Page 230 - Essentials of physical chemistry

192 Essentials of Physical Chemistry

electrons radiate energy so why=how is it possible that electrons can find stable orbitals that do not
radiate energy? Today, we would say that radio antennas radiate because the electronic excitation is
causing the electrons in the metal to change energy levels rapidly but the non-excited energy levels
are stable and do not radiate. These sort of questions were raised about the Bohr model but until
1926 there was no better model. There is no further accurate agreement with experiment except for
spectra of light element ions like He ,Li , etc., one-electron ions of light elements.
þ
2þ
SIGNIFICANCE OF THE BOHR QUANTUM NUMBER n

Today, students in this class will have been exposed to a form of the periodic chart that incorporates
many discoveries since 1913. However, in the early twentieth century the periodic chart was
organized mainly by atomic weights with less organization than we enjoy today. At that time, the
rare gases were believed to be totally inert but their atomic numbers allowed Bohr to postulate an
‘‘electron shell model’’ of atoms heavier than H with occupancies of 2, 8, 8, 18, 18, . . . based largely
on the atomic numbers of the rare gases and the differences between them. For instance, Ne (Z ¼ 10)
has 8 more electrons than He (Z ¼ 2) and Ar (Z ¼ 18) has 8 more electrons than Ne. Then Kr
(Z ¼ 36) has 18 more electrons than Ar, and so on. The Bohr orbits were assigned labels as the K, L,
M, N, . . . shells. Strictly speaking, the Bohr model only has one quantum number, ‘‘n.’’ Later,
Arnold Sommerfeld (1868–1951) postulated elliptical orbitals to introduce additional angular
momentum rules into the shell model. Sommerfeld introduced the l-quantum number, the
m-quantum number, and even the fourth quantum number for spin based on later developments.
For the time being, we will push the Bohr model to its extremes using just the n-quantum number.
In spite of several more recent descriptions of atoms since 1913, the Bohr model survives in some
terminology related to the inner K, L, and M shells as we will soon see, because x-rays come from
transitions between these shells. Although the Bohr model does little to explain chemical bonding in
molecules, it remains a concept related to atoms. Even today, the emblem of the International
Atomic Energy Agency (IAEA) still shows an atom with the Bohr–Sommerfeld elliptical orbits as a
representation of atomic orbitals.

ORBITAL SCREENING
Consider the limitation we have already mentioned in that the Bohr model only applies to
atoms=ions with just one electron. Please note that when there is more than one electron, repulsion
between the electrons alters the energy situation. However, it is possible to adjust the model for
more than one electron, assuming we understand that we are leaving the realm of high accuracy and
merely modeling trends. Probably, every student has done the sodium flame test in freshman
chemistry laboratory. In more sophisticated models, the fluffy yellow flame is believed to be due
to a transition from a 4p orbital back down to a 3s orbital, and the strong yellow ‘‘D’’ line at 589 nm
is in fact two separate lines (a doublet) believed due to two possible spin states of the 4p orbital. The
Bohr model has none of this detail but if we assume that the 4p orbital has nearly the same energy as
the 4s orbital (same shell in the Bohr model), we can treat the Z value as a parameter and fit the
model to a system of an ‘‘ion core’’ of charge (Ze ) with an outer electron in a 4s orbital (assuming it
þ
has the same energy as a 4p orbital; not really true but close). Note 589 nm ¼ 5890 Å. Thus,

1
1 12,398
DE ¼ E 4 E 3 ¼ Z 2 2 ¼ 2:104923 eV:
eff
eff 2 2 (13:6057 eV) ¼ Z (0:661388 eV) ¼
4 3 5890
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2:104923 eV
ffiþ1:784; what does this mean? The
We can solve this for Z eff . Thus, Z eff ¼
0:661388 eV
effective nuclear charge might have been expected to be þ1 assuming that the inner 10 electrons