The iron isotope stife is near the peak of the stability...

The iron isotope stife is near the peak of the stability curve. That is the fundamental reason that iron is more common in the Universe than heavier elements, as the spectra of the Sun and of many other stars reveal. Show that $\$ 6$ has a higher binding energy per nucleon than its neighbors $\$ 0 \mathrm{Mn}$ and $\$ 9 \mathrm{Co}$ . State how your results compare with Figure $44.5 .$

Key Concepts

Binding Energy per Nucleon

Binding energy per nucleon is a measure of the stability of a nucleus, representing the average energy that would be required to remove each nucleon from the nucleus. A higher binding energy per nucleon indicates a more tightly bound and stable nucleus. In this context, comparing iron’s binding energy per nucleon with those of its neighboring nuclei illustrates why iron sits at the maximum of the stability curve, making it more energetically favorable and common in stellar environments.

Nuclear Stability Curve

The nuclear stability curve, often depicted as a plot of binding energy per nucleon versus mass number, reveals a peak around mid-mass numbers, typically near iron. This curve illustrates that nuclei lighter or heavier than this peak are relatively less stable because they have lower binding energies per nucleon. This stability peak explains why fusion processes in stars tend to produce heavier elements up to iron, beyond which energy absorption occurs during fusion, making further fusion processes unfavorable.

Stellar Nucleosynthesis and Element Abundance

In stellar nucleosynthesis, the processes inside stars build up elements through nuclear reactions, with the highest concentration of stable isotopes around the iron peak. The higher binding energy per nucleon of iron compared to its neighbors means that once a star forms iron in its core, further fusion into heavier elements no longer releases energy. This fundamental nuclear property accounts for the observed cosmic abundances of iron and helps to explain why elements heavier than iron are less commonly produced.

Transcript

00:01 In this question we basically want to look at the binding energy per nucleon.

00:06 For iron, 56, as well as its neighbours, which is manganese, as well as cobot.

00:18 So for manganese is 55, coboy is 59.

00:25 To find the binding energy per new clone, so our b .e.

00:32 We have to first find the binding energy, which is, given as the change in mass multiplied by c squared right we change in mass being the mass of the constituents minus away the final mass of the nuclei or mass of the final nuclei that is formed multiply this by c square right this is based on einstein's mass energy relation e equals to mc square so this actually tells us what amount of energy is actually converted from the masses from its initial constituents and this is converted into energy in terms of binding energy right so we then divide this by the number of nucleons a to get the binding energy per new clon for the iron 56 you can first calculate that by taking the initial constituents we have 26 protons so we take 26 multiplied by the mass of the protons this is in terms of atomic mass units then we add the mass of the neutrons right so for iron 56 we have 30 neutrons minus away the mass of the final nuclei which is a 55 934 then we take this multiplied by c square so multiply by c square where c square is given as 931 .5 m e v per u since we are dealing with atomic mass units u so we can get the binding energy in terms of m e v very quickly then we divide everything by the nucleon number which is 56.

03:07 And we should get 8 .765 m e .v per nucleon.

03:21 Next we move on to the neighbors of iron, which is manganese and co -bots.

03:35 Again we do the same process, we take the mass difference.

03:39 So in this case for magnesium we have 25 protons and we have 30 neutrons.

03:50 Ones, minus away the final mass of the nuclei, which is the manganese, which is 24 .938 050, multiplied by c squared, divided by the total number of nucleons, which is 55.

04:23 Then we get about 8 .7, sorry, this is 8 .765.

04:36 For the iron, it shouldn't be 8 .765, but it is actually 8 .790.

04:44 I apologize for that...

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