Quantum Numbers and Atomic Orbitals Overview

Quantum numbers and atomic orbitals are going to be the topics in this lesson, and we'll start with the atomic orbitals actually, and we'll talk about s orbitals, p orbitals, d orbitals. We'll talk about how the shapes here represent what we call wave functions, and how these wave functions are solutions to what we call the Schrodinger equation, ever so briefly, because we're only going to talk about what you need to know for your general chemistry course. We're also going to talk about quantum numbers. Turns out there are four quantum numbers, n, l, m sub l, m sub s. We'll talk about the name of each one, what they tell you, and essentially, an easy way of remembering what they do because they're essentially an electron's address in an atom. My name is Chad, and welcome to Chad's Prep, where my goal is to take the stress out of learning science. Now, in addition to your high school and college science courses, we also offer prep courses for the MCAT, the DAT, and the OAT. You can find those courses at chadsprep.com. Now, this lesson's part of my new general chemistry playlist. I'm releasing several a week throughout the school year, so if you want to be notified every time I post one, subscribe to the channel, click the bell notification. So let's dive into this. So here I've got diagrams of some of the most basic orbitals here, and a couple things you should know is that any orbital in the universe can hold a maximum of two electrons. So you can find orbitals empty, you can find them half occupied with just one electron, or you can find them full being occupied by two electrons, as we'll see. There's s orbitals, p orbitals, d orbitals, there's also f orbitals that we'll definitely talk about in this course. But it turns out there's also g orbitals and h orbitals that you probably won't talk about in this class because we don't actually have any stable elements that have electrons in them normally. But we will talk about the f orbitals, but it turns out the shapes of the f orbitals are more complex that I can't draw them for one. But you typically won't even find illustrations of them in your textbook. But if you Google them, you will find shapes for them. So it turns out where these shapes come from. So it turns out they are mathematical, three-dimensional mathematical equations that we call... wave functions, and this is just the three-dimensional graph. So this is not two-dimensional. Think of this as a three-dimensional sphere. We like to think of the p orbitals as being dumbbell-shaped, but again they're three-dimensional in space. And then the d orbitals are either going to be four-leaf clover-shaped, or they're going to have this kind of toroidal shape, we say. And it turns out that there's just a single s orbital. There are three p orbitals, but there are five d orbitals, and then there are seven F orbitals. And if you see the trend, so again, 1s orbital, 3p orbitals, 5d orbitals, 7f orbitals, 9g orbitals if we cared, 11h orbitals if we cared. They count on the odd numbers. We'll see why when we get to quantum numbers, why exactly that is. But you should know it. Well, for the d orbitals, I just got lazy and didn't draw them all. But four out of the five, turns out they get labeled. dxy, dxz, dyz, dx squared minus y squared. Four out of the five look like a four-leaf clover. What's going to be different though is how they're oriented on the, you know, in space. So, dxy it turns out is in the xy plane, dxz in the xz plane, dyz is in the yz plane. So, and then dx squared minus y squared it turns out is also in the xy plane, but whereas dxy is in between the x and y axes, dx squared minus y squared would be right on the x and y axes. And then dz squared here, so it turns out the big lobes here are going to be on the z axis. and then this lovely circular portion, the donut portion, if you will, is gonna be in the xy plane. Cool, so I just wanted to distinguish that, but I wanted you to realize that even though I'm lazy, there are five d orbitals, even though I only drew two of them here. So, but these four lobes represent a single orbital, and again, you could hold a maximum of two electrons in an orbital like that. Okay, one thing you should know is that I'm giving you the most basic version of each of these orbitals. So, The first time you'll ever hit the lowest shell number that actually has an s orbital is the first shell. The lowest shell number that has p orbitals is the second shell. The lowest shell number that has d orbitals is the third shell. But as you go higher and higher, it actually turns out that those same type of orbitals that you encounter again get a little more complex, a little more complicated. So like when you go to 2s, it's a bigger sphere, okay? But it also turns out has what's called a node in it. And it's kind of like, you know, putting a softball inside a basketball. so to speak, and the air space in the middle is going to be a node where you don't find the electrons. And you could find them in the softball region in the middle, or you could find them in the outer region, but there's a hollow gap in the middle that you don't find them. It's really weird. And they get more complex as you go up. As you go up to like 3s, you'd have two of those nodes. We call them radial