Introduction to Quantum Numbers Part II

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Introduction to Chemistry: Structures and Solutions

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Duke University

Introduction to Chemistry: Structures and Solutions

59 notes

This is an introductory course for students with limited background in chemistry; basic concepts such as atomic and molecular structure, solutions, phases of matter, and quantitative problem solving will be emphasized with the goal of preparing students for further study in chemistry.

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Quantum Number, Lewis Dot Structure

<p>Welcome to week 2! This week we will introduce quantum numbers, more information about periodicity, chemical bonding concepts including Lewis dot structures, resonance, bond order. Continue using the discussion forum for any questions, concerns, discussions, and suggestions. We value your feedback very much!</p> <p>I hope you already found out how we captured the course logo from Colored Flames. Wasn't that fun to watch? Kudos to those of you who browsed the course, watched videos, submitted exercises, and/or posted on the discussion forum! Keep the great work going please.</p>

Introduction to Quantum Numbers Part I17:12

Introduction to Quantum Numbers Part II25:29

Rencontrer les enseignants

Prof. Dorian A. Canelas
Assistant Professor of the Practice
Chemistry

Greetings.

Let's continue to describe each electron in an atom using quantum numbers.

Remember that in quantum theory

the electron is a particle that behaves as a wave.

The Quantum numbers are used in the wave equation that describes the energy of

the electron.

We've already discussed the first two quantum numbers, but

we need four quantum numbers to fully describe the condition of the electron.

The ones that we discussed in the last video include N the principal

quantum number, we also see N in a number of equations such as

the Rydberg equation and L the orbital angular

momentum quantum number which tells us something about the orbital shape.

Now we're going to discuss ml, which is the magnetic orbital quantum number,

which gives us information about the orbital orientation.

And ms, which is the spin quantum number and tells us about the electron spin.

Remember in the last lecture we went through how the quantum

number n is used to describe the primary energy of the electron.

The larger the value of n, which here is 1 2 3 and 4, the further

away from the nucleus that electron probably is spending most of its time.

L gave us information about the shape of the shell or the sub-level.

Now shell is not really the best descriptive word here.

It's actually a cloud.

If you're really interested in this,

you should check out the online website called the Orbitron.

I'm going to write that up here because this is such a great website.

Orbitron, which is run by Mark Wintersgroup out of the University of

Sheffield.

It's really a wonderful website to look out.

L tells us something again about the shape.

And if l equals 0, then we have an s orbital which has a round shape

or a spherical shape of the shell.

If you look in the Orbitron website you can see that

there are actually some nodes inside that sphere that we can't see.

So node are regions where the electron density goes down to a very low value.

But if we just look at a crude picture of the outside of the sub-level we can see

that an s orbital spherical in shape.

Which is pretty easy to remember, s for spherical.

Once the value for n is 2 or

higher, then we can have more complicated shapes for the orbitals.

And orbital remember is a place where the electron can spend its time.

So for example if n equals 2, we can have three different p orbitals.

Each of those p orbitals has a different orientation,

which you'll see today as described by ML.

This particular type of drawing of a p orbital,

which you'll see frequently in textbooks, shows a p orbital as being fairly skinny.

Sort of a dumbbell shape with a sharp point in the middle

where the nucleus resides.

As it turns out, the real p orbital is quite a bit wider than this.

Perhaps is more like this.

And it doesn't have this point where the nucleus is.

Why is the p orbital shaded on one side and not the other?

Sometimes you'll see it just shown as different colors.

It's shaded on the different sides of the p orbital to show that the wave

function has a different sign on the different sides of the p orbital.

In other words, you might think of, and this is not the most accurate way to show

it, but you might think of the wave on the side that's shaded here

as looking like that, where it goes up to be positive first, and

the wave on the other side you can think of as going the other direction.

So the wave has the same wavelength, it has the same amplitude, it has

the same frequency, but it has a different sign on different sides of the orbital.

And that becomes very important when you start blending these orbitals later,

perhaps if you take a higher level chemistry class

in molecular orbital theory.

Now, of course, the p orbitals can also be present at higher sublevels.

So we could have the 3p orbitals,

which of course would have all three orientations again.

And the 4p orbitals which would have all orientations again.

Each time because, n is getting larger here, going from 2 to 3 to 4,

the orbital is getting slightly larger but because it's a p orbital,

it retains the same shape.

Now ml is the magnetic orbital quantum number.

It's what tells us the orientation of the orbital.

It can indicate the number of orbitals in each sublevel.

It has integral values that are bounded by l.

