Shielding effect and Slater's rules

What is shielding effect?

Valence electrons for a multi-electron atom are attracted by the nucleus of an atom and repelled by the electrons from inner-shells. Slater's rules used for calculating shielding.
Shielding effect and Slater's rules
Shielding electrons and Slater's rules
The combined effect of this attractive and repulsive force acting on the valence electrons is that the Valence electrons experience less attraction from the nucleus of an atom. This is known as the shielding effect.

Slater's set some empirical rules to calculate the shielding or screening of various electrons present in different orbitals of an atom or an ion.
Once we get the value of screening constant it is easy enough to find an effective nuclear charge (Zeff).

Shielding electrons in nS, nP level

Slater's rules are applicable for calculating screening constant and Z effective of an electron in atom, ions or molecule and the rules are
  1. The first to do is to write out the electronic configuration of elements of the atom or ion in the following order and grouping.
    (1S) (2S, 2P) (3S, 3P) (3d) (4S, 4P) (4d) (4f) (5S, 5P) etc.
    It may be noted that so far as the shielding effect is concerned the S and P electron belonging to the same principal quantum shell have the same effect as advocated by Slater.
    (1S)²(2S, 2P)⁸(3S)¹
  2. Electron in a certain nS, nP level is screened only by electrons at the same level and by the electrons of lower energy levels of an atom.
    Electrons lying above nS, nP level do not screen any nS, nP electron to any extent.
  3. Electrons of higher energy levels of an atom no shielding effect on any lower-energy level of an atom.
    Thus for calculating the screening constant of the valence electron of a sodium atom, the electronic configuration of the sodium atom is
    (1S)² (2S, 2P)⁸ (3S)¹
    Calculation of the screening constant, the valence electron of an atom will be excluded from our calculation.
    Thus for calculating the screening constant of the 2P electron of the sodium atom, one 2P electron, and one 3S electron will be excluded from our calculation.
  4. Electrons lying nS and nP atomic level shield a valence electron in the same group by 0.35 each. This is also true for the electrons of the nd or nf atomic level of atom or ion.
  5. Electrons belonging to one lower quantum shell, that is (n-1) shell shield the valence electron by 0.85 each.
  6. Electrons belonging to (n-2) or still lower quantum shell shields the valence electron by 1.0 each. This means the shielding effect is complete.
Shielding effect in nS, nP subshell of an atom
Shielding effect in nS, nP subshell

Effective nuclear charge of sodium atom

Screening constant (σ) = (2 × 1) + (8 × 0.85) + (0 × 0.35)
= 8.8.

∴ Z effective of sodium atom = (11 - 8.8)
= 2.2.

Question
Calculate the effective nuclear charge of the hydrogen atom.

Answer
The hydrogen atom has a single 1S valence electron. There is no other electron to screen it from the nuclear charge of a single proton.
Thus, σ = 0 and Zeff = 1.0 - 0 = 1.0.

Thus hydrogen electron sees the full nuclear charge of the nucleus of an atom, that is the electron is totally exposed to the proton.

Sodium, potassium, and magnesium ion

Electronic configuration of sodium (Na⁺) ion
(1S)²(2S, 2P)⁸
Screening constant of sodium ion
= (2 × 0.85) + (8 × 0.35)
= 4.5.
Effective nuclear charge of sodium ion
= (11 - 4.5)
= 4.5.

Electronic configuration of potassium ion
(1S)²(2S, 2P)⁸(3S, 3P)⁸
Screening constant (σ) for potassium ion = (2× 1) + (8 × 0.85) + (8 × 0.35)
= 11.6.
Effective nuclear charge of potassium ion
= (19 - 11.6)
= 7.40.

Electronic configuration of magnesium (Mg⁺²) ion
(1S)²(2S, 2P)⁸
Screening constant of magnesium ion
= (2 × 0.85) + (8 × 0.85)
= 4.50.
Effective nuclear charge of Magnesium ion
= (12 - 4.50)
= 7.50.

Valence electron of fluorine and fluoride ion

Electronic configuration of fluorine atom
(1S)²(2S, 2P)⁷

Screening constant (σ) of fluorine atom
= (2 × 0.85) + (6 × 0.35)
= 3.80.

