Question

Rank the following lists from smallest on the left to largest on the right. If two entries have equal value, state equal value. (a) Zeff for the Li 1s, 2s, and 2p orbitals. (b) Zeff for the Li2+ 1s, 2s, and 2p orbitals. (c) r for the F 1s, 2s, and 2p orbitals. (d) |Zeff(2s) – Zeff(2p) for Li, F, and Ne. (e) For an H atom, the uncertainty of the e location, for the e in the 1s, 3p, or 4p state. (f) For an H atom, uncertainty of linear momentum, for the e- in the 1s, 3p, or 4p state. (g) For an H atom, the angular momentum, for the electron in the 1s, 3p, or 4p state. (h) For an H atom, uncertainty of angular momentum, for the e in the 1s, 3p, or 4p state. (i) The outermost electron mg value for a Li, Be, or B atom. (j) The maximum possible mi value for an electron in a As, P, or N atom. (k) The maximum possible number of radial nodes for an electron in a K, Na, or Al atom. (1) The maximum possible number of angular nodes for an electron in a K, Ti, or As atom.

Answer 1

Background for parts (a), (b), and (d)

Z_eff or effective nuclear charge is the net positive charge experienced by an electron. Basically it takes into account the repulsion offered by electrons in the lower shells.

Z_eff = Z - S; where Z is the atomic number or number of protons; while S is the average amount of electron density between the nucleus and the electron on which Z_eff is to be calculated.

S is different for different shaped orbitals. For instance, s-orbitals are better shielders than p-orbitals. Slater's rules state that s- and p-electrons of the same shell as that of electron for which Zeff is being calculated (i.e. electron in question) contribute 0.35 electrons. For shell which is one energy level lower than electron in question, the contribution in 0.85 and for shells which have a difference of 2 or more shells from the electron in question, the contribution of each electron is 1.

(a) For Li: Z=3

For its 1s orbital, S=0; therefore Zeff = 3-0 =3

For its 2s orbital, there are two 1s electrons between nucleus and 2s. Each will contribute 0.85; i.e. S=0.85 +0.85 = 1.7; therefore Zeff = 3-1.7 = 1.3

For its 2p orbital, contibution of two 1s electrons = 1.7 (as above)

contribution of one 2s electron = 0 (because in the neutral Li atom, only 2s electron can jump to 2p orbital.

Therefore Zeff = 3 - 1.7 = 1.3

Hence, Z_eff(for 2p) = Z_eff (for 2s) < Z_eff (for 1s)

(b) On a similar line as done in part (a) but keeping in mind that Li2+ has only one electron and there will be no shielding whether it is in 1s or 2s or 2p

Hence, Z_eff(for 2p) = Z_eff (for 2s) = Z_eff (for 1s)

(c) 'r' is the radial distribution function or the probability of finding an electron at a fixed distance from the nucleus. As the electron moves away from the nucleus, its energy increases, the size of orbitals also increases and the probability of finding it at a fixed distance from the nucleus must decrease.

Hence, r(2p) < r(2s) < r(1s)

(d) |Z_eff(2s) - Z_eff(2p)| for Li is zero

Z_eff(2s) for F = Z - S = 9 - (0.85+0.85) = 7.3

Z_eff (2p) for F = Z - S = 9 - (0.85+0.85+0.35) = 6.95

Therefore |Z_eff(2s) - Z_eff(2p)| for F = |7.3-6.95| = 0.35

Similiarly, for Ne: |Z_eff(2s) - Z_eff(2p)| = 0.35 (because atomic number cancels out in the overall calculation and only contribution of 2s electron makes the net difference)

Hence, |Z_eff(2s) - Z_eff(2p)| for Li < |Z_eff(2s) - Z_eff(2p)| for F = |Z_eff(2s) - Z_eff(2p)| for Ne

(e) The uncertainty of electron location increases with increasing distance from the nucleus

Hence, uncertainty of e- ( in 1s < in 3p < in 4p)

(f) The uncertainty of linear momentum and uncertainty in its position are related by Heisenberg's uncertainty principle. If one of them is accurate, the uncertainty in the other becomes more. Combining this fact with the reasoning in the part (e)

Hence, uncertainty in linear momentum for an electron in ( 4p < 3p < 1s)

(g) Angular momentum of an electron is given by the formula, mvr = nh/2π ( according to Bohr's theory )

Therefore, higher the principal quantum number, higher the angular momentum

Hence, Angular momentum for an electron in( 1s < 3p < 4p )

(h) The order remains the same as that of the part (f) which concerns linear momentum and for the same reason.

(i) m_s (spin quantum number) values or spin quantum number's values acquire +1/2 and -1/2 values. So their magnitude remains equal.

(j) m_l (magnetic quantum number) ranges from -l to +l and l is dependent on n which in turn is dependent on atomic number.

Therefore, the bigger the atom, the maximum possible value of m_l increases

Hence, the order is maximum possible value of m_l for (N < P < As)