Calculating Binding Energy for C13 Neutron Removal

[Krushnaraj Pandya]

Homework Statement

My textbook says the binding energy per nucleon is the energy needed to remove one nucleon from the nucleus, In a subsequent numerical which is as follows "The binding energy per nucleon for C12 is 7.68 MeV and that for C13 is 7.47 MeV. Calculate the energy required to remove a neutron from C13".The solution given is-"13 X 7.47 - 12 X 7.68 = 4.95 MeV".

Homework Equations

The Attempt at a Solution

According to the definition provided of binding energy per nucleon the answer should be just 7.47 MeV, instead they calculated the Q value- Why did they do that and what blunder am I making? I'd really appreciate your help, thank you.

 

[Drakkith]

My textbook says the binding energy per nucleon is the energy needed to remove one nucleon from the nucleus,

Yes, there does appear to be a discrepancy at first glance, doesn't there? Unfortunately you're the victim of a bad definition. A better definition would be that binding energy is the energy required to completely 'disassemble' the nucleus. So it's the energy required to separate all the nucleons, not just remove one.

 

[Krushnaraj Pandya]

Yes, there does appear to be a discrepancy at first glance, doesn't there? Unfortunately you're the victim of a bad definition. A better definition would be that binding energy is the energy required to completely 'disassemble' the nucleus. So it's the energy required to separate all the nucleons, not just remove one.

That was the definition given for binding energy.
Note that the definition I refer to is for the binding energy per nucleon. I know Binding energy is the energy required to separate all the nucleons (=Δmc^2),
what then is the correct definition of Binding energy per nucleon?

 

[Drakkith]

That was the definition given for binding energy.
Note that the definition I refer to is for the binding energy per nucleon. I know Binding energy is the energy required to separate all the nucleons (=Δmc^2),
what then is the correct definition of Binding energy per nucleon?

The energy per nucleon required to completely disassemble the nucleus.

 

[Krushnaraj Pandya]

The energy per nucleon required to completely disassemble the nucleus.

Still not clear to me, per nucleon as in we remove the nucleons one by one and add up all the energies required in each step? What's the difference between total binding energy then since it is also the energy required to disassemble the nucleus

 

[Drakkith]

Still not clear to me, per nucleon as in we remove the nucleons one by one and add up all the energies required in each step? What's the difference between total binding energy then since it is also the energy required to disassemble the nucleus

It doesn't matter if you remove the nucleons one at a time or all at once. The total energy remains the same. The difference between the total binding energy and the binding energy per nucleon is right in their names. Total binding energy is the total amount of energy required. The binding energy per nucleon is the total divided by the number of nucleons.

Note that the actual energy you would spend to remove successive nucleons, one at a time, changes as you remove more and more nucleons. Consider lithium-6, with a binding energy of around 5.3 MeV per nucleon. If you remove a proton and a neutron you get helium-4, with a binding energy of around 7.075 MeV per nucleon. Thus the energy spent to remove successive nucleons must change, otherwise every element would have identical binding energy per nucleon. That is, if it took 5.3 MeV to remove every single nucleon, one at a time, from a nucleus of lithium-6, then after removing a proton and a neutron our binding energy per nucleon should still be 5.3 MeV. But it's not. It has risen to around 7 MeV. That means it took less than 5.3 MeV to remove that first nucleon. The total binding energy of lithium-6 is around 31.8 MeV, while helium-4 is 28.3 MeV, for a difference between them of 3.5 MeV. So it actually took far less than 5.3 MeV to remove not just one, but two nucleons from lithium-6.

This is similar to removing electrons from their atomic orbitals. The first electron usually takes far less energy to remove than the next electron, which requires less energy to remove than the next electron, and so on... But the total energy required to remove the electrons from an atom is identical for all atoms of that type. We can take this total energy and divide by the number of electrons to get the binding energy per electron. While this doesn't tell us the amount of energy required to remove any specific electron, it immediately gives us an idea of how tightly bound the electrons are to the nucleus. An analogous situation occurs for nucleons and nuclei.

Does that make sense?

 

[SammyS]

The energy per nucleon required to completely disassemble the nucleus.

Still not clear to me, per nucleon as in we remove the nucleons one by one and add up all the energies required in each step? What's the difference between total binding energy then since it is also the energy required to disassemble the nucleus

Would it be clearer to say the following ?

