AP Chemistry Quantum Mechanics & Electron Orbitals

Hello, and welcome back to Educator.com, and welcome back to AP Chemistry.0000

Today, we are going to continue our discussion of the electromagnetic spectrum and light, and we are actually going to be discussing quantum mechanics and electron orbitals.0004

We are going to be talking about what the actual electronic structure of the atom is, based on quantum mechanics, which is a development of about 100 years ago, roughly.0013

Before I actually get into the quantum-mechanical model of the atom, I want to sort of continue--finish up--the tail end of what we were discussing last time.0024

We had discussed light, the electromagnetic spectrum, and this notion that energy is quantized.0032

I'm going to mention one more thing, one more piece of experimental data that sort of caused this shift in thinking.0038

What I am going to talk about, before I get into the quantum mechanics, is something called line spectra.0047

Let's just jump in: so I'm going to draw some pictures here.0051

So, if I have just a regular light source (we'll just call this a regular light source), and if I shine it at a screen with a slit inside, so that only a single beam of light passes through, and if I pass this through a prism, well, you know what is going to happen==the same thing that happens after a rain, when you see a rainbow, or when you pass normal white light through a prism.0055

You are actually going to break it up into its spectrum of colors.0082

This is visible light: so you are going to have the red, orange, yellow, blue, green, indigo, violet--it is a continuum.0088

In other words, all of the colors from red to violet are represented in a continuum; there are no gaps.0094

You have all of these colors that you see; OK--these are not waves...well, they are waves, but it's just that everything is full here.0105

So now, if we do the same thing, but this time if we have a tube, and in this tube, if we have some hydrogen gas, and of course, we have wires running through; and if we ignite this hydrogen gas--spark it up a little bit--what happens is (it's pure hydrogen gas--nothing else in there): it actually gives off a little bit of light.0113

What is happening is: when we ignite it, we actually end up splitting the hydrogen molecule into 2 hydrogen atoms; and then, what happens is: the energy, that excess energy of the split, actually ends up promoting the electrons of the individual hydrogen atoms up to higher energy levels.0134

Well, they don't like being up there; when they fall back down to their normal (what we call "ground") state, they actually give off their energy as light.0156

The same thing: so this is also a light source--pure hydrogen--and again, we do the same thing: we have a little bit of a slit: we allow one beam to pass through; and when we pass this one through a prism, something very, very interesting happens.0163

What you end up with, actually, are lines of different colors.0178

The normal white light, when it passes through, gives you the whole continuum of red, orange, yellow, blue, green, indigo, violet; and as each one sort of passes into the other one, you get individual different hues; you get some purple; you get some red-orange; you get some blue-green...things like that.0198

Here, you are going to get a red line; you are going to get a yellow line; and you are going to get a blue or violet line; and that is it.0214

Everything else--there is nothing there; there are no colors here; it is completely black.0222

This is actually kind of interesting, because basically, what is happening is: these electrons are jumping up to certain energy levels, and when they fall back down, they are not just giving off energy of any frequency (which is the different colors, because visible light is just part of the electromagnetic spectrum, and it has different frequencies).0226

It is not just giving all of it; you are not going to see this whole thing, a whole bunch of lines; what you see are very specific numbers of lines.0250

That means that they are actually jumping up from a certain energy level to another energy level--not all of the little spaces in between.0257

It is not a continuum; so it turns out, it is more proof that energy is quantized--that only specific amounts of energy...the electron can jump up only according to certain steps.0266

It is not a continuum; in other words, you are not going to see a function like this--a nice, continuous function; what you are going to get is a jump up, and jump up, and jump up again...something like that.0280

That is quantum--that is why we call it a quantum: quantum is the Latin word for packet: so energy comes in certain basic, discrete units.0294

That is it: there is a fundamental unit; nothing smaller than that, nothing bigger than that; that is the unit, and it comes in increments of that unit.0302

