Find the difference in atomic mass between the two isotopes

[Ailiniel]

Homework Statement

Two isotopes of a certain element have binding energies that differ by 5.03 MeV. The isotope with the larger binding energy contains one more neutron than the other isotope. Find the difference in atomic mass between the two isotopes.

Homework Equations

1u=931.5MeV

The Attempt at a Solution

mass defect = m
c = speed of light
binding energy= mc^2
5.03 MeV = binding energy 1 - binding energy 2

I don't know where to go from here.

The solution manual says if you convert 5.03MeV to atomic mass unit (u) and then subtract that from the atomic mass unit of a neutron (1.008665u-0.00540u), then that will be your answer. I guess the real question here is that I can't relate how you get the difference in atomic mass by just subtracting from the neutron.

 

[ehild]

Think: What are isotopes?

ehild

 

[Ailiniel]

Isotopes differ by the number of nucleons but have the same number of protons. Ok, I'm starting to see it.

So, why do you subtract the nucleon atomic mass from the atomic mass difference? Isn't the difference in mass between two different isotopes already there? Or is it asking for the difference in mass between two same isotopes?

 

[ehild]

The atomic mass of an isotope differs from the mass of its its constituent nucleons by the binding energy/c².
The nucleus of the first isotope has Z protons and N neutrons, the second one consists of Z protons and N+1 neutrons. The mass of the nucleus is

M=Z m_p+N m_n-B/c²

The binding energy of the second one is higher, as the extra neutron is also bound to the nucleus. Of course, the mass is also greater because of that extra neutron.

Write the equations for the masses and subtract them to get the mass difference between the isotope nuclei.

ehild

 

[asteriatic]

To find the difference in atomic mass between the two isotopes, we need to use the equation E=mc^2, where E is the binding energy, m is the mass defect, and c is the speed of light. We can rearrange this equation to solve for the mass defect: m=E/c^2.

We are given that the binding energies of the two isotopes differ by 5.03 MeV. So, we can set up the following equation:

5.03 MeV = m1c^2 - m2c^2

We also know that the isotope with the larger binding energy contains one more neutron than the other. This means that the difference in binding energy is equal to the binding energy of one neutron, which is approximately 0.00540 u (provided in the solution manual).

Now, we can substitute this value into our equation:

5.03 MeV = m1c^2 - m2c^2
0.00540 u = m1 - m2

We can then use the conversion factor 1u=931.5 MeV to convert the binding energy (in MeV) to mass (in u). This gives us:

0.00540 u = m1 - m2
0.00540 u x (931.5 MeV/1u) = m1 - m2
5.03 MeV = m1 - m2

Now, we can equate the two equations and solve for the difference in atomic mass (m1 - m2):

m1 - m2 = 5.03 MeV

Therefore, the difference in atomic mass between the two isotopes is 5.03 MeV.