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.

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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