nodes and stuff, and not super important at this level. So, but if you take a more advanced like physical chemistry class, or if you're taking maybe a majors class, they might go through something like that. So, but the same thing happens with the p orbitals and d orbitals. If you go from 2p, if you go and talk about the 3p. these orbitals get bigger, but again, they have these radial nodes incorporated into their shapes and stuff. So I'm not going to say any more than what I just said about them. I just want you to realize that there are more complicated shapes when you get to the higher shell versions of these same types of orbitals. All right. Now, one thing I did say, though, is that as you go from like 1s to 2s to 3s, they get bigger. And if you recall, then, that means the electrons that are going to occupy them get higher energy because, again, The closer an electron is to the nucleus, the lower its energy, as we learned a couple lessons ago. So it turns out then that the lower the shell number, the smaller the orbitals, and the lower the energy of the electrons that are in those orbitals. And so back when we studied the Bohr model of the atom, Bohr thought electrons went around the nucleus in these two-dimensional circular orbits. What's really true we find out though is that it's actually these three-dimensional orbitals that electrons actually live, and they have a wave-like property that Bohr really didn't take into account. Alright, so these wave functions in some way shape or form describe where an electron may be found. Okay, so in addition to knowing these shapes, now we've got to talk about these lovely quantum numbers. It turns out there are four of them. There's n, there's l, there's m sub l, and there's m sub s. You should know the name of each of these quantum numbers, you should know what they tell you, and you should know what range of values are possible. So if we look at n first, so n is going to tell you, or n is called, I should say, the principal quantum number. And it's going to tell you what shell its electron is found in, what shell the orbital, I guess, really is in that the electron is found in. And in this case, you have like the first shell, the second shell, the third shell. And again, these are the same n values that correspond to the Bohr model of atom, like the first orbit, the second orbit, the third orbit, according to Mr. Bohr there. So in this case, we're going to say that they give you the shell number. And if we look at the range of values that are possible, so in this case the lowest shell is shell number one, and then it turns out just like with Bohr's orbits could go up to the infinitieth orbit, so can the shells. The first shell all the way up to infinity. So you should know though that in the ground state we only have atoms with electrons up to like the seventh shell, but again we can verify that the additional shells do indeed exist. by looking at different electron transitions where electrons are actually promoted up into those higher energy orbitals. Okay, so that's our principal quantum number. Next on the list is L here, and L is often referred to as our azimuthal quantum number. And your azimuthal quantum number is going to tell you the subshell. And you can kind of think of this as a code here. So the subshell In this case could be S, P, D, or F, or again technically G or H, but you're never gonna see it. So S, P, D, or F, and it's a code here. So 0 is gonna tell you that it's an... let's put this in blue actually, just like it is on your study guide. So when L equals 0, a value of 0, that tells you you're an S subshell. When L equals 1, it tells you you're in a P subshell. When L equals 2, it tells you you're in a D subshell. And when L equals 3, it tells you you're in an F subshell. So these quantum numbers are going to describe, and I say you, but it's really describing where an electron is found in an atom. And so if I say L equals zero, you're supposed to know, oh, that electron is in an S orbital somewhere. So if I say L equals one, you're supposed to know, oh, that electron's found in a P orbital somewhere. And we call it the P subshell because in any P subshell, there are actually three P orbitals, one on the X axis, one on the Y axis, one on the Z axis. And when L equals one, the electron that has that value of L equals one has an electron in one of those three orbitals and somewhere in the entire p subshell. All right when l equals 2 that electron is in the d subshell one of the five d orbitals and when l equals 3 that electron's in the f subshell one of the seven f orbitals. All right if we look at range of values here so we like to define the range of values for l in terms of n so and it turns out it can start from a minimum value of zero up to a maximum of n minus 1. So we'll see the profound implications of this here in a sec. But in this case, notice the lowest value you could have is 0. We saw that. But the highest value it takes depends on what shell you're in. If you're in shell number 1, i.e. n equals 1, well, 1 minus 1 is 0, and 0 up to 0 means the only values it can take is 0. And so here from 0 to n minus 1, all the integer values in between, and I should have said that up here as well. Your range for the shell number is any integer value from 1 to infinity. So 1, 2, 3, 4, but