Question 29.2: The Binding Energy of the Deuteron Goal Calculate the bindin...
The Binding Energy of the Deuteron
Goal Calculate the binding energy of a nucleus.
Problem The nucleus of the deuterium atom, called the deuteron, consists of a proton and a neutron. Calculate the deuteron’s binding energy in MeV, given that its atomic mass-that is, the mass of a deuterium nucleus plus an electron-is 2.014102 \mathrm{u}.
Strategy Calculate the sum of the masses of the individual particles and subtract the mass of the combined particle. The masses of the neutral atoms can be used instead of the nuclei because the electron masses cancel. Use the values from Table 29.4 or Table B of the appendix. The mass of an atom given in Appendix B includes the mass of Z electrons, where Z is the atom’s atomic number.
To find the binding energy, first sum the masses of the hydrogen atom and neutron and subtract the mass of the deuteron:
\begin{aligned} \Delta m & =\left(m_{p}+m_{n}\right)-m_{d} \\ & =(1.007825 \mathrm{u}+1.008665 \mathrm{u})-2.014102 \mathrm{u} \\ & =0.002388 \mathrm{u} \end{aligned}
Convert this mass difference to its equivalent in MeV:
E_{b}=(0.002388 \mathrm{u}) \frac{931.5 \mathrm{MeV}}{1 \mathrm{u}}=2.224 \mathrm{MeV}
Remarks This result tells us that to separate a deuteron into a proton and a neutron, it’s necessary to add 2.224 MeV of energy to the deuteron to overcome the attractive nuclear force between the proton and the neutron. One way of supplying the deuteron with this energy is by bombarding it with energetic particles.
If the binding energy of a nucleus were zero, the nucleus would separate into its constituent protons and neutrons without the addition of any energy; that is, it would spontaneously break apart.