Nuclear Binding Energy: What Does it Really Mean?

[Precursor]

Binding energy of a nucleus is defined as the energy required to separate all of its protons and neutrons and move them infinitely far apart. Does that mean it requires an infinite amount of energy to liberate the nucleons and move them a distance infinitely apart? That doesn't make any sense because it's impossible to provide an infinite amount of energy. Or am I taking the definition to literal? Can someone please clarify this?

 

[Hunterbender]

Binding energy is (*surprise!*) the energy necc. to keep the molecule together. Or precisely, the total energy of a bound system (the nucleus) is less than the combined energy of the separate molecule is the binding energy. Each electron, proton, neutron has an energy equivalence. In addition, there is some energy used to keep these parts intact as one. (sort of like glue). But this force of attraction also is a form of energy, which we call binding energy.

*the statement "move them to infinitely apart" simply mean break them apart, NOT necc. moving them to the edge of the universe. In addition, the statement assumes that universe has no retarding force, so I guess...moving them across space won't take any additional energy.

Hope it helps

 

[heth]

Precursor> What do you know about inverse square laws? This might help.

 

[Precursor]

Precursor> What do you know about inverse square laws? This might help.

So, according to the inverse square law, there is always a force acting between charged particles, but the farther away from each other the weaker the force. But at an extremely far distance away, the force acting between the charged particles is negligible. Does that mean the binding energy is finite?

 

[Hunterbender]

I am going to read Heth's mind and see if I can continue where he's going.

So, as the distance increase, the force between 2 particles becomes less and less until it's negligible (as you said). Then the work necc. to keep them farther apart toward infinity also becomes negligible.

After all, binding energy (when you really go down into the basis) is just a force (strong force I believe). [there are 4 well known accepted forces: gravity, Electromagnetic, Strong, and weak (nuclear decay and stuff)]

 

[Precursor]

ISo, as the distance increase, the force between 2 particles becomes less and less until it's negligible (as you said). Then the work necc. to keep them farther apart toward infinity also becomes negligible.]

But doesn't the force between 2 particles become negligible only when they are at an infinite distance apart, which makes the work neessary to move them that far apart, an infinite amount of energy? This seems to be going in circles...

 

[Delphi51]

But doesn't the force between 2 particles become negligible only when they are at an infinite distance apart, which makes the work neessary to move them that far apart, an infinite amount of energy?

It appears that way, but instinct doesn't work well with non-linear things.
You really need calculus to figure it out. The work necessary is the integral (as r goes from initial value R to infinity) of F dr. If you have done integration, you'll find it easy. And finite.

 

[Precursor]

I think you've convinced me Delphi51. But it would be even better if you can show me the proof with integration, even though I have not done integration before . Thanks everyone.

 

[IPart]

Oh my God!

Precursor: this is a good question. The fallacy is in your assumption that the amount of energy required to separate the nucleons infinitely apart (which is a theoretical construct ..but can be physically understood as being some perfected realization of the situation where the nucleons are separated outside of each other's force fields) is itself infinite. This energy is still finite.

How do you understand this physically? Consider this example: suppose you roll down a ball on a frictionless surface ..following Newton's law, it will head off to infinity eventually. But you would have only expended finite energy. The point is that you should not assume that sending things off to infinity will require infinite energy.

Before I proceed - we are talking about NUCLEAR forces here ..in fact, Coulomb forces would cause repulsion, rather than cause the nucleons to be bound together.

The energy deficit can be equated to the MASS deficit through Einstein's equation.

As far as I can tell, the question you're asking has nothing to do with non-linear effects.

Once all this has sunk in I can tell you more about what really binding energy physically means then.

 

[Precursor]

IPart: Thanks a lot for that clarification. So you're trying to say that only a finite amount of energy is required to liberate the nucleons in the nucleus. In relation to the physical example you provided, are you trying to say that the liberated nucleons move to infinity by themselves? What causes that if that actually happens? And how are nuclear forces important here, as you were trying to explain?

When thinking of physics concepts, I tend to think about them "physically" as you mentioned. Your post really struck me for that. Is it incorrect to think of physics in a physical mindframe?

 

[heth]

Sorry - wasn't on the forum for a couple of days. What I was trying to allude to is that when you're calculating the energy (or escape velocity) required to escape to infinity from, say, the Earth's surface, the fact that the force drops off with distance means that you don't require an infinite amount of energy / have an infinite escape velocity. That's probably something you've calculated at some point?

Although the strong force (the force that binds protons and neutrons together in a nucleus) is not an inverse square force, it also drops off with distance. You can use the same argument for the non-infinite energy being required to separate protons / neutrons as you can for non-infinite energy being required to separate a spaceship from the Earth, even if the maths involved is different.

 

[Delphi51]

the fact that the force drops off with distance means that you don't require an infinite amount of energy / have an infinite escape velocity.

More to it than just dropping off. If the force dropped off as F = k/r, then the energy would be integral from r=R to infinity of k/r*dr = ln(infinity) - ln R = infinity.

The fact that it drops off as F = k/r^2 (what I meant by non-linear, where one's physical intuition is unreliable) that makes it come out to less than infinity: E = Integral from r=R to infinity of k/r^2*dr = k/R.