As I mentioned, the shells themselves can
be divided into orbitals, each of which contains
two electrons. Orbitals are characterised
by shape that is produced when the region
surrounding the nucleus is plotted in which
it’s typically regarded that a 95% chance
exists of finding the electrons.
The orbital shapes that we will come across
are s, p, d and f. We’re mostly going to
focus on s, p and d orbitals.
The analogy to use is that each shell unlocks
if you like an additional orbital type. So,
in other words, for the first shell you can
only get the s-type orbital. For the second
shell you can get the s- and the p-type orbitals.
And for the third shell you can get s, p and
d. And for the fourth it’s possible to have
access to all four, bearing in mind each individual
orbital can only contain two electrons. And
we’ll talk about this in more depth a little
The orbital quantum numbers are given for
s orbitals as 0, 1, 2 and 3. And the nomenclature
in quantum mechanics, where n was for principal
quantum number, is that the orbital quantum
number carries the letter l. So, as we’ll
see, it’s possible to actually assign the
specific designation of an electron just using
The simplest orbital, as you can see here,
is where l, or the orbital quantum number,
is equal to 0. And it is spherical.
Here you can see three particular types of
shells of orbitals: the 1s, the 2s and the
3s. Each of these is spherical but, as you
can see, each of these is of a different
size. As of course we move further away from
the nucleus, of course the electrons themselves
or the probability of finding them is also
more distant from the nucleus. Also bear in
mind it’s spherical and so the x, y and
z axes there demonstrate the three dimensionality
of this species and the fact that there is
a 95% probability of finding electrons within
Note of course that, if you go back to the
basic quantum mechanics, the theoretical chance
of finding electrons in the nucleus is, of
So, going from left to right, 1s is an s orbital,
which is a spherical orbital which actually
stands for ‘sharp’ rather than ‘spherical,’
which is rather counterintuitive. The 2s is
an s orbital in shell 2 and the 3s is an s
orbital in shell 3. And each of these – the
1s, the 2s and the 3s – can formally contain
p orbitals – these are ‘principal’ orbitals
and this is where you have an orbital quantum
number of 1 or l equals 1 – are more complex.
And remember what I said: as you move up the
shells, it’s possible to accommodate more
and more electrons. The only way to do this
of course is to have more and more different
types of orbital.
And here we can see three of these: the 2py,
the 2px and the 2pz. Note that the number
prior to the py, px and pz correlates to the
principal quantum number which, in this case,
is the lowest one for p orbitals of two. It
is, of course, possible to get 3p and 4p orbitals
but, of course, not possible to get 1p. You
haven’t unlocked them at that point.
Note the similarities between these guys.
They are dumbbell shaped. It’s often the
shape that’s considered. And their orientation
is along the three component axes – that
is the y axes, the x axes and the z axes as
shown in this particular slide.
Crucially, if we look in the centre where
the node is formed, where we have the tiny
spot in the centre is where the nucleus is.
And there we have an electron density of 0.
Now, the question you may be asking is: well
why are they shaded differently? And this
is what I’m going to come to in a moment.
A p orbital has two regions where electrons
may be found on either side. Note what I’ve
done is we’ve got a plane through the node
of the p orbital as shown here. And, along
that plane, there exists no electron density
as it contains the nucleus. But, again, what
is the significance of the negative and the
positive charges? And this is to do with the
phase of electrons.
Now remember what I said: we tried to discount
the idea of treating electrons as particles
because it created a lot of theoretical problems
for us. In fact, we understand that electrons
exist in shells and it can only go up or down
those shells by absorbing or giving off quanta
Equally, we talked about electrons existing
in waves. That’s why we can only talk about
the probability of finding them in particular
parts of an atom. And this is the same analogy
because, in this case, we’re talking about
wave coherence. We’re talking about whether
or not an electron exists as the peak here
of the trough that we’re showing here, the sine wave,
or as the trough.
Right. So, as I mentioned, it’s possible
to treat electrons as waves as well as particles.
And, in the context of the p orbital, this
is very important.
So, if we look at this wave form here, we
can see that it is analogous to the dumbbell
shape of our p orbital where we see in the
centre node the chances of finding an electron
are 0 but the chances of finding an electron
either side are reasonably high.
So, if we think of an electron as a wave with
positive and negative regions, we can think
about the ideas of coherence, i.e. constructive
or destructive interference.
So, as there are three p orbitals, there needs
to be a way of telling them apart because,
if you look here, you can see that they are
degenerate. By that, what I’m talking about
here is the idea that you could superimpose
one on the other by a simple rotation, around
And so a way of defining an electron in a
given p orbital is via the magnetic quantum
number or ml, okay? Not to be confused with
ms. That is a different one which we’ll
come onto a little later. So this helps us
to define the direction of an orbital as well
as its type.
So here we can see we’ve got the orbital
running along the y axis and across the x
axis and then, finally, along the z axis.
So d orbitals: these have a more complex set
of shapes and they have the orbital quantum
number of 2. The d in the orbital stands for
‘diffuse’ and they have two nodal plans
and come in sets of five. The ml, or magnetic
orbital number, can be -2, -1, 0, 1 or 2.
And don’t just take it and accept it as
read. There is actually an equation which
we’ll come onto in the next lecture which
explains how you can determine this yourselves.
The lowest energy shell containing d orbitals
is n = 3. Prior to that, they haven’t been
Degeneracy: this is what I alluded to in the
case of the p orbitals and, as we’ll see
a little later on, in the d orbitals. These
are orbitals with the same energy. When they
have the same energy and they have the same
orientation, they’re regarded as degenerate.
In other words, they’re superimposable onto
each other. And this is the case for p and
also some of the d orbitals. And degeneracy
is based on the idea that, whilst they have
the same energy and the electrons – or the
probability of finding them – is the same
distance from the nucleus, they point in different
directions. By rotating them 90° in the three
axes, however, they would all be identical.
They’re all superimposable and therefore
they’re termed degenerate.