Identifying the intermediate nucleons

The intermediate nucleons in the carbon-nitrogen cycle include: - Nitrogen-13 (\(_{7}^{13}\mathrm{N}\)) - Carbon-12 (\(_{6}^{12}\mathrm{C}\)) - Carbon-13 (\(_{6}^{13}\mathrm{C}\)) - Nitrogen-14 (\(_{7}^{14}\mathrm{N}\)) - Oxygen-15 (\(_{8}^{15}\mathrm{O}\)) These intermediate nuclei help to transform four hydrogen nuclei into one helium nucleus.

Calculating the energy released per mole of hydrogen nuclei

To calculate the amount of energy released per mole of hydrogen nuclei in the overall reaction, we need to determine the mass difference between the reactants and products. The overall reaction involves the transformation of four hydrogen nuclei into one helium nucleus: \[4_{1}^{1}\mathrm{H} \rightarrow 4 \mathrm{He}\] Now, we calculate the mass difference: Mass difference = (Mass of four hydrogen nuclei) - (Mass of one helium nucleus) Using the given atomic masses: Mass of four hydrogen nuclei = 4 * 1.00782 u Mass of one helium nucleus = 4.00260 u Mass difference = (4 * 1.00782 u) - 4.00260 u Mass difference = 0.03128 u To convert this mass difference into energy, we use the famous Einstein's mass-energy equivalence equation, E=mc^2, and the following conversion factors: 1 u = 1.66 x 10^{-27} kg c = 3.00 x 10^8 m/s (speed of light) Energy released = (Mass difference) * (1.66 x 10^{-27} kg/u) * (3.00 x 10^8 m/s)^2 Energy released = 4.894 x 10^{-12} J (per nucleus) 1 mole = 6.022 x 10^23 nuclei Energy released per mole = (4.894 x 10^{-12} J) * (6.022 x 10^23 nuclei/mol) Energy released per mole ≈ 2.95 x 10^12 J/mol Hence, the energy released per mole of hydrogen nuclei in the overall carbon-nitrogen cycle reaction is approximately 2.95 x 10^12 Joules/mol.

Key concepts

Nuclear Fusion

Nuclear fusion is the process where two or more atomic nuclei combine to form a heavier nucleus, releasing energy in the process. This phenomenon is fundamental to the energy production in stars. In the sun, hydrogen nuclei (protons) are fused to form helium, releasing tremendous amounts of energy. The carbon-nitrogen cycle is one of the nuclear fusion pathways that occur in stars like the sun.
It involves a series of reactions where carbon-12 acts as a catalyst, facilitating the fusion of hydrogen into helium. The energy from this process is the result of the conversion of a small amount of mass into energy, as described by Einstein's equation, \(E=mc^2\). This process sustains the radiation and warmth that we receive from the sun, highlighting fusion's pivotal role in stellar energy production.

Energy Calculation

Energy calculations in nuclear fusion reactions involve understanding the mass-energy equivalence principle. In the carbon-nitrogen cycle, energy release is calculated by determining the mass difference between reactants and products.
For our example, four hydrogen nuclei are converted into one helium nucleus, transforming some of the mass into energy. This can be calculated by examining the difference in atomic masses before and after the reaction.
Given atomic masses for hydrogen and helium, the mass difference is found to be 0.03128 u per reaction cycle. Using the conversion between atomic mass units (u) and kilograms, one can apply \(E=mc^2\) to find energy per reaction.
Multiplying this energy by Avogadro's number provides insight into energy release per mole, revealing the powerful output of fusion reactions at a macroscopic scale. It's approximately 2.95 x 10^{12} J/mol for this cycle, demonstrating the remarkable energy generation capabilities of nuclear fusion.

Reaction Intermediates

In the carbon-nitrogen cycle, reaction intermediates are crucial components facilitating the overall fusion process. These intermediates are nuclei that form temporarily during the series of fusion reactions.
As the reaction progresses, specific elements such as nitrogen-13 and carbon-13 play intermediary roles.

• Nitrogen-13 serves as a stepping stone in transforming protons into helium nuclei.
• Carbon-12 and carbon-13 act as both starting points and transitional forms during this cyclic process.
• Oxygen-15 and nitrogen-14 appear momentarily, contributing to the continued transformation.

These intermediates ensure the cycle operates efficiently, helping manage matter and energy throughout the fusion process. Intermediary nucleons, although not seen in the final products, are indispensable for sustaining the chain of reactions that produce energy and transform matter in stellar environments.

Stellar Nucleosynthesis

Stellar nucleosynthesis is the process by which stars create new elements through nuclear fusion reactions within their cores. This is how most of the elements we see in the universe were formed.
In every star, including our sun, nucleosynthesis occurs, starting with the simplest element, hydrogen, and through fusion, forming helium and even heavier elements as the star evolves.
In the core of stars, where temperatures and pressures are exceedingly high, nuclear reactions convert lighter elements into heavier ones, releasing energy that fuels the star and contributes to its luminosity. The carbon-nitrogen cycle is an example of this process, demonstrating how hydrogen is transformed into helium, with carbon acting as a catalyst.
Understanding stellar nucleosynthesis gives insight into the life cycle of stars and the chemical evolution of galaxies, offering glimpses into the cosmic origins of the elements constituting planets and living organisms.