not like 3.476. So all integer values. Cool. So if you're in the second shell, n equals 2. Well, 2 minus 1 is 1. And so it can take any integers from 0 up to 1. And so in the second shell, it could be 0 or it could be 1. And in the second shell, there are s and p orbitals only. In the first shell, there were only s orbitals because the only value l could take was 0. In the third shell, you've got 0 up to 3 minus 1, 0 up to 2, so 0, 1, and 2, and 0, 1, and 2 mean that in the third shell there are going to be S, P, and D subshells, i.e. S, P, and D orbitals. And so every time you go to a higher shell, you get something new, it turns out. In the fourth shell, you've got L taking on values of 0 up to 4 minus 1, i.e. 0 to 3, so 0, 1, 2, and 3. Because in the fourth shell, there are S, P, D, and F subshells. Cool, and that's the way it works. And technically, if you go up to the fifth shell, that's where you'd actually encounter the G orbitals, the G subshell. But again, no stable elements actually put any electrons in there, and we don't ever talk about them. So I just bring it up just so you can follow the pattern here. So all you've got to know is up to F orbitals. Cool, but that's how the azimuthal quantum number works. And so... Moving on to m sub L, and m sub L is our magnetic quantum number. So in this case it's going to tell you, it's going to actually identify a specific orbital, but oftentimes we say that it tells you the orientation in space. But it's going to specify an orbital, so it gives us a specific orbital. And so these mean the same thing technically. Like if you look at the three different p orbitals, one's on the x-axis, one's on the y-axis, one's on the z-axis. So when we say that the magnetic quantum number m sub l here specifies the orientation in space, it's gonna tell you if a p orbital's on the x-axis, on the y-axis, or on the z-axis. In that way, it's specifying an individual orbital in that subshell. Same thing with the d orbitals. They differ in their orientations in space. And so when m sub l... specifies an orientation in space, it's actually specifying an orbital in whatever subshell you're in. And it turns out if we take a look at the range of values here, it goes from a minimum of negative L as an integer and all the integer values in between up to positive L. And so just like L is bounded by what value of n you have, well M sub L is going to be bounded by what value of L you have. So if you have L equals 0, L equaling 0 means you're in an S subshell, and in an S subshell there's just one orbital. And so you only have to specify one orbital, and in going from negative 0 to positive 0, there's only 0. And there's only 0 because you only need to specify one orbital for an S subshell. Now if we've got a P subshell, in a P subshell L equals 1, and M sub L could therefore go from negative 1 up to positive 1. That means negative 1, 0, and positive 1. three different values because we have three different p orbitals that we need to specify their orientation in space. And so one of these corresponds to negative one, one of them corresponds to positive one, one of them corresponds to zero. That's beyond the scope of this class though. you just have to know that the three possible values would be negative 1, 0, and 1 for the p orbitals. If we go on to d orbitals, for d orbital, L is equal to 2, which means that M sub L could take on a range of values from negative 2 to positive 2, which means negative 2, negative 1, 0, positive 1, positive 2, and there's five different values because there are 1, 2, 3, 4, 5 different d orbitals. So again, if L equals 3... L equaling 3 means F orbital. There are 7 F orbitals. And that's going to allow us to get 7 different values now for M sub L. So from negative 3 to positive 3. So that means negative 3, negative 2, negative 1, 0, positive 1, positive 2, positive 3. 7 different values because each of those F orbitals is going to get a specific value of M sub L because it's identifying or specifying an individual orbital. Cool. Now it turns out that these 3 quantum numbers, so we're kind of, you know, deriving them based on what they actually tell us, and they're kind of giving us an electron's address it turns out, but they actually come from the solutions of the Schrodinger equation. They fall right out of the Schrodinger equation, but this last one, m sub s, called the spin quantum number, does not come from the Schrodinger equation. I've seen that tested on a timer too. It's the only one that doesn't come from the Schrodinger equation, but it turns out that an electron has a spin associated with it, and what that spin actually is, I can't... actually tell you, but it is a fundamental property of matter. So particles of matter, these individual subatomic particles, have spin associated with them much of the time, and this spin is weird. Now it turns out that something with spin can interact with a magnetic field when it's moving, and usually we attribute that to charged particles and things of this sort in a weird way, and the idea is that if an electron was rotating around the nucleus in one direction, it would have a magnetic field associated with it, and if it was going around the nucleus in the opposite direction, it would actually have a magnetic field associated with it in the opposite