OK, so let me write this down: so this means: the electron in the hydrogen atom is absorbing energy (the energy that we get from the current that we run through that hydrogen gas) of only specific frequencies, which it emits as light when the electron returns to its lowest energy state (what we call the ground state).0312

So, the electron in a hydrogen atom basically spends all of its time...most of its time...in the ground state.0368

If I want to promote it to a higher energy level, I have to put some energy into it: either I have to heat it up, or I have to ignite it with a spark--something like that.0388

It is promoted up: it doesn't like to be at the higher energy levels--it prefers to be at its ground state; it will fall back down.0398

That excess energy that it gained in going up, it will release as light: this is a very, very, very common them that we'll revisit several times over the next couple of lessons.0403

OK, so let's give a little bit of a pictorial of this: I wonder if I should do it on this...yes, that is fine; I'll go ahead and do it on this page.0414

Something like this: so we have this one level, another level...maybe that is another level...here is a ground state: so from here to here, and we said that the change in energy is equal to Planck's constant, times the frequency of the particular light that it absorbs and/or emits.0425

If it is going to absorb at a certain frequency, when it drops back down, that is the frequency it is going to emit: that is the whole point.0449

It is not going to be any more than that or any less than that; that is the frequency it is going to emit.0455

So, this change here, in going to maybe the first level up--well, that is going to be the first energy level; so it equals Planck's constant times the frequency of whatever light that happens to be.0459

This is going to be the second level; so the change in energy there is going to equal Planck's constant times that particular frequency, whatever it happens to be when it drops back down.0473

The energy is definitely quantized: it is not just going to jump up to every single level in between--it is not like the real numbers.0486

OK, the real numbers, from 0 to infinity--no matter where you drop your hand, or drop a knife, on the real numbers, you will always land on top of a real number.0494

The real numbers form something called a continuum; there are no gaps in the real-number system.0503

Energy is not like that: energy has very discrete units; so you can actually drop your knife somewhere--like, let's say, in between here--and you will hit a point where there is no energy value.0509

It has to have specific values: that is the whole idea.0521

OK, well, so let's move on: now, a model for the hydrogen atom was developed, and an equation was derived for these various energy levels of the orbits.0526

That is what it is: it is basically just a series of orbits: this is how we sort of think about it when we first learn about this stuff in, let's say, middle school or high school.0572

We think about the protons in the center; we think about these electrons travelling around the protons in circular or elliptical orbits (if you want to), simply as a way of getting it to have some sort of a picture that we can play with.0580

Now, that is not what is really happening; and we will describe what is really happening in just a minute; but that is a good way of thinking about it.0594

These orbits are sort of fixed of a certain radius, so one energy level has one radius; a little further out, there is another orbit that has an energy level; it has another radius.0600

So, when we say "orbit," we are talking about energy level; and that is what we want to concentrate on.0612

When we say "orbit," in our mind, we are picturing planets going around a sun...electrons going around the central nucleus.0616

That is fine, if you want to think about it; but what those orbits really are is: they are energy levels.0624

It is the energy of the electron in that orbit that matters, not really where it is, because, as it turns out, we don't really know specifically where it is.0629

We have a pretty good guess of where it is, but we can't really say specifically "it's there" or "it's there" or "it's there."0640

But, we can say specifically how much energy that electron has, and that is what is important.0646

In physics, it is energy that matters.0652

OK, so I'm going to draw a little picture: so let's say we have this central thing; so we have 1, 2, 3, so this is the positive...so any time...the energy associated with a given orbital is equal to -2.178x10^-18, times Z²/n².0656

n is an integer (in other words, 1, 2, 3, 4, 5, 6, 7, 8, 9--positive integer), and Z is the nuclear charge.0695

For a hydrogen atom, it's just 1, because there is only 1 proton.0708

This is in Joules.0712

So, for example, if this is 1; this is 2; this is n=3; so what happens is, well: if an electron is promoted from 1 to 2, and then it drops back down, it is going to give a certain line in the spectrum; it's going to give it off as light of a certain color.0714

