Atomic Orbitals
Bohr’s model worked well for hydrogen, the simplest atom, but didn’t work very well for
any others. In the early 1900s Schrödinger developed a more involved model and set of
equations that better described atoms by using quantum mechanical concepts. His model
introduced a mathematical description of the electron’s motion called a wave functionor
atomic orbital. Squaring the wave function (orbital) gives the volume of space in which
the probability of finding the electron is high. This is commonly referred to as the electron
cloud (electron density).
Schrödinger’s equation required the use of three quantum numbersto describe each
electron within an atom, corresponding to the orbital size, shape, and orientation in space.
It was also found that a quantum number concerning the spin of the electron was needed.
The first quantum number is the principal quantum number (n). It describes the
energy (related to size) of the orbital and relative distance from the nucleus. The allowed (by
the mathematics of the Schrödinger equation) values are positive integers (1, 2, 3, 4, etc.).
The smaller the value of n, the closer the orbital is to the nucleus. The number n is some-
times called the atom’s shell.
The second quantum number is the angular momentum quantum number (l). Its
value is related to the principal quantum number and has allowed values of 0 up to (n– 1).
For example, if n=3, then the possible values of lwould be 0,1, and 2 (3 – 1). This value
of ldefines the shape of the orbital:- If l=0, the orbital is called an s orbital and has a spherical shape with the nucleus at the
center of the sphere. The greater the value of n, the larger the sphere. - If l=1, the orbital is called a p orbital and has two lobes of high electron density on
either side of the nucleus. This makes for an hourglass or dumbbell shape. - If l=2, the orbital is a d orbital and can have variety of shapes.
- If l=3, the orbital is an f orbital, with more complex shapes.
Figure 10.3 shows the shapes of the s, p, and d orbitals. These are sometimes called
sublevelsor subshells.
The third quantum number is the magnetic quantum number (ml). It describes the
orientation of the orbital around the nucleus. The possible values of m 1 depend on the value
of the angular momentum quantum number, l. The allowed values for mlare –lthrough
zero to +l. For example, for l=2 the possible values of mlwould be –2, –1, 0, +1, +2. This
is why, for example, if l=1 (a p orbital), then there are three p orbitals corresponding to
mlvalues of –1, 0, +1. This is also shown in Figure 10.3.
The fourth quantum number, the spin quantum number (ms), indicates the direction
the electron is spinning. There are only two possible values for ms, +^1 ⁄ 2 and –^1 ⁄ 2.
The quantum numbers for the six electrons in carbon would be:QUANTUM FIRST SECOND THIRD FOURTH FIFTH SIXTH
NUMBER ELECTRON ELECTRON ELECTRON ELECTRON ELECTRON ELECTRON
n 11 22 2 2
l 00 00 1 1
ml 00 00 1 0
ms +^1 ⁄ 2 −^1 ⁄ 2 +^1 ⁄ 2 −^1 ⁄ 2 +^1 ⁄ 2 +^1 / 2
Therefore, the electron configuration of carbon is 1s^2 2s^2 2p^2.140 STEP 4. Review the Knowledge You Need to Score High