3.4 Atomic Orbitals

At the turn of the 20^th century, physicists were understandably pretty proud of their scientific achievements. With the laws of Newton and Maxwell's equations, many physicists felt about all that was left in knowing about the universe was to pin down a few constants to more decimal places. Nobel prize winning scientists like Max Planck, Albert Einstein, Niels Bohr, Louis De Broglie, Werner Heisenberg began unravelling a deeper mystery by considering something called wave-particle duality of light and matter. Before this wave-particle duality idea took hold, physicists in the early 20^th century used the equations of classical physics to describe the motion of electrons in atoms. It turned out classical physics fell apart in describing electronic behavior in atoms. Scientists needed a new approach that took the wave behavior of the electron into account. In 1926, an Austrian physicist, Erwin Schrödinger (1887–1961; Nobel Prize in Physics, 1933), developed wave mechanics, a mathematical technique that describes the relationship between the motion of a particle that exhibits wavelike properties (such as an electron) and its allowed energies. In doing so, Schrödinger developed the theory of quantum mechanicsA theory developed by Erwin Schrödinger that describes the energies and spatial distributions of electrons in atoms and molecules., which is used today to describe the energies and spatial distributions of electrons in atoms and molecules.

Although quantum mechanics uses sophisticated mathematics, you do not need to understand the mathematical details to follow our discussion of its general conclusions. We focus on the properties of the wave functions that are the solutions of Schrödinger's equations.

Orbital Shapes

The solutions of Schrödinger's equation give us regions of space called orbitals. An orbital is the quantum mechanical refinement of Bohr’s orbit. In contrast to Bohr's concept of a simple circular orbit with a fixed radius, orbitals are mathematically derived regions of space with different probabilities of finding up to 2 electrons. These regions of space, called orbitals, are distinguished by letters of the alphabet. The sequence of letters is s, p, d, f, g...alphabetically skipping the vowels. For a given value of n, the shell number (also referred to as "energy level"), there are that same number of sublevels (also referred to as subshells).

Figure 3.4(a) Atomic Orbitals

(a) The lone s orbital is spherical in distribution. (b) The three p orbitals are shaped like dumbbells, and each one points in a different direction. (c) The five d orbitals are rosette in shape, except for the dz² orbital, which is a "dumbbell + torus" combination. They are all oriented in different directions.

Table 3.4(1)Electron Distribution in Atoms

Shell number	Subshell name	Subshell max e-	Shell max e-
1	1s	2	2
2	2s	2	2 + 6 = 8
	2p	6
3	3s	2	2 + 6 + 10 = 18
	3p	6
	3d	10
4	4s	2	2 + 6 + 10 + 14 = 32
	4p	6
	4d	10
	4f	14
5	5s	2	2 + 6 + 10 + 14 + 18 = 50
	5p	6
	5d	10
	5f	14
	5g	18

Summary

Because of the discovery of wave–particle duality, scientists developed quantum mechanics, which uses wave functions (Ψ) to describe the mathematical relationship between the motion of electrons in atoms and molecules and their energies.

Each wave function describes a particular spatial distribution of an electron in an atom, an atomic orbital.

The four chemically important types of atomic orbital are assigned letters (s, p, d, f). The orbitals designated s orbitals and are spherically symmetrical, with the greatest probability of finding the electron occurring at the nucleus. The orbitals designated p orbitals have a dumbbell shape. The orbitals designated d orbitals have more complex shapes with at least two nodal surfaces. The orbitals designated f orbitals are still more complex.

Key Takeaway

• There is a relationship between the motions of electrons in atoms and molecules and their energies that is described by quantum mechanics.

Example 3.4-1

1. How many subshells are possible for n = 3? What are they?

2. State the number of orbitals and electrons that can occupy each subshell.

   1. 2s
   2. 3p
   3. 4d
   4. 6f

3. How many orbitals and subshells are found within the principal shell n = 6? How do these orbital energies compare with those for n = 4?

Answers

1. Three subshells, with s, p, and d.

2. 1. 2 electrons; 1 orbital
   2. 6 electrons; 3 orbitals
   3. 10 electrons; 5 orbitals
   4. 14 electrons; 7 orbitals

3. A principal shell with n = 6 contains six subshells. These subshells contain 1, 3, 5, 7, 9, and 11 orbitals, respectively, for a total of 36 orbitals. The energies of the orbitals with n = 6 are higher than those of the corresponding orbitals with the same letter for n = 4.