If an electron is promoted to level 3, and it drops back down, it is going to give, let's say, 3 (actually, these are going to be a little further)...it's going to give another line, and so forth.0745

That is what that spectrum represents: it represents the electron jumping to a higher energy level, which is represented by an integer (1, 2, 3, 4, 5)--let's say 1 to 2, 1 to 3, 1 to 4; let's say it happens to jump from 1 to 2, and then maybe from 2 to 6--there is some excess energy.0765

And then, from 6, it drops all the way back down to 1; it is going to release energy, and you are going to get a different line on the spectrum; that is the whole idea.0781

The more energy you put into it, the more lines that are going to show up in the spectrum, because there is going to be more excitation.0789

This equation is one that you can generally use; it works pretty well; this is based on the Bohr model of the hydrogen atom.0796

Now, the Bohr model of the hydrogen atom is not the model that we use; the quantum mechanical model is what we use; but it actually ended up working pretty well for the time being.0804

So, for all practical purposes, it actually works pretty nicely.0812

We'll just go ahead and do one example of that, and then we'll put it behind us; but it's good to know.0817

Example 1: Calculate the energy required to excite the hydrogen electron from level 1 to level 3, and also calculate the wavelength of light that must be absorbed to effect this excitation.0823

OK, well, let's go ahead and start: so the energy 1 is equal to -2.178x10^-18, times 1 squared over 1 squared, equals -2.178x10^-18 Joules.0892

OK, so energy level 3 equals -2.178x10^-18 times 1 squared over 3 squared, equals -2.42x10^-19 Joules.0913

Yes, OK: ΔE, the change in energy, equals E₃ -E₁ (the difference in the energy levels to go from E₁ to E₃): you notice, at E₁: the energy at the 1 level is not 0.0939

OK, it is -2.178x10^-18, so it still has some energy; it is not a 0 value at the ground state.0952

There is still some energy there, OK?0963

The difference is 2.42x10^-18 Joules.0966

Now, we want the change in energy: well, we also know that the change in energy is equal to Planck's constant, times the frequency of the light, which is equal to Planck's constant times the speed of light over λ.0976

I'm sorry, we said we are looking for the wavelength: so wavelength is equal to hc/ΔE; ΔE is that thing; so we get 6.626x10^-34 Joule-seconds, times 3x10⁸ meters per second, divided by 2.42x10^-18 Joules.0997

Joules goes with Joules; second goes with second; we are left with wavelength; we are left with meter; so it confirms it, and our wavelength is going to be 8.2x10^-8.1027

This is in the ultraviolet range.1044

This means that, if I want to excite a hydrogen electron--the electron in a hydrogen atom--from level 1 ground state, to a level 3, that means I have to hit it with some ultraviolet light.1047

Now, not just any ultraviolet light: ultraviolet has a range; specifically, I have to hit it with 8.2x10^-8...light that has a wavelength of that.1058

Or, I can go ahead and put it back into the equation to find the frequency.1068

Frequency equals speed of light over wavelength, right?1072

OK, so that is it--just some basic equations to play with: the real idea that we want you to take away from this is that electrons are travelling around this nucleus; they have different energies; you can think of them as orbits that go further and further away.1076

As you go further and further away, those electrons have greater energy, and you can actually...electrons can jump from one energy level to another, if they absorb energy of a specific frequency, because energy is quantized.1092

They can't just make any leap: it has to be specific frequencies of energy that allow it--specific energies of light that promote it--and when it promotes it out, it drops back to a lower energy level; it emits that as light.1106

Now, you can actually test this, if you want, tonight.1120

When you are in your room getting ready to go to sleep, why don't you go ahead and just climb into bed, turn off all the lights, and (if it happens to be a reasonably dry day--this works better in the west than it does in the east, where it tends to be a little bit more humid), just rip the covers off you.1125

