the topic of this video is development
of quantum theory understanding quantum
theory of electrons and atoms
the learning objectives are on the
screen so go ahead and pause the video
now so you can write those down in your notes
notes
to start we can revisit this figure
with the bohr model like atom
with a central positively charged
nucleus and discrete energy states
accessible to
an electron in the atom and we
characterize these discrete energy
states using integer
values n equals 1 2 3 and
in theory any integer values
up to infinity now
it turns out that we do need more
to to describe the the
probability of finding electrons in um
atoms uh but before i jump into
that i'll just briefly comment a little
bit more on
these values of n it turns out that
these are really
important integers
in describing electrons
in atoms and so this these n values are
actually what we
refer to as principal quantum numbers
and i'll revisit that in a second but
but essentially
what this can help us to do is think about
about
atomic orbitals and the probability of finding
finding
electrons in an atom so let's go ahead
and throw in the definition of atomic orbital
orbital
a general region in
within which an electron is most
probable to reside so
we're just talking about probability of
finding an electron in a certain region
of space
around a nucleus
and within an atom there are discrete
energy levels
available to them and so if we think
about well what does that look like then
in three-dimensional space um
uh what we can use to help us think
about this in three-dimensional space is
to figure out the left
um so the the figure that i'm
that i'm uh sort of tracing in red down
on the lower left of this figure
shows the probability of finding
electrons some distance
in picometers from the nucleus what this
looks like
over here is this small little um blue
sphere okay so we can think about that
probability in three dimensional space
appearing as a sphere now if we that's
just the
um the when n equals one what if we go
to n equals two
well there's still a probability of finding
finding
uh an electron at the smaller
region of space down here where we had n
equals one
but now we've also at n equals two we
have the ability to go further out
uh further away from the nucleus but
you'll notice that there does there's
this dip down here this
node where at some point there's zero
probability of finding electron even
though you can find it on either side
of that node what does it look like in
three dimensional space
imagine a small sphere
of probability of finding electron in
the center some
node of of of empty space around it
where there is
close to zero probability or zero uh just
just
zero probability of an electron and then
another bigger region of space
outside of that and you can keep
building successively on this
um with nodes in between we have n
equals one down here
n equals two here separated by a node
between the two n equals three
here so the the principal quantum number
is really helping us to think about the general
general
size and energy of an orbital
so before i jump into what the actual
shapes of atomic orbitals can look like
i'm going to build a table and we will
go through
not just the principal quantum number
but the other three quantum numbers
that will help us to describe electrons
in atomic orbitals
okay so i have written out quantum
number symbol
allowed values and meaning this is going
to be the basis
of a table that you can draw in your notes
notes
and like i mentioned before there are
four quantum numbers that we need to describe
describe
the um any given electron within
any given atomic orbital the first
quantum number is what we've already
been talking about the symbol is going
to go over here
n uh so this is what
the allowed values here are
n equals 1 2 3
all the way up to any integer value up
to infinity
the meaning here is going to be the size
size and relative energy size and
relative energy
of the orbital so so we start with the
principal quantum number um the next
uh what we're going to be doing here is
as i introduce quantum numbers we get
much we get more and more specific
okay so the principal quantum number is
is is telling us something about the
size and the relative energy
that's pretty vague so how much more can
we uh do we need to know
another one is called the um so we can
get more specific and this is
secondary or traditionally called the angular
momentum quantum
number okay so angular momentum is is definitely
definitely
the the a very common way to refer to
this quantum number
um it is lowercase l
and the allowed values for l are l can equal
equal
0 1 2
and really any integer value up to n
minus 1. so it can be at most
n minus 1 n being the principal quantum
number keeping in mind that n's lowest value
value
is is 1. um and i'll
walk through examples for this in a
second but what this refers to now is
the actual shape of the orbital so this
is interesting so now principal quantum
number is telling us
size relative energy what's another
layer of specificity well shape okay so
we know the size and the
relative energy but what's the shape of
the orbital that's what the
angular momentum quantum number helps us
to determine um
or describe the next one is going to be
called the magnetic
quantum number this has the symbol
lowercase m subscript lowercase
l and the values of of
m sub l here or the magnetic quantum number
number
is going to be m sub l equals negative
l our angular momentum quantum number um
to positive l um
in integer
steps so what does this mean it means
that if
l for example is 1 then m sub l
is going to be negative 1
zero or one okay so it can take the
value of the
of negative l up to positive l in single integer
integer
steps what does magnetic quantum number
refer to
this is referring to the spatial orientation
of whatever shape that we were talking
about and that shape
is being defined by the size and
relative energy so you see how there's
this progression here size and relative energy
energy
it uh is is pretty broad but then it can become
become
um more specific with shape and shape we
might want to know okay we know the
