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## Q. 13.8

One of the processes by which supergiant stars produce heavier masses is the reaction ${ }^{4} He +{ }^{12} C \rightarrow{ }^{16} O +\gamma$. (a) How much energy is expended in this reaction? (b) Explain why such a fusion reaction cannot occur in low-mass stars.

Strategy (a) To find the energy expended we find the Q value from Equation (13.7). (b) We consider the progression of a star’s life to determine what fusion reactions can occur.

$Q=M_{x} c^{2}+M_{X} c^{2}-\left(M_{y} c^{2}+M_{Y} c^{2}\right)=K_{y}+K_{Y}-K_{x}$ (13.7)

## Verified Solution

(a) We look up the masses in Appendix 8 to find the Q value:

\begin{aligned}Q &=\left[M\left({ }^{4} He \right)+M\left({ }^{12} C \right)-M\left({ }^{16} O \right)\right] c^{2} \\Q &=(4.002603 u +12.0 u -15.994915 u ) c^{2} \\&=0.007688 u \cdot c^{2} \\Q &=0.007688 u \cdot c^{2}\left(931.5 \frac{ MeV }{ u \cdot c^{2}}\right)=7.2 MeV\end{aligned}

(b) Stars begin by burning hydrogen to form helium. As the helium is exhausted, the stars contract to higher densities and temperatures, which then allows helium to burn. Because there are no mass-5 and mass-8 stable nuclides, there is a hurdle to get beyond mass-4 nuclides. The hurdle is cleared only if the star is large enough so that as it contracts, helium can burn to create heavier masses. Our sun can produce carbon, but the burning of alpha particles and carbon requires temperatures as high as 300 million K in order to overcome the Coulomb barrier and undergo nuclear fusion. Our sun does not reach this temperature.