When you rip the covers off you, the static electricity--what you will end up with--you will get this little light show.1142

What is happening is that, as you pull things away, you are ripping electrons off; you are actually exciting electrons.1147

They are jumping up, and when they drop back down, they are going to give off their excess energy as light.1153

OK, so now, let us go ahead and move on to quantum mechanics and the quantum mechanical model of the electrons in an atom.1160

Quantum mechanics: all right, so we are just going to start off with the fundamental axiom of quantum mechanics, which is the Schrodinger equation.1174

OK, so the Schrodinger equation: basically, Ervine Schrodinger...remember, we talked about how particles have wave-like properties.1188

Well, the idea was: if a particle has a wave-like property--an electron has a wave-like property--well, there must be some wave equation that is going to govern its behavior.1202

Well, that is what Ervine Schrodinger did; he basically just took what he knew, and with a certain basic set of axioms, he came up with an equation which should, hopefully, govern the behavior of an electron as it wanders around a proton.1213

That is what the Schrodinger equation is: it is an equation that treats the electron as a wave.1233

So, as a wave, it is actually a wave equation; and our solutions are going to be...many of the solutions to the Schrodinger equation are going to be periodic functions; so they are going to actually be waves.1242

That is it: instead of thinking about it as a particle, the way we do with normal classical mechanics, the Newtonian mechanics that you have sort of been doing up until this point (or that you are doing now in your courses--your corresponding physics courses), what Ervine Schrodinger did is: he decided to treat it from its wave-like properties.1255

Because it is a wave-like property, well, you can develop a wave equation for it.1273

I'm just going to go ahead and write out the wave equation; I want you to see what it actually looks like.1278

You don't actually have to know this, but I would like you to see what it looks like; it is actually incredibly beautiful and reasonably simple--at least, simpler than other equations that you are going to run across in your studies.1283

It looks like this: minus h (that is Planck's constant, by the way), over 8 pi squared m (m is the mass of the electron); I'm going to use a regular derivative...d squared of f, over d, x squared, plus V(x), times f, equals E times f.1297

OK, so let me tell you what this means: now, you know (at least, you should have heard somewhere in your basic physics course) that the total energy of a system is made up of its kinetic and potential energy.1338

That is exactly what is going on here: what this says is: you can think of this, sort of, as the kinetic energy; this V is the potential energy, and this is the total energy.1349

So again, I am just trying to give you a nice, broad view of what this looks like.1362

What it says is this: the total energy of a particular system is equal to its potential, plus its kinetic; and there is a relationship that exists.1366

This relationship happens to do with this (so this is what we are actually looking for): in an algebraic equation, like something like x²+6x+6, when you solve that, you are looking for some specific number that satisfies that equation.1380

Well, when you move on to differential equations, it is the same thing; this is kind of like an algebraic equation; you can think of this as a variable, this as a variable, this as a variable, x; except now, instead of looking for a number, we are actually looking for a function that satisfies this equation.1393

That is it: it is just a higher level, but you are still doing the same thing: you are still looking for an unknown.1412

In this case, it is an unknown function: well, what is interesting here is that, in the theory of differential and partial differential equations, you don't just get one solution; you often get many solutions...an infinite number of solutions.1419

OK, here is what is interesting: I'm going to rewrite this in a form that is a little bit more tractable: h times f equals E times f; this h is kind of like this whole thing.1433

OK, it is something called an operator; you don't need to know what that is, but this is also another representation of it.1451

So, when we solve this differential equation, here is what ends up happening: this is what is important.1457

The solutions to this equation, which are functions, are the energy levels that the electron in a hydrogen atom can occupy.1465

We call these functions orbitals.1501

When we say "Where is the electron?", there is the electron; the energy level that an electron occupies is represented by a function--a function which happens to be a solution of the Schrodinger equation.1516

