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Atomic (electronic) configuration
The walkthrough + explanations
Slide 2 – Atomic shells
•
Currently 7 known shells
– K, L, M, N, O, P, Q from inside to outside
• Each shell comprises of different kinds of orbitals
– 4 known orbitals: sharp, principal, diffuse and fundamental
– There is a proposed g orbital for the 8th period
K corresponds to the first shell, L corresponds to the second shell, M corresponds to the third shell
etc. We do not use these letters for writing electronic configuration because things like Pp can cause
confusion when written and so period numbers are used instead. The names sharp, principal, diffuse
and fundamental come from earlier times when atomic subshells were studied by looking at spectral
lines. Each type of orbital would produce a different kind of spectral line – sharp, principal, diffuse or
fundamental, hence the name. g is just a continuation from f. j will not be used; it will go directly
from i to k.
Slide 3 – Shape of atomic orbitals
•
•
•
No real shape, just a probability density
Possible to define a region in which the electron can be found >90% of the time
Wavefunction given by:
3
𝜓𝑛𝑙𝑚 = (√(
𝑙
(𝑛 − 𝑙 − 1)! −𝑍𝑟
2𝑍
2𝑍𝑟
2𝑍𝑟
2𝑙+1
𝑛𝑎𝜇
𝑒
𝐿
)
(
)
(
)) 𝑌𝑙𝑚 (𝜃, 𝜙)
𝑛−𝑙−1
𝑛𝑎𝜇 2𝑛[(𝑛 + 𝑙)!]3
𝑛𝑎𝜇
𝑛𝑎𝜇
• Square of wavefunction gives probability of electron at a certain point from the nucleus.
Due to Heisenberg’s uncertainty principle, it is impossible to define exactly where an electron is at a
given moment in time. However, it is possible to estimate the probability of an electron occurring at
a certain place, by squaring the electron’s wavefunction as given above. From there it is possible to
define a region in which the electron will occur more than 90% of the time, and so the orbitals have
a very rough shape but not a defined one.
The wavefunction, using the spherical coordinates (𝑟, 𝜃, 𝜙) is derived from the equation 𝜓𝑛𝑙𝑚 =
𝑅𝑛𝑙 (𝑟)𝑌𝑙𝑚 (𝜃, 𝜙) where n, l, m are quantum numbers, 𝑌𝑙𝑚 (𝜃, 𝜙)is a spherical harmonic and 𝑅𝑛𝑙 (𝑟) is
3
2𝑍
equal to the huge monster of √(𝑛𝑎 )
𝜇
ℏ2
4𝜋𝜀0
where 𝜀0
𝜇𝑒 2
(𝑛𝑢𝑐𝑙𝑒𝑢𝑠 𝑚𝑎𝑠𝑠)(𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑚𝑎𝑠𝑠)
, and
(𝑛𝑢𝑐𝑙𝑒𝑢𝑠 𝑚𝑎𝑠𝑠)+(𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑚𝑎𝑠𝑠)
number, 𝑎𝜇 being
(𝑛−𝑙−1)!
2𝑛[(𝑛+𝑙)!]3
𝑒
−𝑍𝑟
𝑛𝑎𝜇
2𝑍𝑟
𝑙
2𝑍𝑟
(𝑛𝑎 ) 𝐿2𝑙+1
𝑛−𝑙−1 (𝑛𝑎 ), where Z is the atomic
𝜇
𝜇
is the permeability of vacuum (defined as
1
𝑐 2 (4𝜋×10−7 )
F m-1), 𝜇 =
L the generalized Laguerre polynomial. I will not go into details
about the Laguerre polynomials and spherical harmonics here.
Slide 4 – Electrons in orbitals
•
•
•
There are varying kinds of each orbital:
s – 1, p – 3, d – 5, f – 7
Each orbital can hold 2 electrons
– Pauli’s exclusion principle states that no 2 particles can occupy the same state at the
same time
– Only 2 possible values for spin
– Hence only 2 electrons with opposite spins can occupy the same orbital, else rule
will be violated
• Maximum number of electrons per type of orbital:
s – 2, p – 6, d – 10, f – 14
The s orbital is a spherical orbital, symmetrical on all sides, hence no matter how you orient it the
probability distribution is the same, so there is only 1 kind of s orbital. The p orbitals, being very
roughly dumb-bell shaped, can be oriented in the x, y and z directions – the 3 dimensions we know
of.
The d orbitals get more complex. 4 of them look like two of the p orbitals stuck together at right
angles, while the last one is something like a p orbital with a ring around the nucleus some distance
away. The first 4 can be oriented in either the x-y plane, x-z plane, y-z plane or x2-y2 plane. They are
called dxy , dxz , dyz and dx2 −y2 respectively. Their axes occupy a 2-dimensional plane – i.e. a flat
surface. The last one is the dz2 , which is a different shape from the other orbitals.
The f orbitals are even more complex. The following pictures are taken from the Grand Orbital Table
at http://www.orbitals.com/orb/orbtable.htm, where the program Orbital Viewer is available for
free and can view many orbitals for periods up to 30.
From left to right above are the 𝑓𝑧3 , 𝑓𝑥𝑧2 , 𝑓𝑥𝑦𝑧 and 𝑓𝑥(𝑥 2 −3𝑦2 ) orbitals. There is one other similar
orbital for the latter 3 orbitals, while the first one is unique, making a total of 7 f orbitals.
