Nuclear Binding Energy and Decay Constant

[alzaeem]

Homework Statement

I have my GCSE A level Board exam in three days and i am a bit confused about this topic, there is a usual question where they give you data about the energy released when a nucleus decays, and they ask you to suggest with a reason which one has a greater binding energy, or greater decays constant.

question 1:
the beta decay of Strontium produces Yittrium and releases 5.5MeV of energy, suggest which nucleus (Sr or Y) has the greater binding energy.

question 2:
two isotopes of Radium (224 and 226) undergo alpha-particle decay, the energy of their a-particles is 5.68MeV and 4.78MeV respectively, suggest which of the two isotopes has the greater decay constant.

so could anyone tell me exactly how to go about answering these questions, and pls don't give me a link to a wikipedia page, i just need a quick clarification.
thanks a lot.

 

[malawi_glenn]

here we don't give answers if you don't show us your work/thoughts.

answer these questions for us:
1. What is the mass of a nucleus, and what is the bining energy of a nucleus?
2. what is decay, and why does it take place?
3. What is the relationship between decay-probability and energy released in alpa decay?
4. What does the "decay constant" represent?

the purpose of these forums is to help people become better scientists, in making them think logical and scientific. Not just give the right answers like a machine..

 

[alzaeem]

to answer your questions from what i know:

1. mass of a nucleus is the sum of the masses of its nucleons minus the mass defect.
2. decays takes place to make the nucleus more stable as the product will have a higher binding energy per nucleon.
3. i have no idea
4. decay constant is the probability of decay of a nucleus per unit time.

as a matter of fact i do know the answers to my questions, it is the why that i don't get. the correct answers are Yittrium for question 1 and Radium 224 for question 2. What confuses me the most is why the products have a higher binding energy, i mean if the prduct has higher binding energy then shouldn't energy be supplied instead of released in a nuclear decay/fission/fusion?

 

[malawi_glenn]

you have the right answers on my questions too, except that #3 is that the higher energy of alpha particle, the higher decay prob. It has to do with the tunnel effect in the quantum mechanical view on alpha decay.

The mass defect...

The mass of a nucleus is mass of its constituents minus the binnding energy of the nucleus. --> M(A,Z) = Zm_p + (A-Z)m_n - Bind_energy

Decay takes place if the daugher nuclei is more stable, in other word have more bindig energy per nucleon.

In the case for beta decay, the # of nucleons is the same, so what nuclei do you think have the most binding energy?

Just put up an equation:

E_f = E_i =>

Z1m_p + (A-Z)1m_n - Bind_energy1 = Z2m_p + (A-Z)2m_n - Bind_energy2 + Q + m_e

{this is a bit simplification! , we need to take account for the difference in neutron and proton + electron mass, but this works fenomenological!. Q is the energy released in the reaction}

We see that the masses of the parent and daugher nucleis constituents are the same (remember, A isunaffected in beta decay), and left is only:

Bind_energy2 - Bind_energy1 = Q

if Q is positve (has to be, otherwise the decay does not take place)

Which Bind_energy is biggest?

What "happens" is that mass is transformed into binding energy, so when something gets more binding energy, it is the same way of saying that the mass gets smaller. And if the mass gets smaller, the energy must go somewere. And that is the energy released in the decay, Q-value.

 

[alzaeem]

so what you are saying is that Binding energy is not energy stored in the nucleus, it is just the lost mass converted to energy and that this energy is released in the form of heat, so the more energy released the greater is the binding energy, did i get it correctly?
and thanks a lot for your great explanation.

 

[malawi_glenn]

well in a very approximative way we can say so...

take a look again on the definition of nuclear mass and binding energy, and try to find out how the things are related.