Question
The purpose of this problem is to show in three ways that the binding energy of the electron in a hydrogen atom is negligible compared with the masses of the proton and electron. (a) Calculate the mass equivalent in u of the 13.6-eV binding energy of an electron in a hydrogen atom, and compare this with the mass of the hydrogen atom obtained from Appendix A. (b) Subtract the mass of the proton given in Table 31.2 from the mass of the hydrogen atom given in Appendix A. You will find the difference is equal to the electron’s mass to three digits, implying the binding energy is small in comparison. (c) Take the ratio of the binding energy of the electron (13.6 eV) to the energy equivalent of the electron’s mass (0.511 MeV). (d) Discuss how your answers confirm the stated purpose of this problem.
1. $-6.9\times 10^{9}\%$
2. The difference is nearly the rest mass of a free electron, to four significant figures.
3. $2.7\times 10^{-5}$
4. Part (a) shows that the electron binding energy ($BE_e$) is negligible compared to the combined mass of a proton and electron. Part (b) shows that the difference in mass between a hydrogen atom and a proton is essentially the mass of an unbound electron, which indicates that the electron binding energy didn't really affect the bound electron mass in the atom. Part (c) directly compares $BE_e$ to $m_e$ and shows that $BE_e$ is smaller by 5 orders of magnitude.
Solution Video