Q29 of 29 Page 79

The Bohr model for the H-atom relies on the Coulomb’s law of electrostatics. Coulomb’s law has not directly been verified for very short distances of the order of angstroms. Supposing Coulomb’s law between two opposite charge +q1, –q2 is modified to



Calculate in such a case, the ground state energy of a H-atom, if ε = 0.1, R0 = 1Å.


For



And we know by Bohr angular momentum formula,


-------(1) where


Now, the centripetal force keeps the electron revolving around the nucleus, ∴



Now putting the values, we get,



This is the radius of the orbit of the electron in ground state


∴


∴


On putting the values, we get,


Now for P.E. from to r



On putting the values, we get,


P.E.=


For modified case,


total energy at ground state =


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25

Deuterium was discovered in 1932 by Harold Urey by measuring the small change in wavelength for a particular transition in 1H and 2H. This is because, the wavelength of transition depend to a certain extent on the nuclear mass. If nuclear motion is taken into account then the electrons and nucleus revolve around their common center of mass. Such a system is equivalent to a single particle with a reduced mass μ, revolving around the nucleus at a distance equal to the electron-nucleus separation. Here μ = meM/ (me + M) where M is the nuclear mass and me is the electronic mass. Estimate the percentage difference in wavelength for the 1st line of the Lyman series in 1H and 2H. (Mass of 1H nucleus is 1.6725 × 10–27 kg, Mass of 2H nucleus is 3.3374 × 10–27 kg, Mass of electron = 9.109 × 10-31 kg.)

 26

If a proton had a radius R and the charge was uniformly distributed, calculate using Bohr theory, the ground state energy of a H-atom when (i) R = 0.1Å, and (ii) R = 10 Å.

 27

In the Auger process an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an n = 4 Auger electron emitted by Chromium by absorbing the energy from a n = 2 to n = 1 transition.

 28

The inverse square law in electrostatics is for the force between an electron and a proton. The dependence of |F| can be understood in quantum theory as being due to the fact that the ‘particle’ of light (photon) is massless. If photons had a mass mp, force would be modified to where and .

Estimate the change in the ground state energy of a H-atom if mp were 10–6 times the mass of an electron.