Question

Stars generate energy through nuclear fusion. A fusion reaction is exoergic (i.e., it gives off energy) if the product has a lower rest energy than the original nuclei. The binding energy of a nucleus is defined as the difference between the rest energy that the individual particles would have if they were not bound in a nucleus and the rest energy of the nucleus itself. So, stated another way, fusion reactions are exoergic if the binding energy of the product is greater than the binding energy of the original nuclei.

Therefore, to calculate the binding energy of a nucleus, you will need to know the masses of a free proton and a free neutron. Normally, the mass of neutral hydrogen is used instead of the mass of a free proton. This substitution works because usually the mass of the product nucleus is given as the mass of the neutral atom with that nucleus, and so the energy related to electrons on each side of the equation should cancel. The relevant quantities are as follows:

mass of neutral hydrogren ¹₁H=1.007825u,

mass of neutron =1.008665u,

and c²=931.5 MeV/u.

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Part A

The mass of neutral ¹⁶₈O=15.994915 u. Find the binding energy for ¹⁶₈O.

Express your answer in millions of electron volts (MeV) to four significant figures.

Part B

Find the binding energy per nucleon for ¹⁶₈O.

Express your answer in millions of electron volts per nucleon (MeV/nucleon) to four significant figures.

Part C

The mass of neutral ⁵⁶₂₆Fe=55.934939 u. Find the binding energy per nucleon for ⁵⁶₂₆Fe.

Express your answer in millions of electron volts per nucleon (MeV/nucleon) to four significant figures.

Part D

The mass of neutral ²⁰⁸₈₂Pb=207.976627 u. Find the binding energy per nucleon of ²⁰⁸₈₂Pb.

Express your answer in millions of electron volts per nucleon (MeV/nucleon) to four significant figures.

Part E

Consider the binding energy per nucleon for ¹⁶₈O and ⁵⁶₂₆Fe. Would you expect the fusion reactions leading from ¹⁶₈O to ⁵⁶₂₆Fe to be exoergic (giving off energy), endoergic (absorbing energy), or neither?

Part F

Consider the binding energy per nucleon for ⁵⁶₂₆Fe and ²⁰⁸₈₂Pb. Would you expect the fusion reactions leading from ⁵⁶₂₆Fe to ²⁰⁸₈₂Pb to be exoergic, endoergic, or neither?

Answer 1

A. mass defect = 8(1.007825 + 1.008665)) - 15.994915

binding energy = 931.5 MeV/amu x 0.137005 amu = 127.6201 MeV

B. binding energy per nucleons = 127.6201MeV/16 nucleons = 7.9762 MeV/nucleons

C. Mass of protons and neutrons = 26(1.007825u) + 30(1.008665u) = 56.4634u
Mass of Iron 56 = 55.934939u
Mass Defect = 56.4634 - 55.934939 = 0.528461 u
Binding Energy = 0.528461* 931.5 MeV = 492.2614Mev
Binding energy per nucleon = 492.2614 / 56 = 8.7903 MeV/nucleon

D. Mass of protons and neutrons = 82(1.007825u) + 126(1.008665u) = 209.733444u
Mass of Pb = 207.976627 u
Mass Defect = 209.73344 - 207.976627 = 1.756813 u
Binding Energy = 1.756813* 931.5 MeV = 1636.4713Mev
Binding energy per nucleon = 1636.4713/208 = 7.8676MeV/nucleon

E. Binding energy per nucleonof product - Binding energy per nucleon of reactant = 8.7903 - 7.9762

= positive so the reaction will be endoergic.

F. Binding energy per nucleonof product - Binding energy per nucleon of reactant = 7.8676 - 8.7903

= negative so the reaction will be exoergic.