Why Does Calculating Nuclear Binding Energy Result in Negative Q-Values?

In summary, the conversation discusses the calculation of binding energy for Thorium and Radon using a given equation. The student initially gets a negative value for the Q-value, but after clarification, realizes their mistake and correctly calculates a positive value. The Q-value is defined as the energy released in a process and is calculated by taking the difference between the initial and final binding energies. The conversation also explains the relationship between binding energy and mass, and how it is used in the calculation of Q-value.
  • #1
Brewer
212
0

Homework Statement


Question-1.jpg



Homework Equations


Given in the question


The Attempt at a Solution


Using the numbers and equation given in the question, I got a binding energy of Thorium of 1743.64MeV, and a binding energy of Radon (?) of 1720.88MeV

I've come across a number of this style question before (most notably in the last piece of coursework I handed in before Christmas and haven't gotten back to check feedback for), and I always approach the question in the same way.

First I plug the various numbers into the equation to get the binding energies of the atoms. Then I say that the binding energy of Th is equal to that of Ra and the alpha particle, and the difference between the two will be the energy released in the decay. However when I set about the question this way I always get a negative value of Q, and I can't see why that is. Surely if the question states that energy is released I'm expecting a positive answer? I seem to remember a peer saying something about why its negative when we were doing the coursework I previously mentioned, but I didn't catch it properly and I can't remember what was said anyway.

If any of you could explain this to me (in fairly simple terms) then I would appreciate it. If the answer is not supposed to be negative then if you could point me in the direction of the correct answer then that would be appreciated as well.

Thanks in advance for any help.

Brewer
 
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  • #2
Can you show us how you calculated your Q-value? So we can say what you did wrong or if you did right.

But I can say that your Q.values look okay, by comparing to experimental ones.

The Q-value is defined as: Binding energy final - binding energy initial, I think your misstakes lies there.
 
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  • #3
That makes sense I think. I think I was just doing it backwards. I was saying that:

B(Th) = B(Ra) + B(alpha) + Q, which I can now see is the reverse of what you said. Can you explain why this is like this, or is it just one of those things that's defined like that, and so is that with no questions asked (much like how I perceive the delta function!)

Thank you for your help though.
 
  • #4
As a result that leaves me with an answer of Q = 5.54MeV. Which is just the modulus of the answer I was getting before.
 
  • #5
Well the Q-value is defined as the energy relased in a process.

[tex] Q = \sum M_{initial} - \sum M_{final} [/tex]

And we know that Mass = energy, if the mass after the reaction is less then initial mass: energy is realseased (Q is positive).

Now the the binding energy is defined as the difference between the total mass of all parts and the total nucleus (Z,A):
[tex] B(Z,A) = Z\cdot m_p + (A-Z)\cdot m_n - M(Z,A) [/tex]

In the reaction for the Q value you just put this into it and viola:
let the reaction be (Z,A) to (Z-2,A-4) + (2,4)

[tex] Q = M(Z,A) -(M(Z-2,A-4) + M(2.4)) = Z\cdot m_p + (A-Z)\cdot m_n -B(Z,A) - ((Z-2)\cdot m_p + (A-4-(Z-2))\cdot m_n -B(Z-2,A-4) + [/tex]
[tex] 2\cdot m_p + 2\cdot m_n - B(2,4)) = \sum B_{final} - \sum B_{initial} [/tex]
 
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FAQ: Why Does Calculating Nuclear Binding Energy Result in Negative Q-Values?

1. What is nuclear binding energy?

Nuclear binding energy is the amount of energy required to hold the nucleus of an atom together. It is the energy that is released when a nucleus is formed from its individual protons and neutrons.

2. How is nuclear binding energy calculated?

Nuclear binding energy is calculated using Einstein's famous equation, E=mc², where E is energy, m is mass, and c is the speed of light. To determine the nuclear binding energy, the mass of the individual protons and neutrons is compared to the mass of the nucleus as a whole.

3. Why is nuclear binding energy important?

Nuclear binding energy is important because it is what holds the nucleus of an atom together. Without it, the nucleus would break apart, and the atom would no longer be stable. It is also a crucial factor in nuclear reactions and the production of energy.

4. How does nuclear binding energy affect an atom's stability?

Higher nuclear binding energy results in a more stable atom. When the nucleus has a high binding energy, it is harder for the atom to lose its structure and become unstable. This is why elements with larger atomic numbers generally have a higher nuclear binding energy and are more stable.

5. How does nuclear binding energy relate to nuclear power?

Nuclear binding energy is the source of energy in nuclear power plants. When the nucleus of an atom is split through nuclear fission, a large amount of binding energy is released as heat and used to generate electricity. The higher the nuclear binding energy of an atom, the more energy it can produce through nuclear reactions.

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