These functions are reasonably complex, but actually, they look complex symbolically; they are actually not.1533

They are actually reasonably straightforward, and not as counterintuitive as many would have you believe, when it comes to discussing quantum mechanics as a whole.1540

People want to make quantum mechanics seem like it's completely left-field--it's completely counterintuitive--it is not.1549

A lot of it just makes pretty good sense; yes, there are elements of it that are a little bit counterintuitive, but not as much as people make it out to be.1555

The complication in quantum mechanics is really more a question of symbolism, and sort of a busy whole lot of symbolism, than it is actual problems with understanding what is going on.1564

If you are ever in a position to actually take a quantum mechanics course, I do hope you avail yourself of that; it's extraordinarily, extraordinarily beautiful.1577

So anyway, this is how we want to start to think: so, when we say an orbital, we think of electrons travelling in orbits around; but really, what they are is: they are electrons that are occupying a certain energy.1586

As it turns out, if this is the nucleus, an electron can be here, or here, or here, or here; it doesn't get further and further away.1601

On average, on average, higher energy means further away; but an electron of higher energy can be here; it can have a higher energy than an electron here--because again, we don't know specifically where the electron is.1610

It can be anywhere: so these functions--these orbitals--they represent energies.1623

That is why you saw that E, times the function on the right-hand side of the equation; that is what that means.1627

We are calculating energies for these orbitals.1634

OK, now, there is a heavy statistical component to quantum mechanics.1637

OK, so there is a heavy statistical component to quantum mechanics: this is where we start to get into a little bit of the counterintuitive part.1671

When we square the wave functions (the solutions to the Schrodinger equation that we get--the orbitals--I'm going to put orbitals in parentheses here), we get a number.1684

This number is the probability that the electron of that energy will be at that location in space.1710

OK, so let's say this again: "when we take the square of the wave function"...so we have this wave function (whatever it is), and it represents the energy of the electron in that orbital.1743

We get a particular number when we take the square of it; well, this number represents the probability that we will actually find the electron of that energy in a particular region of space.1755

So, as it turns out, when we talk about the electron being here or here or here or here or here, we are not saying that it is really there; we are saying that chances are, if you were to actually be able to find it and trap it, you would find it, most likely, here.1767

This is the difference between quantum mechanics and classical mechanics, and this is the fundamental and only difference, really.1784

Classical mechanics is what we call deterministic, meaning any equation that you come up with can tell you where a particle is going to be two minutes from now, five minutes from now, two years from now...once I actually solve my function.1791

Quantum mechanics is not like that; by the very nature of these high speeds and the tiny, tiny...well, by the very nature of these particles and their wave behavior, it is not deterministic; it is probabilistic.1806

We can't say specifically that a particle is here or here, or it's travelling this fast or that fast.1820

I can tell you how much energy it has (that is nice--that is good; actually, for most purposes, that is perfectly fine).1825

But, I can't tell you where it is going to be; I can tell you most likely where it is going to be--that is it; that is as good as I'm going to get with quantum mechanics.1831

It is a little disturbing, because it is very...that is the part that is counterintuitive.1841

It is not counterintuitive by its very nature; it is counterintuitive because it goes against our experience.1845

Our experience is that, when we leave a set of keys on a desk, when we come back the next morning, it is going to be exactly where we left it.1851

Quantum mechanics doesn't behave that way.1858

It isn't deterministic; you can't determine that it is going to be there.1860

You don't know; it is a "toss-up"--it is a gamble.1864

OK, so let's go ahead and move on to some more practical aspects.1868

Let's talk about something called quantum numbers.1875

OK, quantum numbers: all right, so when we solve the Schrodinger equation and find these different orbitals (right?--we said there are many solutions to this differential equation), integers show up in these equations.1878

It is kind of interesting to think that the behavior of electrons, neutrons, protons, and things like that is actually governed by the integral number system--1, 2, 3, 4, 5, 6, 7--that is actually kind of extraordinary, when you think about it.1929