direction. Well, it turns out that these don't even have to be rotating around the nucleus, and they don't really go around in nice circular orbits anyways, and even when they're not rotating around a nucleus, they still interact with a magnetic field and have a characteristic magnetic field associated with them that allows them to interact with a magnetic field, and so it turns out it... it's not that they're spinning around the nucleus in any way shape or form it's just some fundamental property that they have so we call it spin and based on the idea that maybe it's revolving around the nucleus spinning around the nucleus but it turns out it has nothing to do with that it's just a fundamental inherent property of electrons and it's not just electrons most particles have a spin associated with them okay it turns out that this spin is actually what we're going to represent here. It takes on two possible It is either plus one half or minus one half. So we don't have this range of values. There's just two values possible. So it turns out there's only two types of spin. So, and you can kind of think of this. There's only two types of charge in the universe. And we identified them as positive and negative charges. Now notice, that's just what somebody called them. We could have called them left charges and right charges, or up charges and down charges, or blue charges and green charges, or we could have called them anything and just known they were opposites. So, well here, we've got these opposite spins, but it turns out they have actual numerical values associated with them that are indeed opposites. So, plus one half and minus one half, and we're going to find out that we're going to represent electrons here in a little bit with arrows like so. And we're going to have one arrow point up and one arrow point down, and these arrows represent electrons, and we represent the opposite spin quantum numbers for them by representing the arrows pointing in opposite directions. Now, we've got these lovely four quantum numbers here. And in this case, we know their names, we know their symbols, we know what they tell us, and now we know the range of values they can take. And there's a few different kind of questions you might have to answer. Now, one thing we've got to give a prelude to the next lesson, and that's called the, we've got to talk about what's called the Pauli Exclusion Principle. And it essentially excludes any two electrons from having the same four quantum numbers. So... every electron in an atom is described by some combination of four quantum numbers. And no two electrons in an atom can have exactly the same set of four quantum numbers. What that ultimately means, because again, the set of four quantum numbers ultimately describes an electron's address, it means that no two electrons can live in the same orbital and have the same spin, as it turns out. Now, what do I mean by these being like an electron's address? Well, think about Think about this. Let's say I gave you my address and I said that I live in state number 48. And so I didn't tell you what state I lived in. I just assigned a number value to my state. Well, if you've heard me say it in other videos, I live in Arizona. And the reason I call it state 48, that was a code because Arizona was the 48th state to be made a state in the union, if you will. So that's where that code comes from. And then maybe I tell you, hey, I live in city number 7. So, and again, you might... start doing some research and find out, you know, what was the seventh city in Arizona? And I have no idea if this is correct, but I live in Tempe. So we'd be like, oh, the seventh city in Arizona was, you know, Tempe. And I'm just giving you a code. And so I'm like, I live in state 48. I live in city number seven. And so, and then I might tell you, I live in street 104. And then we might do some research and find out what was the 104th street to be paved in the city of Tempe or something like that. And then I might finally give you a house number. And so instead of giving you my address in normal terms like house number, street, city, state, I'm giving you numerical values as a code for this. Well we're doing the same thing here. When I give you a set of four quantum numbers, it's a code for where an electron lives. And we'll take a look at how this works here. All right, so here's a common diagram we use to represent the energy location for the different subshells and orbitals in an atom. And the lowest energy orbital is a 1s, so this box represents the orbital. And then we're going to draw arrows inside these boxes to represent the electrons. So we might put an electron like here, and then when the next one turns out we'd have to draw would be spin down. So it turns out one spin up, one spin down, and again that corresponds to the opposite values of the spin quantum number as we'll see. And based on where an electron is found somewhere in this diagram, we can assign values to n, l, m sub l, and m sub s. What you'll find is that most of the time you can assign a specific value to n and l, But you might have a range of values that are possible for m sub l, and you might not be able to specify which it is. And then m sub s, you're probably just going to know it's either plus 1 half or minus 1 half, and good chance you