Question

The carbon cycle, a stellar burning process secondary to the proton cycle, consists of the following sequential series of nuclear reactions and decays:
12C + p → 13N + γ, with 1.9 MeV of energy released;
13N → 13C + e+ + v, with 1.2 MeV of energy released;
13C + p → 14N + γ, with 7.6 MeV of energy released;
14N + p → 150 + γ, with 7.4 MeV of energy released;
15O → 15N + e+ + v, with 1.7 MeV of energy released;
15N + p → 12C + 4He, with 5.0 MeV of energy released.
(a) What is the net effect of one cycle, and how much energy is produced per cycle?
(b) Consider a star of mass 3·1030 kg, consisting mainly of hydrogen, with 0.1 percent of its mass in 12С. The characteristic duration of a carbon cycle in the star is 5·106 yr. Estimate the energy produced each year by the carbon cycle, assuming that each carbon nucleus in the star acts as a catalyst for the cycle.
(Answer: (a) 24.8 MeV released, (b) 7.5·1047 MeV (1.2·1034 J).)

Answer 1

(A) The total energy produced per cycle can be calculated by just adding up all the sequential series of nuclear reactions and decays.

1.9 + 1.2 + 7.6 + 7.4 + 1.7 + 5.0 = 24.8 MeV

Therefore 24.8 MeV is Released or (24.8 * 1.6 * 10^-13J = 39.68* 10^-13J)

(B) Mass of Star = 3 * 10³⁰

0.1 Percent = 3 * 10³⁰ * 0.1 * 10^-2

Therefore,

Mass of Carbon = 3 * 10²⁷

Total Number of Carbon Atoms Calculation

Using Mole Concept, in 12 g of Carbon, We have 6.023 * 10²³ Atoms of Carbon

Hence, in 3 *10²⁷ kg of Carbon , we will have 6.023 * 10²³* 3 * 10²⁷ /12

Final Answer : 3.0115 * 10⁵⁰ Atoms

Now , we already know Energy produced per cycle.

Hence, if we calculate Energy produced for full cycle of all atoms, we get 3.0115 * 10⁵⁰ * 39.68 * 10^-13J = 1.1949 * 10³⁴ J

But this is for 5 years.

Hence, for one year, it is 2.3 * 10³³ J