Energy Release through Fission and SEMG

[TFM]

Homework Statement

On neutron-capture induced fission, $$^{235}_{92}U$$ typically splits into two new “fission product" nuclei with masses in the ratio 1:1.4. These are born with the same proton to neutron ratio as the original uranium, so they have too many neutrons to be stable at their mass number and are highly radioactive. Energy is released in two stages:

First an intermediate or prompt release leading to radioactive fission products in their ground state;
And then a much slower release via the beta and gamma decays of the fission product nuclei, which continue until they become stable.

Use the semi-empirical mass equation to estimate the magnitudes of the energy release per fission in each of the two stages. You may take the final Z/A ratios from appropriate known stable nuclei.

Homework Equations

SEMG: $$M(Z,A) = ZM_p +(A-Z)M_n - a_vA + a_sA^{2/3} + a_C\frac{Z^2}{A^{1/3}} + a_A\frac{(A-2Z)^2}{A} + \left(\frac{-1}{\frac{0}{1}}\right)\frac{a_P}{A^{1/2}}$$

Energy-Mass Relationship: $$E = mc^2$$

The Attempt at a Solution

Okay I have worked out that the Fission goes like so:

Firstly:

$$^{235}_{92}U + ^1_0n_1 \rightarrow ^{236}_{92}U \rightarrow ^{98}_{36}Kr_{62} + ^{138}_{50}Sn_{88}$$

And then through a series of beta decays:

$$^{98}_{36}Kr_{62} \rightarrow ^{98}_{42}Mo_{57}$$

$$^{138}_{50}Kr_{88} \rightarrow ^{138}_{58}Mo_{80}$$

However I am slightly unsure how to work out the energy using the SEMG.

I think I have to put in E = mc^2 into SEMG, and rearrange for E:

$$E = c^2ZM_p + c^2(A-Z)M_n - c^2a_vA + c^2a_sA^{2/3} + c^2a_C\frac{Z^2}{A^{1/3}} + c^2a_A\frac{(A-2Z)^2}{A} + c^2\left(\frac{-1}{\frac{0}{1}}\right)\frac{a_P}{A^{1/2}}$$

Then work out the energy for each part and then work out the difference, ie
Work out the energy in U, Mo, Kr

Then work out the Energy relaesed from U compared to Kr and Sn, and then Kr to Mo and Sn to Ce?

Does this make Sense?

Thanks in Advanced,

TFM

 

[Jhero]

you are on the right track in terms of using the semi-empirical mass equation to estimate the energy release in each stage of neutron-capture induced fission. However, there are a few things to consider in your approach.

Firstly, your initial fission reaction is correct, but it is important to note that the resulting fission products are not in their ground state. They will undergo a series of beta and gamma decays to reach their ground state, which is the second stage of energy release.

In terms of using the semi-empirical mass equation, you are correct in using E=mc^2, but you will need to use the mass values of each nucleus in the equation. These can be found in a table of nuclear masses. Additionally, you will need to consider the binding energy of each nucleus, which is the energy released when a nucleus is formed from its constituent particles. This can also be found in a table of nuclear masses.

To calculate the energy release in each stage, you will need to subtract the total mass of the initial nucleus (U-235) from the total mass of the final nuclei (Kr-98 and Sn-138 for the first stage, and Mo-98 and Mo-138 for the second stage). The difference in mass will give you the mass defect, which can then be used in E=mc^2 to calculate the energy release.

In terms of comparing the energy release in each stage, you can simply subtract the energy release in the first stage from the energy release in the second stage to get the total energy release per fission. You can then compare this to the energy release in other fission reactions to see how it compares.

I hope this helps in your calculations. Good luck!