Sizes of Electron Shells and Atoms

Consider the radial electron density distributions for He, Ne and Ar, shown in Figure 7.3 of your book.

The electron density in He shows a broad maximum at a distance of ~ 0.32Å from the nucleus. This clearly shows that the 1s orbital is not a well defined circular orbit as suggested by Bohr, rather there is a reasonable probability of finding an electron over a reasonably wide distance range.

The curve for Ne shows two peaks, a sharp one at ~ 0.08Å from the nucleus and a second maximum at ~ 0.35Å. The first peak corresponds to the maximum in the electron density of the n = 1 shell, and the second peak to the maximum in the electron density of the n = 2 shell.

The electron density distribution for Ar now has three peaks, corresponding to the n = 1, 2 and 3 shells. Comparing all three curves leads to the following conclusions:

The size of a shell decreases as the atomic number increases. This is a consequence of the increased number of protons in the nucleus, which leads to a stronger Coulombic attraction with the electrons.

Atomic size (proportional to the distance over which there is appreciable electron density) increases as we go from He to Ne to Ar. At first this might seem counter to our first conclusion, that increasing the number of protons draws the electrons closer to the nucleus. However, we need to remember that for an atom like Argon the n = 1 and n = 2 electrons (10 core electrons in all) are located on average between the n = 3 electrons and the nucleus. This shields the electrons in the third shell from the full positive charge of the nucleus.

Combining these two observations tells us that when moving down a column in the periodic table:

• The core electrons become more tightly bound and the core orbitals smaller as the atomic number increases
• The outmost value of n increases, which leads to an increase in the size of the atom.

You might look at the values of atomic radii in the above table and wonder how such values relate to the electron density distribution. After all atoms are not hard spheres, with precisely defined radii. Nonetheless, atomic radii are tabulated and serve as a useful predictive concept in chemistry. It is important to know values of atomic radii are determined.

Using diffraction methods, we can accurately measure the distance between nuclei. If we make the assumption that atoms are spheres that touch each other:

r = ½ d_N-N

For example

In Fe the distance between atoms is 2.48Å ® Atomic radius, r_Fe = 1.24Å

In an I₂ molecule, the I – I distance is 2.66Å ® Atomic radius, r_I = 1.33Å

The atomic radii are often called the covalent radii (because the bonds between atoms are purely covalent). In ionic compounds, where there are cations and anions, ionic radii are used. Cationic radii are smaller and anionic radii are larger than covalent radii.

For example:

It is not surprising that adding an electron to atom to form an anion, without changing the charge of the nucleus, will lead to an increase in the radius. By a similar argument removing an electron to form a cation, increases overall positive charge and leads to a decrease in the radius.

Periodic Trends in Atomic Radii

1. As you go down a column (group)

• Principle quantum number increases
• Atomic radius increases

For example the radii of the halogens are as follows:

• F = 0.71Å
• Cl = 0.99Å
• Br = 1.14Å
• I = 1.33Å

2. As you move left to right across a period

• Effective nuclear charge increases
• Atomic radius decreases

For example

• Na = 1.86Å
• Al = 1.43Å
• S = 1.04Å
• Ar = 0.97Å

We’ve already discussed the reasons why size increases as you move downward in the periodic table, but the fact that radii decrease as you move left to right across the periodic table may seem a bit surprising. This is caused by the inability of electrons in the outer shell to fully shield other electrons in the same shell from the charge of the nucleus (which of course increases as you move left to right).

• Electrons with lower n shield the outer electrons quite well.
• Electrons with same n and lower l shield the outer electrons partially.
• Electrons in the outer subshell shield each other rather poorly.

Slater developed a semi-quantitative set of rules for estimating the effective nuclear charge that the outermost electrons feel. [Note you are not expected to know Slater’s rules for exams and quizzes. I include them only to help your understanding of the material.]

1. Write out the electron configuration of the element in the following order and groupings: (1s) (2s,2p) (3s,3p) (3d) (4s, 4p) (4f) (5s, 5p), etc.

2. Electrons in any group to the right of the (ns, np) group contribute nothing to the shielding constant.

3. All of the other electrons in the (ns, np) group shield the valence electron to an extent of 0.35 each.

4. All electrons in the n-1 shell shield to an extent of 0.85 each.

5. All electrons in the n-2 or lower shells shield completely (shielding = 1).

6. When the outer electrons are in either an f or a d subshell, all electrons to the left shield completely (shielding = 1).

Using Slater’s rules we can estimate the effective nuclear charge seen by the first 18 elements, using the following formula

Z_Eff = Z – S

Where Z_Eff is the effective nuclear charge, Z is the actual charge on the nucleus, and S is the shielding constant (each electron makes a contribution to the shielding constant according to the above rules).

You can see that as we move left to right across a period the outer electrons feel a stronger attraction to the nucleus (it seems to have a greater positive charge) and the atomic radius decreases.

Slaters rule #6 tells us that there are special rules for d and f electrons. This is because electrons in these orbitals spend a greater proportion of their time away from the nucleus, compared to electrons in the s and p orbitals. This is particularly true for the electrons in f-orbitals, and as a consequence f-electrons do a particularly poor job of shielding outer electrons from the nucleus. This leads to an effect called the lanthanide contraction.

Lanthanide Contraction ® The 4d (Y-Cd) and 5d (La-Hg) elements tend to have similar radii because of the poor shielding of the f electrons.

As a final note keep in mind that in general moving down a column leads to a larger effect than moving to the left.

Example

Arrange Se, P, As, S in order of increasing atomic radius.

S (1.04 Å) < P (1.10 Å) < Se (1.17 Å) < As (1.21 Å)

One last point to leave you with. Atomic radii are averages of internuclear separations observed in a variety of substances, not absolute values.