It can go from positive l to minus l.

I think the best way to illustrate this is with an example.

How many 5p orbitals exist?

Well first we have to remind ourselves what the value is for l,

if we have a p orbital.

Here's a little table I put together to remind us of that.

Remember if it's a s orbital then l is 0.

If it's a p orbital then l is 1.

So l equals 1, which means ml then following our rules,

can go from positive 1 down to negative 1, as integer values including 0.

So ml here can be 1, 0 or minus 1.

That means that there are three 5 p orbitals.

Every electron in an atom has a unique set of quantum numbers.

And the value form l tells us the orientation of space of

the sublevel orbital where a particular electron resides,

whether it's in a P orbital or a D orbital or an F orbital.

All of those orbitals have some directionality to them, and

ml tells us which of those orbitals the electron is in.

Let me show you what I mean.

So remember the electron probability regions and

sublevels are now considered to be clouds.

And we know that there are three distinct p orbitals at every level.

That gives us three distinct values for ml.

So here's one of the orbitals, here's a second one, and here's a third one.

So remember, for the p orbital,

l = 1 and mL = 1, 0, or -1.

We don't have a way to match up each of these numbers for

an mL with a particular orientation because remember the orientation depends

on where we are in space relative to the atom.

So we cannot for example say that 0 is this one.

We can't say that.

There's an element of uncertainty with regards to mL.

We can know m and we can know l, but we can't for

certain know the value of mL for an exact particular electron.

There's this element of uncertainty in Quantum Theory.

So we begin to review ml designates the orientation of the orbitals in

the sublevel.

It tells us how many of each type of orbitals there are in the sublevel.

And it gives us an idea of the different directionality of those sublevels.

Here's one for you to try.

How many 3D orbitals exist?

So, this time l equals two.

So, if you think about what the boundaries are for ML.

Then you should be able to figure out how many 3D orbitals exist.

Go ahead and try to answer that now.

Thank you for submitting your answer.

Hopefully, you realize that if l equals 2, ml can equal two,

one, 0, minus one one or minus two, which means that there are five D orbitals.

The d orbitals look a little bit different from the s and p orbitals.

There are five of them, and they start to look like clover leaves.

Again, remember the shading shows the phase of

the wave function in that particular area of the orbital.

So, we can imagine the d orbital turning in different directions.

And then the third one looks very strange and I don't actually

completely myself understand how this relates to the first four.

But these are the five d orbitals sketched as crude drawings of the electron cloud.

As I said before, if you're really interested in this,

check out the orbitron.

It's a wonderful website.

The final quantum number that we're going to learn about,

is ms, the spin quantum number.

So, the electron is not only a particle that behaves as a wave, but

it is a charged particle that is spinning on its axis.

It can spin either in the clockwise direction or

in the counterclockwise direction.

If you have a background in physics,

you know that a spinning charged particle creates a magnetic field.

For the electron on the left, the magnetic field that's created is pointed up.

And for the electron on the right,

the magnetic field that's created is pointed down.

The spin quantum number tells us something about the difference

between these two electrons.

If we have the magnetic field pointed up,

we say that it has a spin value of plus one half.

And if the electron is spinning in the opposite direction and creates a magnetic

field that is pointed down, we say that the MS value is minus one half,

So, now we have four quantum numbers, n, l, ml, and

ms, and with those four quantum numbers,

we can describe any individual electron in an atom.

There are some additional guidelines about electrons.

Electrons obviously can't be in exactly the same place at exactly the same time.

We can't have two electrons occupy one spot.

Pauli said, that no two electrons in an atom can have the same set

of quantum numbers, and this is called the Pauli Exclusion Principle.

Therefore, any of these orbitals can accommodate up to two electrons.

But, if there are two electrons in the same orbital,

they must have opposite spin.

In this class, we've already been drawing electrons as arrows and

we've been showing the orbitals as crude lines.

And some of you might have noticed that when I put two of the arrows on a single

line, I tend to show them having

opposite directions of which way the arrows are pointing.

This is just a way of showing that

the electrons that are in the same orbital must have opposite spin.

Okay, it's time to apply some of what we know through some practice.

Here's an example.

Like many of the examples in chemistry, we have energy increasing on

the y axis, so the bottom of this screen is low energy.

The nucleus of course for

this particular atom that I've depicted is down way below the bottom of the screen.

And here, I'm only showing the electrons.

The electrons are shown according to the orbitals that they occupy and

I'm also showing their relative spans,

whether they have a plus one half or minus one half span.

Somehow I lost a line here, but I'm sure it'll come back.

