Chemistry · Ch 4 — Structure of Atom
Atomic Orbitals and Quantum Numbers
Atomic Orbitals and Quantum Numbers
Because many different wave functions solve the Schrödinger equation for a given atom, an atom possesses many atomic orbitals, and these orbitals — really just names for particular solutions — form the basis of the whole quantum-mechanical picture of electronic structure. Each orbital is designated by three quantum numbers, n, l and mₗ, while each individual ELECTRON occupying an orbital is further assigned a fourth, mₛ. The principal quantum number n is a positive integer () that identifies the shell; all orbitals sharing one value of n belong to the same shell, and as n increases the number of orbitals allowed in that shell grows as . Shells are traditionally labelled K, L, M, N, ... for respectively, and both the physical size of a shell and (for hydrogen-like species) its energy increase — becoming less negative — as n increases. In multi-electron atoms, by contrast, orbital energy depends on both n and a second quantum number, l. This azimuthal (or subsidiary) quantum number l identifies the SUBSHELL within a given shell: orbitals sharing the same n but different l values belong to different subshells, the number of subshells in a shell equals n itself, and l ranges over whole-number values from 0 up to . Subshells with are given the letters s, p, d, f, ... respectively — so, for example, the K shell () has only one subshell, (i.e. 1s). Finally, the magnetic orbital quantum number mₗ specifies the spatial ORIENTATION of a particular orbital within its subshell: for a subshell of azimuthal number l, there are allowed values of mₗ, running $-l, -(l-1 …
Table 4.6:
n = 1, shell symbol K, allowed orbitals n² = 1.
n = 2, shell symbol L, allowed orbitals n² = 4.
n = 3, shell symbol M, allowed orbitals n² = 9.
n = 4, shell symbol N, allowed orbitals n² = 16.
(Shell size increases as n increases.) The azimuthal (subsidiary) quantum number l ranges from 0 to (n−1) and gives the subshell within a shell (l = 0,1,2,3,... written s,p,d,f,...); the number of subshells in a shell equals n. The magnetic quantum number mₗ, for a given l, takes (2l+1) values from −l to +l and fixes the spatial orientation of each orbital in …
Table 4.7 (first three shells, showing 2l+1 orbitals per subshell and their sum n²):
K (n=1): total orbitals 1² = 1; 1 subshell, l=0, orbitals 2(0)+1=1; sum = 1.
L (n=2): total orbitals 2² = 4; 2 subshells, l=0 gives 1 orbital and l=1 gives 3 orbitals; sum = 1+3 = 4. …
Table 4.8 — for each shell/subshell, the value of mₗ for every orbital:
K, n=1: subshell 1s (l=0), 1 orbital, mₗ=0.
L, n=2: subshell 2s (l=0), 1 orbital, mₗ=0; subshell 2p (l=1), 3 orbitals, mₗ=−1,0,+1.
M, n=3: subshell 3s (l=0), 1 orbital, mₗ=0; subshell 3p (l=1), 3 orbitals, mₗ=−1,0,+1; subshell 3d (l=2), 5 orbitals, mₗ=−2,−1,0,+1,+2. …
Worked out. Worked example: how many orbitals make the N shell, and how are they distributed by subshell? For the N shell, n=4, so total orbitals = n² = 16. Total subshells = n = 4: s (l=0) has 2(0)+1=1 orbital, p (l=1) has 2(1)+1=3 orbitals, d (l=2) has 2(2)+1=5 orbitals, f (l=3) has 2(3)+1=7 orbitals. 1+3+5+7 = 16, ma …
Worked out. Worked example: an atom has two electrons in its 4s orbital — write all four quantum numbers for each. For 4s, n=4 and the s subshell has l=0; the single s orbital has mₗ=0. The two electrons occupying this one orbital must have opposite spins: electron 1 has n=4, l=0, mₗ=0, mₛ=+1/2; electron 2 has n=4, l=0, mₗ=0, mₛ=−1/2. …