Integers: -1, -2...that is kind of crazy; but...so, when we actually find the solutions...equations...there are integers that show up in these equations.1942

These integers have specific meanings, and we call them quantum numbers.1952

OK, so there are going to be 4 quantum numbers: normally 3--the fourth is sort of an add-on.1957

We will say that we have 4 quantum numbers.1967

These quantum numbers--they tell us exactly in which orbital an electron happens to be: that is what these numbers represent.1970

Because again, these solutions are orbitals; they are energy levels, and they tell us the energy of the particular electron.1981

These atomic numbers are like addresses; you have 4 atomic numbers that actually...3 atomic numbers that give you an address for where an electron can be, and that fourth number will tell you in which direction it happens to be moving--whether it's left or right.1988

Let's go ahead and define them right now.2005

We have n, and n is called the principal quantum number, or the principal number; OK, and it corresponds to the rows of the periodic table.2008

First row, second row, third row, fourth row--those are the primary energy levels.2030

It corresponds to the rows of the periodic table; a little bit later on, we are actually going to discuss a periodic table, so we will be revisiting this.2036

Rows of the periodic table is the primary energy level.2045

As it turns out, the energy levels in an atom have levels to them; so, you might have a certain energy level, but that energy level contains several other energy levels; and those several others contain several others.2060

So, there are layers of energy levels, until we get to what we call an orbital.2072

OK, so n is an integer; it corresponds to the rows of the periodic table; it can be...so n is equal to 1, 2, 3, 4, and so on: 1, 2, 3, 4, 5, 6, 7--positive integers.2077

n can take on any one of those numbers.2093

OK, now, the second quantum number is l, and it is called the angular momentum quantum number, or angular momentum number.2096

I'll call it the angular momentum number; it corresponds to the shape of the orbital in space.2108

And it takes values from 0 to n-1; so if n happens to be 3, the l-values are 0, 1, 2; if n happens to be 5, the l-values happen to be 0, 1, 2, 3, 4; if n is 1, then the l-value is 0.2130

We give these specific names in chemistry: when l is equal to 0, we call that an s sub-orbital.2161

When l is equal to 1, we call it a p sub-orbital.2173

When l is equal to 2, we call it a d; when l is equal to 3, we call it an f; and so on...when l is equal to 4, we call it a g.2178

So, we have specific names for them.2189

OK, so now, let's go ahead and talk about the third quantum number: it is called the magnetic quantum number; this is m_l, and it is called the magnetic number--magnetic quantum number; OK.2192

Let's see: it corresponds to the orientation of the orbital in space.2215

So, the primary tells me what primary level it is; the angular momentum tells me what shape the orbital has; the magnetic tells me the orientation that this particular shape takes in space (in other words, does it lie along the x-axis? does it lie along the z-axis? Is it in between the x- and y-axes? Is it something entirely different?).2238

That is what this number does.2264

And then, of course, last but not least, we have something called the spin quantum number.2266

The spin quantum number--it corresponds to...well, I will actually discuss that in a second; it can be either +1/2 or -1/2; that is it--it can only take those two values.2273

It represents the direction that the electron happens to be spinning in--either this way or that way, clockwise or counterclockwise.2286

Now, that is not really what is happening: spin is an intrinsic property of an electron, the same way that mass is an intrinsic property; but we need to think about it in a way that makes sense to us, and we think about it as an electron being this ball that rotates.2294

That is why we actually call it "spin."2308

OK, now, some general notions before we actually build up the electronic levels: No two electrons can have the same 4 quantum numbers.2312

So, if I have an atom like oxygen, oxygen has 6 electrons altogether; each one of those electrons is identified by a different set of quantum numbers: n, l, m_l, and m_s.2336

Each electron is represented by 4 different quantum numbers.2355

What we are doing: these quantum numbers are a very, very specific and accurate way of identifying where an electron is and how it is behaving.2360

