Question

The Sun's power comes mainly from a series of reactions in which 4 protons are converted overall into a helium nucleus, a = 3He. The first step is the creation of a deuteron, d = 2H, via p + p -> d + e+ + ve.

The neutrino has a typical energy of ~ 0.3 MeV.

i. Calculate the Q value for the pp reaction and hence verify that, from an energy point of view, it can take place. What happens to the two leptons emitted?

ii. Write down the effective reaction for the whole of the pp cycle and hence show that the total energy contribution to the Sun's visible power output per helium nucleus formed is 26.2 MeV.
Assuming that 100 W of energy per m^³ is being produced via this reaction, how many pp fusions are taking place per second per cubic meter? Given that the pp collision rate is 2.5 x 10^⁷ s^-¹per proton, what is the probability of a collision resulting in a fusion? Comment on your result. [The proton number density in the Sun is ~ 5.0 x 10^31 m-³.]

iii. Sketch a curve of the binding energy per nucleon as a function of atomic mass number, and use this to explain why, for a sufficiently massive star, nuclear fusion can proceed up to 2gFe. Briefly state how stars can manufacture nuclei with masses beyond A = 56.

Answer 1

The Sun's energy is produced through the proton-proton cycle of nuclear fusion, converting hydrogen into helium. The binding energy per nucleon curve explains why fusion in stars is efficient up to iron, and heavier elements are created by other processes in massive stars.

Step-by-step explanation:

The Sun's power comes from the process of nuclear fusion, where four protons (hydrogen nuclei) are fused into a helium nucleus in a series of reactions known as the proton-proton cycle. The reactions within this cycle include:

1. ¹H + ¹H → ²H + e+ + ve (0.42 MeV)
2. ¹H + ²H → ³He + γ (5.49 MeV)
3. ³He + ³He → ⁴He + ¹H + ¹H (12.86 MeV)

The overall effect of these reactions leads to the conversion of six protons into one helium nucleus, two protons, and the release of two positrons, which eventually annihilate with electrons to form more gamma rays (y). This adds up to a total energy release of 26.7 MeV, less than the ~0.5 MeV lost to neutrinos. Given a Sun power output of 100 W per m³, and a proton collision rate of 2.5 x 10⁷ s⁻¹ per proton, the fusion rate and the corresponding probability of a pp collision leading to fusion can be calculated.

To answer part iii, the curve of binding energy per nucleon shows that it increases with atomic mass number up to 56 for iron. Stars can create heavier elements through processes like supernovae and neutron capture that occur in later stages of stellar evolution.