The binding energy per nucleon of tritium

[Redwaves]

Homework Statement

The binding energy per nucleon of the $^3 He$ nucleus is 2.6 MeV. Deduce the binding energy per nucleon of tritium from the following information: (1) The mass energies of isolated protons, neutrons and electrons are 938.3 MeV, 939,6 MeV and 0.5 MeV. (2) Tritium decays by the reaction $^3 H ---> ^3 He + e + v$, where kinetic energy of the final particles is 0.0186 MeV. (3) the mass energy of the antineutrino v is negligible on the MeV scale.

Relevant Equations

$E_i = E_f$
$E^2 = \vec{P}^2 c^2 + m^2 c^4$
$E = \gamma mc^2$

Hi,
I know from conservation of energy that $E_i = E_f$
Thus, $M_h c^2 + binding energy = M_{he}c^2+ M_e c^2 + K = M_{he}c^2 + 0.5 MeV + 0.0186 MeV$
If I'm right I have to find $M_{he}c^2$, but something is missing in my understanding, since I don't see how to find that mass energy $M_{he}c^2$.

Is the binding energy the difference between initial energy and final energy?
Is the mass energy of an atom the sum of all the neutrons, proton and electrons and the binding energy?

 

[haruspex]

Is the mass energy of an atom the sum of all the neutrons, proton and electrons and the binding energy?

It relates to all those, but not quite in that way.
Can you define the binding energy?
Hint: it would be better termed the unbinding energy.

 

[Redwaves]

It relates to all those, but not quite in that way.
Can you define the binding energy?
Hint: it would be better termed the unbinding energy.

You are right, but it uses binding in the book.
the unbinding energy is the amount of energy needed to "break" the tritium into the new particles.

 

[haruspex]

You are right, but it uses binding in the book.
the unbinding energy is the amount of energy needed to "break" the tritium into the new particles.

Yes, but that also applies to the helium, right?
So what is the answer to your question that I quoted in post 2?

 

[Redwaves]

Yes, but I don't know the way they are related.
Maybe I wasn't clear, but this is the essence of my question.
I though the mass energy was the sum of all the nucleons, electrons and the binding energy of all nucleons and electrons as well.
I know from the statement that the binding energy for the nucleons of helium is $2.6 \cdot 3$

Is $E_{he} = M_{he}c^2 + (2.6 \cdot 3) + K$ ?

 

[haruspex]

I though the mass energy was the sum of all the nucleons, electrons and the binding energy of all nucleons and electrons as well.

No, you are still getting it backwards.
If the mass energy of an atom is $E_m$ and its binding energy is $E_b$ then you have to add energy $E_b$ to disassemble it into separate nucleons. So if the energy of that collection of nucleons is $E_n$, what equation relates those three energies?
Apply that to both the $^3H$ and the $^3He$.

 

[Redwaves]

What's the difference between $E_m$ and $E_n$. The mass energy is not the energy of all the nucleons ?

$M_{he}c^2 = 2P + 1N - (2.6 \cdot 3) = 2808.4 MeV$
$E_0 = 2808.4 + K + 0.5 + k = 2808.94$
$E_0 = M_H c^2 = 2N + 1P - E_b = 2808.94$
$E_B = 8.56$
$8.56 / 3 = 2.85 MeV$

It sound really good, but I'm not sure to understand why the mass energy is the sum of all nucleons - binding energy.

 

[haruspex]

What's the difference between $E_m$ and $E_n$. The mass energy is not the energy of all the nucleons ?

$M_{he}c^2 = 2P + 1N - (2.6 \cdot 3) = 2808.4 MeV$
$E_0 = 2808.4 + K + 0.5 + k = 2808.94$
$E_0 = M_H c^2 = 2N + 1P - E_b = 2808.94$
$E_B = 8.56$
$8.56 / 3 = 2.85 MeV$

It sound really good, but I'm not sure to understand why the mass energy is the sum of all nucleons - binding energy.

Maybe I am using the terminology incorrectly. By mass energy of an atom or nucleon of mass m I mean $mc^2$.
The mass of an atom is less than the sum of the masses of its nucleons. So its mass energy is less than the sum of the mass energies of its nucleons. The difference is the binding energy.

 

[Redwaves]

The difference is the binding energy.

Is this because it takes that energy to bind the nucleons together. Does it makes sense?

 

[haruspex]

Is this because it takes that energy to bind the nucleons together. Does it makes sense?

No, you still have it backwards!
The binding energy is energy released as a result of the nucleons binding to form the atom. That is why the atom has less mass than the sum of the nucleons that went into making it. To break the atom up again you have to add back the binding energy.

 

[Redwaves]

Ah ! I see. Thanks a lot for your help.