All right, so in this example,

what are the quantum numbers of the circled electron?

So, I'm trying to figure out what are the values for n, l, ml, and ms.

While the circle electron, I see is in the two s orbital.

Well, two is the principle quantum number.

That's n, that part's easy.

The shape of the orbital remember corresponds to the value for l.

And s electron gives us an l value of zero.

If l is zero, then only one orbital of ml is allowed.

Only one value of ml is allowed.

So ml must also be zero.

And finally this electron is shown as a down arrow.

So, that means that ms is minus one half.

So, for this electron, I can write a set of quantum numbers, and

there's only one correct answer.

The set of quantum numbers would be two for n, zero for l,

zero for ml, and minus one half for ms.

And they are often shown in this order.

Okay, so let's review the quantum numbers one more time.

N is the principal quantum number.

It can be a natural number, it can not be zero.

N tells us something about the orbital size.

L is the orbital angular momentum quantum number.

It's restricted by the value of n.

L has to be less than n, and it tells us something about the orbital shape.

Ml is the magnetic orbital quantum number.

It describes the direction of the electron's probability region.

And it is restricted by the value of l.

If l is zero, then ml can only be zero.

And if l is a larger value, then ml can go from l down through zero to minus l,

including zero, giving all integrals as values.

Finally is the spin quantum number.

It indicates which direction the electron is spinning about its axis.

Now, this depends on our orientation in relationship to the electrons.

Which direction do we call down, and which direction do we call up?

Which direction do we call clockwise?

Versus which direction do we call counter clockwise?

I'm usually able to illustrate this pretty well in the classroom by swinging my

arm around clockwise.

I'm swinging it so that it looks clockwise to me.

But, which direction does it look like my pen is moving to you?

Does it look like it's moving clockwise or counter clockwise, from your orientation?

Thank you for your answer.

Hopefully you said well, it looks to me like you're spinning it counter clockwise.

So, when I say the electron's spinning clockwise or counter clockwise or its

giving a plus one half or a minus one half for the direction of the magnetic field.

That is relative to the position of the viewer, isn't it?

That's also true for this direction that we get from ml.

So, ml and ms, we can't know for certain for an electron.

We know that there are different values allowed.

But, we can't be certain about these two things.

The only thing we can determine experimentally for

electrons are the values of n and l.

So, n and l we can determine experimentally.

Ml and ms we what the restrictions are, we know what the allowed values are,

remember the electron is quantized.

But there's an element of uncertainty there.

All right, let's review these orbitals, and

let's look at how the quantum numbers relate to each other.

I'm going to do this by building up the orbitals in terms of their energy.

So the lowest energy principal quantum number we know about is n =1.

If n = 1, then l = 0, and that's an s orbital.

And for an s orbital there's only one allowed value of m l and that's 0.

There's only one s orbital at each principal quantum number.

So there's a picture of my s orbital, and there's a spot when

I can put two electrons, remember I can draw two electrons in that box.

One of them would have a positive spin of one half, and

one of them would have a spin of negative one half.

So every time I draw a box I can put two electrons in the box.

I won't keep doing that, you just happen to note, you just need to remember that.

All right, now what if n = 2?

So that's the only orbital that's allowed for n = 1.

Only one orbital is allowed.

But what if n = 2?

Then what types of orbitals can we have?

If n = 2, I can have two different values for l.

l can be 0 or l can be 1.

If l = 0, that's an s orbital, and if l = 1, that's a p orbital.

For the 2s orbital, there's only one allowed value of ml because sphere's don't

have directionality, do they?

They're spherically symmetrical.

And if l = 1, for the p orbital,

I have three directions of how the p orbital can be oriented.

Let's draw pictures of that.

Here's the 2s orbital.

It's spherical and slightly larger than the 1s orbital.

And then I have three two-piece orbitals.

There they are with their different directions shown.

So, for this particular atom if I was writing in the electrons I could have two

electrons in the 1s orbital.

I could have two electrons in a 2s orbital.

I'm going to switch colors.

I could have six electrons.

I'm drawing these as double headed arrows.

I'm kind of used to drawing them as single headed arrows so I have to adjust.

I could have six electrons in the 2p sub-level.

Two in each of the orbitals.

All right now what if n = 3?

If n = 3, that's a higher energy than n = 1 or n = 2.

l has three values which are allowed, so there's three different shapes which

are allowed for this sublevel m = 3, with l = 0, 1, or 2.

ml has to be 0 if it's an s orbital.

ml has three allowed values for p orbital and

ml has five allowed values for a d orbital.

