32.5 Exercise 19

Another set of reactions that result in the fusing of hydrogen into helium in the Sun and especially in hotter stars is called the carbon cycle. It is
$\begin{matrix}^{\text{12}} \text{C} + ^{1} \text{H } & \rightarrow & ^{\text{13}} \text{N} + \gamma , \\ ^{\text{13}} \text{N } & \rightarrow & ^{\text{13}} \text{C} + e^{+ } + v_{e} , \\ ^{\text{13}} \text{C } + ^{1} \text{H } & \rightarrow & ^{\text{14}} \text{N} + \gamma , \\ ^{\text{14}} \text{N} + ^{1} \text{H } & \rightarrow & ^{\text{15}} \text{O} + \gamma , \\ ^{\text{15}} \text{O } & \rightarrow & ^{\text{15}} \text{N} + e^{+ } + v_{e} , \\ ^{\text{15}} \text{N} + ^{1} \text{H } & \rightarrow & ^{\text{12}} \text{C} + ^{4} \text{He}.\end{matrix}$
Write down the overall effect of the carbon cycle (as was done for the proton-proton cycle in $2 e^{- } + 4 ^{1} \text{H} \rightarrow ^{4} \text{He} + \left(2 v\right)_{e} + 6γ$). Note the number of protons ( $^{1} \text{H}$) required and assume that the positrons ( $e^{+ }$) annihilate electrons to form more $\gamma$ rays.