That is what these quantum numbers tell us.2369

When we solve that Schrodinger equation, we are getting a lot of information about that electron; in fact, we are getting every single bit of information that we need about that system in that solution.2371

Nothing is...well, OK.2381

OK, so a primary level has sublevels; those were the l's; so the n is the primary; the l is the sublevel.2384

And these sublevels have sub-sublevels; those are the m_l.2407

Each sub-sublevel can accommodate 2 electrons, each of opposite spin.2426

OK, and last but not least: a primary has as many (no, let's try this again; don't worry; I know this is a lot of talk and a lot of theory; it is important; we will get down to building them in just a second)...2454

A primary of level n has n sublevels; so the first primary has 1 sublevel; the second primary has 2 sublevels; the third primary has 3 sublevels; the fourth primary has 4 sublevels.2492

Now, some of those sublevels would have sub-sublevels, but the idea is: each primary has that many sublevels.2518

Now, let's go ahead and build it up.2524

OK, so I think I'm going to do this on...let me see...yes, let me start a new page here.2528

OK, so I'm going to have n; I'm going to have l; I'm going to have the sublevel, which corresponds with l; I'm going to have m_l; and last but not least, I'm going to have the number of sub-sublevels.2538

Those are actually the number of orbitals that occupy two electrons, and I'm going to draw them out, too.2573

So, in n=1, we said that l can go from 0 to n-1, so l is only 0.2578

Well, when l is 0, the name for that is...we call it an s sub-orbital.2586

The m_l value--they go from -l to +l; so -0 to +0; so this one is just 0.2593

And it has 1 sub-sublevel; and each sub-sublevel contains...we said can carry 2 electrons of opposite spin.2606

We signify them like this; so, that is it: the primary level; it has an s sublevel; and that s sublevel only has 1 sub-sublevel; and it can occupy 2 electrons--always the case.2618

Let's go to n=2.2641

n=2; l can equal 0 or 1; well, when l equals 0, we said that we call that an s suborbital; when it's 1, it is called a p suborbital.2643

s, 0; actually, let me fill these in later; there is 1 orbital there; a p--this is 1, so our magnetic quantum number can be (it goes from -l to +l, passing through 0): -1, 0, 1.2660

-1, 0, 1; there are 3 here; therefore, there are three sublevels; this magnetic number (the number that you see here--not the actual number itself, but the number--here you see 1; here you see 3)--that tells you how many sub-sublevels there are--how many actual orbitals there are.2685

That is what that number tells you.2706

The second primary level is made up of 2 sublevels: the first sublevel has 1 orbital; the second sublevel has 3 orbitals, for a total of 4 orbitals.2710

Each one of those orbitals can accommodate 2 electrons of opposite spin.2722

OK, let's do #3: l goes from 0 to n-1, so 0, 1, 2; so we have a 0; we have a 1; we have a 2.2730

The 0 is called an s; for l=0, m_l equals 0.2744

This is called a p: for l=p, this equals -1, 0, 1.2751

When l=2, we call it a d suborbital (sublevel, suborbital; some people call it a suborbital), and in d, the m_l runs from -l to +l, passing through 0; so we have -2, -1, 0, 1, 2.2759

We have one s orbital that can carry 2 electrons; we have 3 p orbitals--each can carry 2 electrons; and we have 1, 2, 3, 4, 5 d orbitals; each can carry 2 electrons, for a total of 10.2781

Now, let's do n=4, the fourth primary energy level.2803

It has 0, 1, 2, 3 sublevels--1, 2, 3, 4--see, 4 has 4 sublevels; 3 has 3 sublevels; 2 has 2 sublevels; 1 has 1 sublevel.2808

The 0 is the s; the 1 is the p; this is the d; this is the f.2822

The s is just 0; the p is -1, 0, 1; -2, -1, 0, 1, 2; here we have -3, -2, -1, 0, 1, 2, 3.2829

