In certain stars the carbon cycle is more likely than the proton-proton cycle to be effective in generating energy. This cycle is

We have to show (a) that this cycle of reactions is exactly equivalent in its overall effects to the proton-proton cycle. (b) We have to verify that this cycle, as expected, has the same as the p-p cycle.

Solution:             Click For PDF Version

(a) and (b)

The carbon cycle is

We add the six equations and note that effectively we are dealing with the process

The sum of s in the carbon cycle gives

The two positrons will readily annihilate with two electrons and their mass energy will contribute to net energy generated by fusion of four protons into one helium nucleus. As the rest mass energy of electron (positron) is 0.51 MeV, the effective Q of the carbon cycle is

,

which is also the for the p-p cycle. We note that in the p-p cycle four protons and two electrons get converted into one nucleus of helium and two neutrinos.