Exploring Decrease in Binding Energy per Nucleon in Heavy Nuclei Like U-235

[kihr]

Homework Statement

. Why does the binding energy per nucleon decrease with increase in mass number for heavy nuclei like 235
U?

Homework Equations

The loss in energy of the nucleons when they move from an unbound state to a bound state is the binding energy. The binding energy divided by the mass number gives the binding energy per nucleon.

The Attempt at a Solution

. Suppose individual neutrons and protons in their rest positions (i.e. when separated far away from the nucleus) move towards the nucleus. As they move closer, the potential energy of the protons increases because of the electrostatic force of repulsion between them. As the distance between the nucleons reduces to the point where they begin to be attracted by the strong nuclear forces, a negative potential energy develops due to the nuclear forces, which is higher than the potential energy due to the electrostatic forces between the protons. There is, therefore, a net decrease in the potential energy of all the nucleons This loss of energy when the nucleons come together in the nucleus is equal to the binding energy.
In the case of nuclei of high mass number, the electrostatic potential energy is higher due to the larger number of protons as compared to nuclei of a lower mass number, whereas the negative potential energy due to the nuclear forces remains unchanged as they are only very short range forces. Hence the net decrease in the potential energy of the nucleons in the nucleus is less for heavier nuclei. Since the loss of energy when the nucleons come together in the nucleus is less in this case, the binding energy per nucleon is less for heavier nuclei.


I would like to know whether the reasoning as above is correct. I am, however, a bit confused as a similar line of appraoch would, perhaps, not explain as to why the binding energy per nucleon of nuclei at the lower end is also less than that in the middle range. Is the net potential energy of the nucleons in the bound state negative since they are bound together by the nuclear attractive forces?
Would appreciate some help in clearing my doubts. Thanks.

 

[Jhero]

Your reasoning is correct. The decrease in binding energy per nucleon for heavy nuclei like 235U is due to the increase in electrostatic potential energy as compared to the short-range nuclear forces. This is because the number of protons in the nucleus increases with the mass number, resulting in a stronger repulsive force between them. As you correctly stated, the negative potential energy due to the nuclear forces remains unchanged, but it is not enough to offset the increase in electrostatic potential energy.

As for the lower end of the mass number, the decrease in binding energy per nucleon can be attributed to the fact that these nuclei have relatively few nucleons, resulting in a weaker overall binding force. This can also be explained by the fact that the nuclear forces are short-range, so as the number of nucleons increases, the overall binding force becomes stronger and the binding energy per nucleon increases.

To answer your question, the net potential energy of the nucleons in the bound state is indeed negative, as they are bound together by the attractive nuclear forces. This negative potential energy is what contributes to the overall binding energy of the nucleus.

I hope this helps to clear your doubts. Let me know if you have any further questions.



Scientist