might not know which is which. All right, so let's say we talk about one of these electrons. It doesn't matter which one. Let's just erase one of them and say we're talking about one of those 1s electrons. So that electron, and we might say, what is its set of quantum numbers? Well, it is in shell number one here, and that one is the principal quantum number. And so here n would equal one. So and then l, since we're in an s subshell, you remember, l tells you the code. l equals zero means s, l equals one means p, l equals two means d, l equals three means f. So in this case, because it's an s subshell that that electron is in, l is going to equal 0. And then you recall that m sub l takes on values from negative l to positive l. So in this case, from negative 0 to positive 0 means the only possible value is 0. Cool. And then finally, m sub s. Well again, we said that when you have two electrons in the same orbital, we would draw them as one spin up, one spin down. So, and in this case, when you only draw one of them, though, and it doesn't matter if it's spin up or spin down, we actually don't specify that, you know, spin up means plus one half and spin down means minus one half. There's no specificity there at all. We just know we have one of them. This is either the plus one half spin or the minus one half spin. We don't know which. And so all we can say here is that it's either plus one half or minus one half, and we can't get any more specific to that. Now, what I could tell you is this. If I drew both electrons in this particular orbital, And if I told you the one on the left is the minus one half electron, well, then you'd know that the one on the right would have to be the plus one half electron. And the key is they have to have opposite spins because if they had the same spin, if they were both plus one half or both minus one half, well, they're both in shell number one. They're both in the S orbital. They're both in the same S orbital. And if they had the same spin, then they would have the same four quantum numbers. And again, the Pauli exclusion principle tells us that that is not possible. And it's not possible because Pauli said so, Pauli was just describing nature. It just turns out that nature doesn't put two electrons in the same orbital with the same spin, it's just the way it works. Cool. So that was that electron. So let's say we put an electron somewhere else in this lovely atom. And so a couple things you realize about the orbitals here is that we're going increasing energy as we go up. So notice shell one is lower than shell two, shell two is lower than shell three, and shell three is lower than shell number four of electrons. But also note that, you know, when you get to the higher shells, you got like S's and P's. The P's are higher in energy than the S. The D's are higher than energy than the P, which are higher energy than the S. And then F are higher energy than D, are higher energy than P, are higher energy than S, with one exception. So for any system that only has one electron, it turns out that the way this is represented is not true. And the most notable atom or ion that only has one electron is hydrogen. And for something with only one electron, most notably, again, the only neutral species is hydrogen, It turns out that all the sub-shells and orbitals within a single shell are going to be equal. So instead of having the P higher than S, they would all be at the same level, all the same energy. And that's only true for one electron system. Same thing here, 3S, 3P, 3D. For hydrogen, they're all equal energy. But for any multi-electron system, they split into different energies. And we'll talk about that at a much later date, why that is and stuff like that. So just something you should realize. Alright, so let's say we've got an electron now, and let's put this electron right... here. And so now the questions again are what are the possible, you know, what is the four, or the possible sets of four quantum numbers that that electron could have? Well in this case it's in the 3p, so that means it's in shell number three, so n is going to equal three, and it's in a p subshell, and again l tells you the subshell, and the p subshell corresponds to l equals one. Cool, now here's the deal. We got to go to m sub L, and m sub L takes on a range of values from negative L to positive L, so in this case negative 1 to positive 1, so either negative 1, 0, or positive 1. And on a diagram like this, we don't specify. It could be any of those three, and so we can't actually get it specific here. We can only say that it's either negative 1, 0, or plus 1. It's one of those three. It can't be negative 2, it can't be positive 3, but it's got to be one of these three specific values. but can't get any more specific than that. And then finally m sub s, same thing as before, it's either plus one half or it's minus one half and we can't get any more specific than that either. And so you might get a question on the test that says, you know, multiple choice, let's say, and which of the following is a possible set of quantum numbers for that electron right there or for an electron in a 3p orbital? Well, any answer that doesn't have n equals three is wrong. Any answer that doesn't have l equals one is wrong. But notice for M sub L, there are three