Isotope word question clarification

[opus]

Homework Statement

This is a multiple choice question. I have uploaded a screenshot of the question for reference.

The atomic mass for a particular isotope of an element will:

a) Always be equal to the value obtained by summing the masses of the protons and neutrons present in the atom.

b) Always be less than the masses obtained by summing the masses of the protons and neutrons present in the isotope.

c) Always be greater than the masses obtained by summing the masses of the protons and neutrons present in that isotope.

d) May be the same as, greater than, or less than the masses obtained by summing the masses of the protons and neutrons present in that isotope.

Homework Equations

The Attempt at a Solution

Maybe I don't fully understand the question, but I chose my D because isotopes can have greater masses or lesser masses compared to the atom they're an isotope of. For example, if Carbon loses neutrons, its mass will be less than the vanilla Carbon atom. If Carbon gains neutrons, its mass will be greater than the vanilla Carbon atom. I guess I don't full understand the scope of the question.
Would anyone mind giving me some guidance?https://www.physicsforums.com/attachments/231121 https://www.physicsforums.com/attachments/231121 https://www.physicsforums.com/attachments/231121

 

[Charles Link]

To a good approximation, the atomic mass of an atom (which includes any isotope of that atom) can be estimated by taking the mass of a proton and multiplying by the number of protons, and adding to this the mass of a neutron multiplied by the number of neutrons. Or similarly, just count the number of protons and neutrons and multiply by $1.67 \cdot 10^{-27} \, kg$, because the mass of the proton and neutron are quite similar. $\\$ Because the particles (protons and neutrons) in the atom are in a state of lower energy than they are as free particles, the mass of the atom experiences a loss of mass (it's not a large amount, but it's not zero), as the particles come together to form an atom. $\\$ Edit: The atomic mass of the carbon 12 atom is defined as 12. Thereby, depending on what kind of binding energy the atom has, the individual particles in an atom could on average have more or less binding energy than they do in the $C^{12}$ atom. (According to the answer that was provided, the question is referring to how the masses were determined in the second paragraph, and the info of $C^{12}$ is actually extraneous ).$\\$ With this info, you should be able to select the correct answer. $\\$ See also: https://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl

 

[opus]

Ok so, the sum of the masses of the nucleons is greater outside of the nucleus than it is inside the nucleus. But I don't understand how this applies. We are summing the masses of the nucleons while they're inside the isotope. So how can their masses be anything but that?

 

[Charles Link]

The mass of the atom is just slightly less than the masses of the particles that came together to make the atom. It is impossible to say exactly what the mass of a given proton or neutron is in the atom. As a collective lot, they have lost mass. In coming together, energy is released, and that energy $E=mc^2$ accounts for the loss of mass. The mass change is small, but it is not zero. $\\$ And the answer to (b) is not worded carefully, in that they should be more precise in saying the masses of the protons and neutrons "as determined prior to being in the isotope".

 

[opus]

Ok I understand that part (to the most minimal extent), as I think that's what the concept of nuclear fusion entails. I just couldn't see how that applied to this problem. I think I get the idea now though, although I have to say that I'm not a big fan of the question.
Thank you, Charles.