How do I determine if a certain nuclear decay is allowed?

[Calleguld]

Hi, I am struggling with a question where they want me to determine whether or not three different decay are allowed.

From what I have understood all decays must follow a set of conservation law. These laws are:
1 Conservation of Baryon number
2 Conservation of Lepton number
3 Conservation of electric charge

This is very straight forward when you have simple decays like the decay of a neutron. Where you have:

n -> p+e+anti v_e

But how does it work for nucleons?

For example:

Thorium-222 -> Oxygen-16 + Lead-206

This decay is not allowed as thorium-222 only decays with alpha-decay. But as far as I can see the laws are still followed.

1: 222 -> 16 + 206 = 222
2: 90 -> 8 + 82 = 90
3: 90-90 -> 8-8 + 82-82 = 0

What am I missing? please help!

 

[DuckAmuck]

There's another requirement.
4. Mass+energy must be conserved.

Does Oxygen-16 + Lead-206 have more mass than Thorium-222? If so, then the decay is not allowed.

 

[Calleguld]

There's another requirement.
4. Mass+energy must be conserved.

Does Oxygen-16 + Lead-206 have more mass than Thorium-222? If so, then the decay is not allowed.

I forgot to add that law.

The mass difference is: 222.018468u - (205.974465u + 15.994914u) = 0.0491u

Which would suggest that this decay is allowed.

Thanks for the answer though!

 

[DuckAmuck]

Are you taking into account the difference in mass between neutrons and protons? Recall that neutrons are a bit heavier than protons.

 

[Calleguld]

Are you taking into account the difference in mass between neutrons and protons? Recall that neutrons are a bit heavier than protons.

Yes that difference is accounted for. I got the masses from this site/paper https://www-nds.iaea.org/amdc/ame2012/mass.mas12