Stellar structure - Helium burning

[PaulaS]

1. The problem statement.
Calculate the energy generated per unit mass, if helium burning produces equal amounts (mass fractions) of carbon and oxygen.
mH = 1.672621581 x 10^-27 kg


2. The attempt at a solution
Helium burning produces carbon according to the following reaction:
3He -> C [E(released) = 7.275 MeV]

Helium burning produces oxygen according to the following reactions:
3HE -> C [E(released) = 7.275 MeV]
He + C -> O [E(released) = 7.162 MeV]
______________________________________________ overall reaction for He:
4He -> O [E(released) = 14.437 MeV]


equal mass fractions => X(C) = X(O)
=> m(C)/m(total) = m(O)/m(total)

And then? I have no idea how to continue the exercise. Can anyone provide me with a hint?

Thanks

 

[mark726]

for your question! Let's break down the problem into smaller steps:

1. Determine the mass of carbon and oxygen produced per unit mass of helium burned:
As stated in the problem, the mass of helium (mH) is 1.672621581 x 10^-27 kg. Since the mass fractions of carbon and oxygen are equal, we can assume that the mass of carbon (mC) and oxygen (mO) produced are also equal. Let's represent this mass as mX. Therefore, we can write the following equation:
mC = mO = mX

2. Calculate the energy released per unit mass of helium burned:
From the given reactions, we know that for every 4 units of helium burned, 14.437 MeV of energy is released. Therefore, for 1 unit of helium burned, the energy released would be:
E(released) = (14.437 MeV / 4) = 3.60925 MeV

3. Calculate the energy released per unit mass of carbon and oxygen produced:
Since the mass of carbon and oxygen produced are equal (mC = mO = mX), we can calculate the energy released per unit mass of carbon and oxygen as:
E(released) = 3.60925 MeV / mX

4. Substitute the value of mX from step 1 into the equation from step 3:
E(released) = 3.60925 MeV / mX
= 3.60925 MeV / (mC + mO)
= 3.60925 MeV / (2mX) [since mC = mO = mX]
= 1.804625 MeV / mX

5. Substitute the value of mX into the equation:
mX = mH / 4 [since 4 units of helium is needed to produce 1 unit of carbon and oxygen]
Therefore, mX = (1.672621581 x 10^-27 kg) / 4 = 4.1815539525 x 10^-28 kg

6. Substitute the value of mX into the equation from step 4:
E(released) = 1.804625 MeV / mX
= 1.804625 MeV / (4.1815539525 x 10^-28 kg)
= 4.319330048 x 10^27 Me