This problem demonstrates that the binding energy of the electron in the ground state of a hydrogen atom is much smaller than the rest mass energies of the proton and electron. (a) Calculate the mass equivalent in u of the \(13.6-\mathrm{eV}\) binding energy of an electron in a hydrogen atom, and compare this with the known mass of the hydrogen atom. (b) Subtract the known mass of the proton from the known mass of the hydrogen atom. (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 \(\mathrm{MeV}\) ). (d) Discuss how your answers confirm the stated purpose of this problem.

First, we need to convert the given electron binding energy (13.6 eV) to atomic mass units (u). To do this, we will need the conversion factors: 1 u = 931.5 MeV 1 eV = 1 x 10^{-6} MeV The mass equivalent in u of the 13.6 eV binding energy can be calculated using these conversion factors: Mass equivalent (u) = binding energy (MeV) / 931.5 MeV/u Binding energy (MeV) = 13.6 eV * 1 x 10^{-6} MeV/eV ≈ 1.36 x 10^{-5} MeV Mass equivalent (u) ≈ 1.36 x 10^{-5} MeV / 931.5 MeV/u ≈ 1.46 x 10^{-8} u. Now, we can compare this calculated mass equivalent with the known mass of the hydrogen atom: Mass of hydrogen atom = 1.007825 u

Next, we will subtract the known mass of the proton from the known mass of the hydrogen atom to find the difference: Mass difference = Mass of hydrogen atom - Mass of proton Mass of proton = 1.007276 u Mass difference ≈ 1.007825 u - 1.007276 u ≈ 0.000549 u

Now, we will find the ratio of the given electron binding energy (13.6 eV) to the energy equivalent of the electron's mass (0.511 MeV): Ratio = binding energy (eV) / energy equivalent of the electron's mass (eV) Energy equivalent of electron's mass = 0.511 MeV * 1 x 10^6 eV/MeV = 511000 eV Ratio ≈ 13.6 eV / 511000 eV ≈ 2.66 x 10^{-5}

The results obtained in the previous steps demonstrate that the binding energy of the electron in a hydrogen atom is much smaller than the rest mass energies of the proton and electron. The mass equivalent of the binding energy is approximately 1.46 x 10^{-8} u, which is significantly smaller than the mass of the hydrogen atom (1.007825 u) and the mass of the proton (1.007276 u). The mass difference between the hydrogen atom and the proton is 0.000549 u, which is still much larger than the mass equivalent of the binding energy. Furthermore, the ratio of the binding energy (13.6 eV) to the energy equivalent of the electron's mass (511,000 eV) is 2.66 x 10^{-5}, indicating that the binding energy is much smaller than the electron's rest mass energy. In conclusion, this exercise successfully demonstrates that the binding energy of the electron in the ground state of a hydrogen atom is much smaller when compared to the rest mass energies of the proton and electron.