Chap13 soln

Physical Chemistry

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13 Atomic structure and atomic spectra Solutions to exercises Discussion questions E13.1(b) (1) The principal quantum number, n , determines the energy of a hydrogenic atomic orbital through eqn 13.13. (2) The azimuthal quantum number, l , determines the magnitude of the angular momentum of a hydrogenic atomic orbital through the relation { l(l + 1 ) } 1 / 2 ¯ h . (3) The magnetic quantum number, m l , determines the z-component of the angular momentum of a hydrogenic orbital through the relation m l ¯ h . (4) The spin quantum number, s , determines the magnitude of the spin angular momentum through the relation { s(s + 1 ) } 1 / 2 ¯ h . For a hydrogenic atomic orbitals, s can only be 1 / 2 . (5) The spin quantum number, m s , determines the z-component of the spin angular momentum through the relation m s ¯ h . For hydrogenic atomic orbitals, m s can only be ± 1 / 2. E13.2(b) (a) A boundary surface for a hydrogenic orbital is drawn so as to contain most (say 90%) of the probability density of an electron in that orbital. Its shape varies from orbital to orbital because the electron density distribution is different for different orbitals. (b) The radial distribution function gives the probability that the electron will be found anywhere within a shell of radius r around the nucleus. It gives a better picture of where the electron is likely to be found with respect to the nucleus than the probability density which is the square of the wavefunction. E13.3(b) The first ionization energies increase markedly from Li to Be, decrease slightly from Be to B, again increase markedly from B to N, again decrease slightly from N to O, and finally increase markedly from N to Ne. The general trend is an overall increase of I 1 with atomic number across the period. That is to be expected since the principal quantum number (electron shell) of the outer electron remains the same, while its attraction to the nucleus increases. The slight decrease from Be to B is a reflection of the outer electron being in a higher energy subshell (larger l value) in B than in Be. The slight decrease from N to O is due to the half-filled subshell effect; half-filled sub-shells have increased stability. O has one electron outside of the half-filled p subshell and that electron must pair with another resulting in strong electron–electron repulsions between them. E13.4(b) An electron has a magnetic moment and magnetic field due to its orbital angular momentum. It also has a magnetic moment and magnetic field due to its spin angular momentum. There is an interaction energy between magnetic moments and magnetic fields. That between the spin magnetic moment and the magnetic field generated by the orbital motion is called spin–orbit coupling. The energy of interaction is proportional to the scalar product of the two vectors representing the spin and orbital angular momenta and hence depends upon the orientation of the two vectors. See Fig. 13.29. The total angular momentum of an electron in an atom is the vector sum of the orbital and spin angular momenta as illustrated in Fig. 13.30 and expressed in eqn 13.46. The spin–orbit coupling results in
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