This is (oops, here we go with those crazy lines again--we definitely don't need those): so the s has one orbital; the p has 3 orbitals; the d has 5 orbitals; and the f has 7 orbitals.2848

Notice the pattern: 1, 3, 5, 7; and it is always the case.2873

Notice, s has 1 orbital; s has 1, p has 3; s has 1, p has 3, d has 5; s has 1, p has 3, d has 5, f has 7.2876

That is the pattern: primary, sublevels, sub-sublevels.2889

That is what the quantum numbers represent: each of these can accommodate 2 electrons; I won't go ahead and fill them all in, but you know where we are going.2896

This is the buildup process: on the periodic table, as we go from hydrogen to helium to lithium to beryllium to boron to carbon to nitrogen to oxygen to fluorine to helium to sodium to magnesium, aluminum, and so forth...silicon, phosphorus, sulfur, bromine...as we move down, we are adding electrons.2905

Those electrons are being put into these orbitals.2931

So, each one of these orbitals right here has an address; that address is made up of three numbers.2934

It is made up of the primary, the angular momentum, the magnetic.2943

Each electron in here has a spin; so each electron--any electron--is made up of 4 different quantum numbers; those 4 numbers are the numbers that show up in the solutions to the Schrodinger equation.2950

They tell me where the electron is and in what direction it is moving.2965

OK, so let's go ahead and finish this off here: so, when we say "orbital," we are talking about each one of those--what I am calling a "sub-sublevel."2971

All right, OK; each orbital has 3 quantum numbers associated with it.3019

These three numbers--these three quantum numbers--tell you where an electron is.3044

The fourth number--the fourth quantum number, which is the spin quantum number, m_s, tells you the spin of that electron.3060

Now, in the next lesson, we are going to introduce sort of a shorthand notation; so we are not going to be working specifically with quantum numbers.3081

We will do some problems, especially when we actually do a review for the AP test, and certainly there are questions telling you which quantum numbers are permissible, which ones are not...3087

You just look at the table we just did; you just build it up like that, based on the rules of coming up with the l and the m_l and things like that.3097

And, if certain things are possible, they are possible; if not, they are not; other than that, when we want to know where a particular electron is, we're just going to use the shorthand notation, and we are going to use the primary--those numbers--the n's--and we are going to use the s, p, d, and f, and then we are just going to go ahead and count the total number of electrons in that orbital.3106

So, that is what we will actually start with the next lesson.3128

I know that this was a lot of sort of theoretical stuff, but I didn't concentrate so much on the theory; I wanted you to get an idea of what exactly was going on, so the real take-home lesson here is that these orbitals are just energy levels.3131

When we drop electrons into atoms, as we get more and more--as we run down the periodic table, sort of creating these polyelectronic atoms--bigger and bigger atoms--we are just throwing electrons into these energy levels; that is all that is going on.3150

But, they don't go in randomly; they go in specific places, and each one of these orbitals has an address.3166

For every orbital, you can have 2 electrons; they have to have opposite spin.3172

That is the whole idea.3177

It is not so much a question of electrons flying around in a circular or elliptical orbit, around the nucleus; that is not what is happening.3178

You can think of electrons more like a collection of gnats, if you will; have you ever walked into this sort of cloud of gnats?3190

That is what it is: the electron can be anywhere--it can be close; it could be far; it is the probability.3197

There are certain probabilities that we can say: "It's more likely to be here" or "be there"; those probabilities--that is what gives shape to the orbitals.3204

But, other than that, an orbital is just a mathematical function; that is what it is.3215

We need to think about it pictorially so that we have something in our mind to work with, but you can think of it as just places where electrons stay around, of a given energy level.3219

That is what is important: they are energy levels.3229

OK, with that, I thank you for joining us here at Educator.com.3232

We will see you next time for a further discussion of quantum mechanics and electron configuration; goodbye.3235