Let's go ahead and draw all of those orbitals in.

The lowest energy of those three is 3s orbital.

But the next higher energy is the 3p orbital and there's three of those.

You see how they're a little bit larger than the 2p orbitals?

And then at higher energy still is the 3d orbitals.

And there they are.

So I could have up to ten electrons at the 3d sublevel, couldn't I?

Let's go up even higher.

Now, I've run out of room at the top of my screen, so I'm just going to go

to a new slide, but everything on the new slide is higher energy than this slide.

Here we go, here's the new slide.

Energy is on the y axis again, and this time, n=4,which

mean l has four different values that are allowed from 0 to 3.

I can write out how many values of ml are allowed for each value of l.

So if l = 3, there are seven allowed values for ml, so

there should be seven f-orbitals.

Remember, l is an f-orbital if it equals 3.

Then I can start actually writing a spot that represents the orbital.

There are three of those spots for the p orbital, five of those spots for

the d orbital, seven of those spots for the f orbital.

So as the value for the principle quantum number gets larger,

the variation in the type of shapes we have increases.

Let's relate this now to the periodic table because that's what makes it

really easy to remember.

And remember the chemical elements have this periodicity to them and

we'll really be able to see that when we look at the periodic table.

So the quantum numbers relate to the periodic table.

And we can write an electron configuration which describes the electronic

structure of the atom by listing all the occupied sublevels and

the number of electrons in each of those sublevels.

Remember, experimentally, we can only determine the quantum numbers n and l.

So we can know for

a particular electron, which orbital it's in, what the value is for n,

how far away from the nucleus it is, and what the shape is of that electron cloud.

But we can't determine the orientation of the cloud, and

we can't determine precisely the electron spin.

We can only determine the electron spin relative to our external

frame of reference.

So if we look at the periodic table,

we see that we have what's called the s-block.

There is the p-block over here.

p-block.

So the lightest elements only have s and p types of orbitals.

In the ground state, anyway.

Once we get to some of the heavier elements, some of the transition metals,

then we start to have electrons in those d orbitals.

So this is called the d-block and the inner transition metals in some of

the very heavy late metals and even some of the non-metals,

the very heavy non-metals also have electrons in the f-block.

This was supposed to be green right here.

I'm just going to fix it.

There you go, that one's green, because that's also s.

Somehow it didn't get colored on my slide, but that's okay.

All right, what are the quantum numbers of the circled electron?

This is review, we've already done one like this.

But let's see if you can remember how to do it.

I want you to list what the possible values are for

the quantum numbers, for the circled electrons, and I'll give you a hint.

There's more than one possible set of quantum numbers.

Each set of quantum numbers is going to be a list of n and then l, and

then ml, and then ms.

And there's more than one set of quantum numbers allowed for this electron.

Go ahead and try to answer that now.

Thank you for answering.

This electron is in the 3p sublevel.

It's in the 3p orbital.

So n = 3, l = 1 because l always equals 1 for a p orbital.

M now can be one of three values.

It can be minus 1, 0 or positive 1.

ms we cannot determine exactly experimentally but

we have to agree that if the electron's symbol is an arrow that's pointed up,

we're going to say that's a spin of plus one-half.

So for this electron there are three sets of quantum numbers that would be allowed.

Three possible correct answers.

They all start with 3, 1, and then the third number on the list varies.

If you think back to the beginning of the week,

I asked you to draw a picture of an atom.

I said if we could see an atom, what would it look like?

And many people drew this picture, which looks like Rutherford's model for

the atom.

It gets used a lot as a symbol for atomic energy.

There's a dense nucleus that contains protons and neutrons and

there are some orbiting electrons that have this planetary-like orbit.

But we now know that even though this is a useful model for thinking about the atom,

it's not the best description of what the atoms really look like.

Particularly if we're thinking about what the electrons in the atom are doing.

So I'm going to say this isn't the best model.

It was certainly a useful model and

Rutherford made a huge leap coming up with it.

But now scientists say, well if we really think about what the electrons are doing,

and if we draw the clouds that come from the wave function of the electron,

in hydrogen where there's only one electron, that electron can be

in the different sublevels, right, and it can have different spins.

And that would give us all these different shapes,

these different clouds of electrons.

If you imagined overlaying all of these different clouds,

you would have an idea of what a heavier element might look like.

So quantum theory predicts the condition of the electron.

There are elements of probability and uncertainty there, and

it's a very useful theory for describing some of the experimental

evidence that we have, things like the emission spectral lines that we see.

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