shape but what's the actual orientation
in space
um so we're getting more and more
specific here the final
the fourth and final quantum number that
we need here is the spin
this is m sub s and this can take the value
value
m sub s equals plus or minus one half
the meaning of this is now we're
actually all the way now
we're talking about what is the spin of
the electron
the best maybe not the best way but one
of the ways to describe this is just to
think about an electron as having two possible
possible
um spins spin up or spin down okay so we
can think about that as a vector up or
vector pointing down
the vector pointing up think about that
as uh positive one half vector pointing
down think about that as
negative one half okay so
uh now that we have all four quantum numbers
numbers
we can now use the four quantum numbers
to describe
an electron in any given atomic orbital
and so we'll go ahead and jump into
some of the more common atomic orbitals
that that we might encounter as chemists
and i'll just point out here um
as i was as i was filling out this table
and the meaning
um i didn't yet draw the arrow down to
the spin so
so here from spatial orientation we can
now describe the
electron itself as spin up or spin down
and so again we go
up more generally to the spatial orientation
orientation
to back to the shape back to the size
and relative energy okay so let's take a
look at what this actually
might look like in terms of shapes of
atomic orbitals
so i'm going to give you some examples here
here
let's consider the case where n
equals 1 then by the rules l can be
zero up to n minus one but n minus one
is zero so l
can only be zero at this point so that
means that when the principal quantum
number is one
there is only one shape available to us
that's what this means
and that also means that if there's one
shape m sub l can be negative l to
positive l and single integer steps but here
here
negative l is zero and positive l is
zero so m sub l can only be zero so that
means there's only one shape
okay one shape and
if there's only one shape there's also or
or
only one orientation for this particular
orbital so what does this mean
the s orbital okay
so this s orbital uh is actually
here s
orbital it is just a sphere around the
center point of the atom which is the
nucleus now let's consider the case where
where
n equals 2
well l can still equal zero okay
and in that case this would lead us to
an s
orbital okay but
if l is not zero um
l could be 1 in this particular case
okay so
if l is 1 then we actually have a
different type of orbital
this is now indicative of a different
type of
orbital and we call this the p orbital
and importantly if m sub l
is or if l is equal to 1 then m
sub l the magnetic quantum number
can actually be values of negative one
zero or one so we have three
orientations of this p orbital
and you look over here at our p orbitals
these are our p
orbitals we have
one two three orientations we have an
orbital along the x-axis we have an
orbital along the z-axis and we have an
orbital along the y
notice that the the connotations down
here or the the designation
p x p z p y so we have
um l is equal to one that's a p orbital
and the magnetic quantum number m sub l
indicates that there are three
orientations that we see
let's go to the case where n equals
three now
so if n equals three um l
could equal zero again this is just an s orbital
orbital
l could equal one again this is just a p
orbital we already looked at those
but now we have a new um
we have a new shape of orbital allowed
to us
where the angular momentum quantum
number is two so we're going to call this
this
our d orbital and keep in mind now
if l equals two then m sub l
can equal negative two negative one zero
one and two so we actually have
available to us according to the
magnetic quantum number
and let's take a look at our d orbitals
over here so these down here
are our d orbitals
and we have one two three
four five orientations so the shape
was was um l equals two indicates the d
type orbital m sub l indicates that we
have five available orientations so
these are a little bit more
exotic and you'll probably encounter
these when you take
if and when you take inorganic chemistry
but here you see
that we have different um nodes now so
for the p
orbital we just had a pla one nodal plane
plane
that cut through the p orbitals
okay so we have these nodal planes the s
type orbital has no
nodal planes but now the x type orbitals um
um
some of them have two nodal planes
okay so so these sort of planes that
sort of
intersect through um so that's the case
of d x
d x z and d y z and d
x squared minus y squared we also get
this special orbital called dz squared
which has a donut it looks like a p
orbital with a donut ring around it
okay and finally why not look at
the next type of orbital these types of
orbitals who don't
really think about too heavily until you
think about
things like lanthanides and actinides
this is the case where the principal
quantum number is equal to four l can equal
equal
zero l can equal one l can equal two
and l can equal 3. we've already discussed
discussed
the 0 through 2 case
so when l is equal to 3 m sub l can be
equal to
negative 3 negative 2 negative 1 0
one two or three this is one two
described by these seven
magnetic quantum numbers so what does
that mean we would predict
seven orbitals seven f orbitals okay
orbitals and we have one two three four
5 6 7.
finally one of the last pieces of information
information
needed to use quantum numbers
to describe electrons in atomic orbitals is
is
something known as the poly exclusion
principle so we have
four quantum numbers that we can use to
describe an electron
now the one of the most important things
to remember according to the pauli
exclusion principle
is that no two electrons
in the same atom can have
exactly the same set of
all the four quantum
numbers so in other words in
a single atom you cannot have two
electrons with the same four quantum numbers
numbers
every electron has its own distinct set
of quantum numbers
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