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Chapter 3.3b Quantum Theory Electrons

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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

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

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

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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