Pauli’s exclusion principle states that no 2 fermions can occupy the same quantum state. To put it
into simpler terms in this context, no 2 electrons can be at the same place, having the same
momentum at the same time. (It’s quite obvious actually, but very important.) Each electron has
another property called spin. Spin is basically a form of momentum for the electron and there are
only 2 possible values for the electron. So to fit as many electrons as possible into an orbital, the
electrons must be different. We cannot say anything about the position since we cannot measure it,
so the momentum must be different for all the electrons. Since there are only 2 different values,
thus only 2 electrons can fit into 1 orbital.
Slide 5 – Atomic Configuration
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•
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List out all the orbitals that the atom has as follows:
E.g. Na: 1s 2s 2p 3s
Add the number of electrons in each orbital as a superscript:
Na: 1s22s22p63s1
Filled shells up to the previous period can be replaced by square brackets around a noble gas
E.g. Na: [Ne]3s1
– Sodium’s electron configuration is simply Neon’s + 3s1
For the atomic configuration, the atoms will have orbitals that are ordered in a specific manner, to
be covered in slide 7 under the Aufbau rule. List out the orbitals in the order that the Aufbau rule
defines, then next to each orbital, write as a superscript the number of electrons in each orbital. The
total number of electrons must add to the proton number for a neutral atom.
For an atom that has filled a few periods plus a few extra orbitals but not enough to finish 1 period,
the completely filled periods can be replaced by a noble gas in square brackets. In the example,
sodium can be written as [Ne] 3s1. This is because sodium only differs from neon by the extra 3s1
orbital, so the other orbitals can be replaced by [Ne]. Only noble gases can be put in square brackets
for the configuration.
Slide 6 – A slice of an atom
Just a visualisation of what an atom plus its orbitals might be like.
Slide 7 – Aufbau rule
•
•
•
Aufbau – German, meaning ‘construction’
Rule for filling in orbitals
List out all the orbitals and shells as follows, then they are filled in in this order:
Electron shells have overlapping energies – that is, some orbitals in a shell may have a higher energy
than another one in the next shell. The method above is the way most electron orbitals are filled in,
and it also shows the energy levels of the orbitals relative to each other. Since orbitals with lower
energy levels are filled in first, hence the first orbitals to be filled in have lower energy levels than
those filled in later on.
Slide 8 – Exceptions
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•
•
There a few exceptions to the Aufbau rule starting from the transition metals.
Chromium (24Cr)
– Should be [Ar]4s23d4
– Atomic configuration [Ar]4s13d5
Copper (29Cu)
– Should be [Ar]4s23d9
– Atomic configuration [Ar]4s13d10
• 20 known exceptions in total
As with everything in Science, there are exceptions. The Aufbau rule has exceptions as well – 20 of
them currently known. Chromium, for example, transfers 1 electron from the 4s orbital to the 3d
orbital, and so does copper. The reason behind the anomalies will be explained in slides 10 and 11.
Slide 9 –The exceptions
In this periodic table, elements with anomalous electron configurations are shown with their symbol
in red. Elements whose configurations are not yet known are not shown in this periodic table.
Slides 10&11 – Why are there exceptions?
• Aufbau rule assumes that each orbital has different energy levels
• In reality, this may not be true
• Hence electrons can be located in another orbital with similar energy
• Half-filled and completely filled orbitals may also be more stable
• For the heavier elements, electrons are moving closer to the speed of light
• This causes the mass of the electron to increase and the orbital size to shrink
• Tends to decrease the energy level of the s orbital
• Explains anomalies like mercury, a liquid at room temperature, and the colour of gold
The Aufbau rule assumes that each orbital has fixed energy levels that are discrete for different
atoms, bus this is not true in real life as the effects of Special Relativity play a part. Generally, the
energy levels may overlap a lot, causing electrons to move from one orbital to another. The usual
explanation given is that half-filled and completely filled orbitals may be more stable, but this does
not explain other anomalies.
Special Relativity, as described by Einstein, causes the mass of an object to increase when its velocity
gets closer to the speed of light. In heavy atoms, the electrons in the inner shells are moving closer
and closer to the speed of light. This causes the mass of the electron to increase as follows: 𝑀 =
𝑚0
2
√1−𝑣2
where 𝑚0 is the mass of an electron that is not moving, and v the velocity of the electron
𝑐
𝑍𝑐
which can be approximated by 𝑣 ≈ 137, Z being the proton number. The increase in mass causes the
distance from the electron to the nucleus to shrink as well. In general, this causes the s orbitals to
shrink, and pushes the d and f orbitals outwards. For gold, the distance between the 5d and 6s
orbitals is so low that blue light is absorbed by the electrons which move up from the 5d to the 6s
orbital. This causes the golden colour of gold. For mercury, because the 6s orbital is filled, the
electrons do not participate much in bonding, and the s orbitals also shrink in size, so there is very
little attraction between mercury atoms and this causes mercury to be a liquid at room temperature.
See also
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Hund’s rules – how to fill in orbitals with electrons with spin direction
Special relativity
Pauli’s exclusion principle
Norrby, L. J. (1991). Why Is Mercury Liquid? Accounts of Chemical Research, pp110-113.