Page 249 - Instant notes

The structure of the hydrogen atom 235

one of the three to appear in the formula for the energy of the various allowed states, and
is sufficient to explain the Rydberg series of the hydrogen atom emission spectrum.
In addition to the principal quantum number, n(=1, 2, 3…) are the orbital angular
momentum quantum number, l, which takes the values, l=0, 1, 2…(n−1) and the
magnetic quantum number, m l, which takes the values, m l,=−l, −(l−1),…(l−1), l.
Therefore, for a given value of n there are n allowed values of l, and for a given value of l
there are (2l+1) allowed values of ml. For example, when n=2, 1 can have the value 0, for
which m l can have the value 0 only, or 1, for which m l can have the values −1, 0 and 1.
Each wavefunction, which is specified by a unique set of the 3 quantum numbers, is
called an atomic orbital. All orbitals with the same principal quantum number n are said
to belong to the same shell of the atom. Orbitals with the same value of n but different
values of l are known as the sub-shells of the given shell. The sub-shells are usually
referred to by the letters s (for sub-shells with l=0), p (for sub-shells with l=1), d (for sub-
shells with l=2), f (for sub-shells with l=3) and g, h, j…etc. for larger values of l, if
required. Thus the n=2 shell contains four orbitals (sub-shells), one s orbital and three p
orbitals. Electrons that occupy an s orbital are called s electrons. Similarly, electrons can
be referred to as p, d…electrons.
A fourth quantum number, the electron spin quantum number, m s, is required to
uniquely specify each electronic wavefunction. This quantum number can take the value
+½ or −½ (see Topic G6). No two electron wavefunctions can have the same four
quantum numbers so each atomic orbital can accommodate a maximum of two electrons.
The pattern of allowed combinations of atomic quantum numbers is shown in Table 1.
Table 1. The allowed combinations of quantum
numbers for atoms, and the associated
nomenclature of the corresponding orbitals. Each
orbital can contain a maximum of two electrons,
one each with electron spin quantum numbers +½
and −½

Shell Sub-shell Orbitals Maximum no. of Maximum no. of
electrons per sub-shell electrons per shell
n l designation m l values
value value
n=4 l=3 4f −3, −2, −1, 0, 14
1, 2, 3
l=2 4d −2, −1, 0, 1, 2 10 32
l=1 4p −1, 0, 1 6

l=0 4s 0 2
n=3 l=2 3d −2, −1, 0, 1, 2 10
l=1 3p −1, 0, 1 6 18
l=0 3s 0 2
n=2 l=1 2p −1, 0, 1 6 8