Effective nuclear charge for valence electron of fluorine
= (9 - 3.8)
= 5.20.

Electronic configuration of fluoride ion
(1S)²(2S, 2P)⁸

Screening constant (σ) of  fluoride ion
= (2 × 0.85) + (8 × 0.35)
= 4.50.

Effective nuclear charge of fluoride ion
= (9 - 4.50)
= 4.50.
Question
Comment on the variation in effective nuclear charge for a 2P electron from carbon to oxygen.

Answer
Electronic distribution according to the Slater's rule is:
Carbon (1S)² (2S, 2P)⁴
Nitrogen (1S)² (2S, 2P)⁵
Oxygen (1S)² (2S, 2P)⁶
In carbon, the 2P electron is screened by 1S² 2S² 2P¹ electrons while in nitrogen and oxygen this is done by 1S² 2S² 2P² and 1S² 2S² 2P³ electrons respectively.

Zeff of nitrogen = Zeff of carbon + (1 nuclear charge) - shielding due to one 2P electron
Zeff of nitrogen = Zeff of carbon + 1 - 0.35
= Zeff of carbon + 0.65 and Zeff of oxygen
= Zeff of nitrogen + 0.65

Thus effective nuclear charge will go up by the same amount from carbon to nitrogen and then to oxygen.

Shielding electrons in nd, nf level

The above rules are quite well for estimating the screening constant of S and P orbitals. However when d subshell or f subshell is being shielded the four and five rule replaced by rules for estimation of screening constant.
The replaced rules are, all electrons below the nd subshell or nf subshell contribute 1.0 each towards the screening constant.
Shielding effect in nd, nf subshell of an atom
Shielding effect in nd, nf subshell

4S electron of vanadium and 3d electron of vanadium

Vanadium has atomic number 23 and the electronic configuration according to the Slater's rules is,
(1S)² (2S, 2P)⁸ (3S, 3P)⁸ (3d)3 (4S)²

∴ Screening constant (σ) for 4S electron of vanadium
= (2 ×1.0) + (8×1.0) + (8×0.85) + (3×0.85) + (1×0.35)
= 19.7.

Effective nuclear charge for 4S electron of vanadium
(23 - 19.7)
= 3.3.

Screening constant (σ) for 3d electron of vanadium
= (2 ×1.0) + (8×1.0) + (8×1.0) + (2 ×0.35)
= 18.70.

Effective nuclear charge for 3d electron of vanadium
= (23 - 18.70)
= 4.30

d subshell of vanadium(II) ion

Electronic configuration of vanadium ion
(1S)²(2S, 2P)⁸(3S, 3P)⁸(3d)³

Screening constant (σ)  of vanadium(II) ion
= (2 ×1.0) + (8×1.0) +(8×1.0) + (3×0.35)
= 19.05.

Effective nuclear charge of vanadium(II) ion
= (23 - 19.05)
= 3.95.

4S and 3d energy level of an atom

In the first transition series electron filling up process begins in the 3d level below a filled 4S level. During the ionization process, 4S electron will be lost first. This can be explained with the reference of chromium.
Chromium has atomic number 24 and the electronic configuration
(1S)² (2S, 2P)⁸ (3S, 3P)⁸ (3d)⁵ (4S)¹

∴ Screening constant (σ) for 4S electron of chromium
= (2×1.0) + (8×1.0) + (8×0.85) + (5×0.85) + (0×0.35)
= 21.05.

Effective nuclear charge for 4S electron of chromium atom
= (24 - 21.05)
= 2.95.

∴ Screening constant (σ) for 3d electron of chromium atom
= (2×1.0) + (8×1.0) + (8×1.0) + (3×0.85) + (4×0.35)
= 19.40

Effective nuclear charge for 4S electron of chromium atom
= (24 - 19.40)
= 4.60.
Shielding of 3d subshell electron is lower than 4S subshell. Thus for the first transition series, 3d electron is more tightly held than 4S electron.
Hence during the ionization 4S subshell lost electron in the 3d subshell.

Shielding effect and Slater's rules, shielding electrons of d subshell, shielding of sodium, magnesium, potassium, and vanadium for study chemistry

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