Atomic radius

Definition. The atomic radius of an atom is the distance from the centre of its nucleus to the outermost shell that contains a valence electron (Figure 3.1a).

Why it can't be measured directly. An isolated, non-bonded atom's electron cloud has no sharp boundary, so its radius cannot be measured directly. In practice, except for the noble gases (which use a van der Waals radius instead, since they do not form ordinary bonds), atomic radius is reported as either a covalent radius or a metallic radius, depending on the type of bonding the atom actually engages in.

Covalent radius. This is one-half of the internuclear distance between two identical atoms joined by a single covalent bond (Figure 3.1b), a distance that can be measured experimentally by X-ray diffraction. Worked example: the internuclear distance in Cl2 is 1.98 Angstrom (Figure 3.1c), so $d_{Cl-Cl} = r_{Cl} + r_{Cl} = 2\,r_{Cl}$, giving $r_{Cl} = 1.98 / 2 = 0.99$ Angstrom. Because forming a covalent bond involves the overlap of atomic orbitals, it pulls the two nuclei slightly closer together than their "true" atomic radii would suggest -- so the covalent radius is always a little shorter than the actual atomic radius.

For a hetero-nuclear diatomic molecule A-B, the individual covalent radii can still be extracted from the measured internuclear distance $d_{A-B}$ using the Schomaker-Stevenson relation:

where $\chi_A$ and $\chi_B$ are the Pauling electronegativities of A and B (with $\chi_A > \chi_B$ by convention), and radii are in Angstrom. Worked example: for H-Cl, $d_{H-Cl} = 1.28$ Angstrom, $r_{Cl} = 0.99$ Angstrom, $\chi_{Cl} = 3$, $\chi_H = 2.1$:

Metallic radius. This is one-half of the distance between two adjacent metal atoms in a closely packed metallic crystal lattice. Worked example: in solid copper the distance between adjacent Cu atoms is 2.56 Angstrom, so the metallic radius of copper is $2.56/2 = 1.28$ Angstrom. (Metallic radius can also be worked out from a crystal's unit-cell length -- the detailed procedure is covered in the Class XII solid state unit.)

Trend across a period: atomic radius decreases. Moving left to right, each successive element adds its new valence electron to the same shell while simultaneously adding a proton to the nucleus. The growing nuclear charge pulls the (same-shell) valence electrons in more tightly, so atomic radius shrinks steadily across a period.

Effective nuclear charge and Slater's rules. Electrons in inner shells repel the valence electrons even as the nucleus attracts them, partly cancelling the nucleus's pull -- this partial cancellation is the shielding (screening) effect, and the net attractive charge actually felt by a valence electron is the effective nuclear charge, $Z_{eff} = Z - S$, where $S$ is the screening constant. Slater's rules estimate $S$ in four steps:

1. Write the atom's electron configuration and regroup it as $(1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)\ldots$
2. Locate the group containing the electron of interest. Any electron in a group to the right of this (higher in the list) contributes nothing to shielding it. Every other electron within the same group contributes 0.35 to $S$ (0.30 if the electron of interest is a 1s electron).
3. For an electron of interest that is an s or p electron: each electron one group inward, $(n-1)$, contributes 0.85; each electron two or more groups inward, $(n-2)$ and beyond, contributes the full 1.00. For an electron of interest that is a d or f electron: every electron in any group to its left contributes the full 1.00 (Table 3.12).
4. Sum all these contributions to get $S$.

Worked example (scandium, Z = 21, configuration $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^1$, regrouped as $(1s)^2(2s,2p)^8(3s,3p)^8(3d)^1(4s)^2$):

Effective nuclear charge on the 4s electron: the (4s) group itself contributes $1 \times 0.35 = 0.35$ (one other 4s electron); the $(n-1)$ group $(3s,3p,3d)$ has 9 electrons contributing $9 \times 0.85 = 7.65$; the $(n-2)$ and lower groups $(1s,2s,2p)$ have 10 electrons contributing $10 \times 1.00 = 10.00$. $S = 0.35 + 7.65 + 10.00 = 18.00$, so $Z_{eff} = 21 - 18 = 3$. …