For more information, please see full course syllabus of AP Chemistry

For more information, please see full course syllabus of AP Chemistry

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### Quantum Mechanics & Electron Orbitals

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Quantum Mechanics & Electron Orbitals
- Quantum Mechanics & Electron Orbitals Overview
- Electron Orbital and Energy Levels for the Hydrogen Atom
- Example 1
- Quantum Mechanics: Schrodinger Equation
- Quantum Numbers Overview
- Principal Quantum Numbers
- Angular Momentum Numbers
- Magnetic Quantum Numbers
- Spin Quantum Numbers
- Primary Level, Sublevels, and Sub-Sub-Levels
- Example
- Orbital & Quantum Numbers

- Intro 0:00
- Quantum Mechanics & Electron Orbitals 0:51
- Quantum Mechanics & Electron Orbitals Overview
- Electron Orbital and Energy Levels for the Hydrogen Atom
- Example 1
- Quantum Mechanics: Schrodinger Equation
- Quantum Numbers Overview
- Principal Quantum Numbers
- Angular Momentum Numbers
- Magnetic Quantum Numbers
- Spin Quantum Numbers
- Primary Level, Sublevels, and Sub-Sub-Levels
- Example
- Orbital & Quantum Numbers

### AP Chemistry Online Prep Course

### Transcription: 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 Z^{2}/n^{2}.*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 E _{3} -E_{1} (the difference in the energy levels to go from E_{1} to E_{3}): you notice, at E_{1}: 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 3x10^{8} 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 x ^{2}+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

2 answers

Last reply by: Gaurav Kumar

Thu Oct 15, 2015 11:49 AM

Post by Gaurav Kumar on October 12, 2015

Professor Hovasapian,

I am confused to how you found the magnetic quantum numbers. Can you please explain a little more?

Thank you so much

0 answers

Post by Isaac Martinez on April 12, 2015

Good day Professor Hovasapian,

I am a little concerned about your final answer on example one. I think you might have incorrectly labeled the power on the wavelength. E3+ -2.42 X 10^-19j. When you calculated for the wavelength of the light you wrote -2.42 X 10^-18j.

Thank you so much for your lecture :)

2 answers

Last reply by: Professor Hovasapian

Fri Feb 27, 2015 1:38 AM

Post by aimun amatul-hayee on February 24, 2015

Hey Mr. Hovaspian,

So the n stands for the energy levels, and l is the sublevel, and ml is the number of sub levels, so how will you calculate the amount of electrons in a configuration, such as 2,1,3,1

1 answer

Last reply by: Professor Hovasapian

Thu May 8, 2014 2:52 AM

Post by Isela Morales on May 7, 2014

I know you have heard this before, but I have to say it again....

YOU ARE AMAZING PROFESSOR!!!!!!! educator has great teachers, but you are THE BEST!!!! Thank you for helping me understand the subject. By the way, I love your hair ;) please let it grow out, makes you look like a young Einstein.

0 answers

Post by Julie Mohamed on May 5, 2014

you should do organic chemistry. i really think you are a wonderful professor

1 answer

Last reply by: Professor Hovasapian

Sun Jul 14, 2013 5:37 PM

Post by KyungYeop Kim on July 13, 2013

great lecture professor Raffi! After watching this, I can't wait to learn more about Quantum Mechanics. By the way, I saw that your lecture on [Physical Science] is coming out soon. (Which I'm very excited about) Would it be suitable for a HS student like me? I was hoping maybe you could lecture on physics, but that seems improbable since there's already two on this site. But I'll keep my fingers crossed :)

1 answer

Last reply by: Professor Hovasapian

Sun Jun 23, 2013 5:10 PM

Post by Archimedes S on June 23, 2013

Hi Professor Hovasapian. Will the energy you put into the atom be the same energy you get out? For instance, in example 1, if I hit it with the UV light to excite the electron to the 3rd energy level, when it drops back down will the light it emits be the same as the incoming UV?

1 answer

Last reply by: Professor Hovasapian

Thu May 16, 2013 2:46 AM

Post by Rajendran Rajaram on May 15, 2013

hello raffi,

Isnt The charge in E in 1.936 X 10^-18 joules? different websites tell me different things

I hope all is well with you,

1 answer

Last reply by: Professor Hovasapian

Tue Oct 16, 2012 1:31 AM

Post by Abdihakim Mohamed on October 15, 2012

awesome way to explain :) love it!

3 answers

Last reply by: Professor Hovasapian

Mon Oct 29, 2012 4:48 AM

Post by Riley Argue on October 8, 2012

I agree with "Shadd" :)

0 answers

Post by Shadd Watson on May 17, 2012

Mr. Hovasapian, you really make me excited to learn chemistry. Your clear understanding of the topics comes through in your clear teaching as well. Thanks so much for your videos!

1 answer

Last reply by: Professor Hovasapian

Mon Jul 16, 2012 12:26 AM

Post by Jeff Walter on April 5, 2012

Isn't the change in "E" equal to 1.936 x 10^-18 Joules?