The binding energy per nucleon is the average energy per nucleon that's required to completely disassemble the nucleus.​

.

 

[Drakkith]

Would it be clearer to say the following ?

The binding energy per nucleon is the average energy per nucleon that's required to completely disassemble the nucleus.​

.

Sure. I just don't talk good!

 

[SammyS]

Sure. I just don't talk good!

That made me chuckle !Your explanations were clear enough to me!

 

[Krushnaraj Pandya]

It doesn't matter if you remove the nucleons one at a time or all at once. The total energy remains the same. The difference between the total binding energy and the binding energy per nucleon is right in their names. Total binding energy is the total amount of energy required. The binding energy per nucleon is the total divided by the number of nucleons.

Note that the actual energy you would spend to remove successive nucleons, one at a time, changes as you remove more and more nucleons. Consider lithium-6, with a binding energy of around 5.3 MeV per nucleon. If you remove a proton and a neutron you get helium-4, with a binding energy of around 7.075 MeV per nucleon. Thus the energy spent to remove successive nucleons must change, otherwise every element would have identical binding energy per nucleon. That is, if it took 5.3 MeV to remove every single nucleon, one at a time, from a nucleus of lithium-6, then after removing a proton and a neutron our binding energy per nucleon should still be 5.3 MeV. But it's not. It has risen to around 7 MeV. That means it took less than 5.3 MeV to remove that first nucleon. The total binding energy of lithium-6 is around 31.8 MeV, while helium-4 is 28.3 MeV, for a difference between them of 3.5 MeV. So it actually took far less than 5.3 MeV to remove not just one, but two nucleons from lithium-6.

This is similar to removing electrons from their atomic orbitals. The first electron usually takes far less energy to remove than the next electron, which requires less energy to remove than the next electron, and so on... But the total energy required to remove the electrons from an atom is identical for all atoms of that type. We can take this total energy and divide by the number of electrons to get the binding energy per electron. While this doesn't tell us the amount of energy required to remove any specific electron, it immediately gives us an idea of how tightly bound the electrons are to the nucleus. An analogous situation occurs for nucleons and nuclei.

Does that make sense?

Ah, that makes things much clearer- thank you :D
I understand the definitions clearly now.

Would it be clearer to say the following ?

The binding energy per nucleon is the average energy per nucleon that's required to completely disassemble the nucleus.​

.

Thank you for clearing it up further,
the only thing I don't understand now is the conceptual reason behind why we calculate Q values to represent the energy (I'm always confused between whether the reactants have higher binding energy or the products in a fusion or fission reaction, or who has the higher rest mass etc. If there is an intuitive way to understand and keep track of the energetics I'd really appreciate it)

 

[Krushnaraj Pandya]

Also a question that has perpetually confused me and that several people have given me different answers to is whether binding energy is positive or negative- By definition it is the energy required to break apart a nucleus and therefore should be positive but in nuclear reactions I saw it being compared to potential energy, such that in the products it becomes more negative and thus results in stability and release of energy(=Q), I don't understand which logic to follow

 

[Drakkith]

Also a question that has perpetually confused me and that several people have given me different answers to is whether binding energy is positive or negative- By definition it is the energy required to break apart a nucleus and therefore should be positive but in nuclear reactions I saw it being compared to potential energy, such that in the products it becomes more negative and thus results in stability and release of energy(=Q), I don't understand which logic to follow

I believe binding energy is always positive by definition. But the difference in the binding energy between the reactants and products can be either positive or negative. It is this difference that determines whether or not you get a release of energy from a reaction.

 

[Krushnaraj Pandya]

I believe binding energy is always positive by definition. But the difference in the binding energy between the reactants and products can be either positive or negative. It is this difference that determines whether or not you get a release of energy from a reaction.

So total binding energy of products should be higher (therefore, BE(products)-BE(reactants) should be positive) since it'd require more energy to break them (i.e. more stable), am I correct?

 

[Drakkith]

So total binding energy of products should be higher (therefore, BE(products)-BE(reactants) should be positive) since it'd require more energy to break them (i.e. more stable), am I correct?

It depends on the reaction.

 

[Krushnaraj Pandya]

It depends on the reaction.

For a spontaneous one?

 

[Drakkith]

For a spontaneous one?

Then it will release energy, so the reactants have to have less binding energy than the products.

 

[Krushnaraj Pandya]

Thank you very much, Its crystal clear now :D