possible correct answers here. Either negative one, zero, or positive one. And then for M sub S, there are two possible correct answers, plus one half and minus one half. And you've got to find the combination of four that are all within this range. So with three different options here and two different options here, there would actually be six possible correct answers they could include on your answer sheet. Obviously, they're only going to include one of those options as a correct answer. Cool. The other way you could see quantum numbers showing up is they might just start giving you different combinations of quantum numbers and asking which of the following is even possible. So which of the following is a possible set of quantum numbers? So let's say we start messing with this a little bit. And so I first tell you that n equals 4. Is that possible? Yes. But what if I told you it was n equals negative 4? Well again, n only takes on a range of values from 1 at the lowest up to infinity. it can't be a negative number. So that whatever else we fill in for L, m sub L and m sub s, this would not be a possible set of quantum numbers. But as long as I just said n equals plain old four, positive four, okay, so far so good. Now, what values could L have here? Well, in this case, you got you remember that L takes on a range of values from zero at the minimum to a maximum of n minus one, in this case to a maximum of three. And so in this case, if I put It's not, let's just say four again. Well, then again, no matter what I fill in for m sub l and m sub s, you'd know that this is not a possible set of quantum numbers because with n equals four, l could be zero, one, two, or three, but it couldn't be anything greater than that. So that wouldn't be possible anymore. Okay, so let's say I put one of the possible numbers. Let's say I put l equals two. And again, it could have been zero, one, two, or three, but no higher than three. So this is possible though. Okay, so then we move on to m sub l. And in this case, m sub l depends on the value of l in the same way that l depended on the value of n. And for m sub l it takes on a range of values from negative l to positive l, those integer values in between. So negative 2, negative 1, 0, plus 1, plus 2. So if here again I put minus 4, well that's outside the range of possible values. And again no matter what I fill in for m sub s, this would not be a possible set of quantum numbers for anything. Again, outside that range of negative 2 to positive 2. But once again, if I go in and say, how about negative 1, that works. It could have been negative 2, negative 1, 0, plus 1, or plus 2. Any of those would have been possible. And then finally, for m sub s. So there's only two values possible. And so if I say, hey, what if it's plus one? Well, that's not one of the possible values. Again, the values for m sub s are either plus one half or minus one half. And this is usually the first place I look. And so usually on a multiple choice question, I'll start with m sub s. And any answer that doesn't have plus one half or minus one half, I just get rid of them. Then I go look at n. And usually that's not where they're going to get you. But if I have any negative numbers or zero for n, that's out. It's got to be any positive integer. And then I go to l and I compare l to n. And I say, does this go from zero to a maximum of n minus one? And then I go and compare l to m sub l. And does m sub l range from negative l? to positive L. And slowly but surely you're going to find out that you're going to eliminate all the wrong choices and be left with one that actually fits this criteria. Cool, but that's essentially what you need to understand regarding quantum numbers up to this stage of the game. And this will start to make even more sense when we start looking at electron configurations in the next lesson. Now if you found this lesson helpful, would you give me a thumbs up? Best thing you can do to make sure YouTube shares this lesson with other students. And if you are looking for general chemistry practice problems, So quizzes, chapter tests, practice final exams, or a general chemistry final exam rapid review, then check out my General Chemistry Master Course. I'll leave a link in the description. A free trial is available. Happy studying.

Turns out there are four quantum numbers, n, l, m sub l, m sub s. We'll talk about the name of each one, what they tell you, and essentially, an easy way of remembering what they do because they're essentially an electron's address in an atom. My name is Chad, and welcome to Chad's Prep, where my goal is to take the stress out of learning science.

Now, in addition to your high school and college science courses, we also offer prep courses for the MCAT, the DAT, and the OAT. You can find those courses at chadsprep.com. Now, this lesson's part of my new general chemistry playlist. I'm releasing several a week throughout the school year, so if you want to be notified every time I post one, subscribe to the channel, click the bell notification.

So let's dive into this. So here I've got diagrams of some of the most basic orbitals here, and a couple things you should know is that any orbital in the universe can hold a maximum of two electrons. So you can find orbitals empty, you can find them half occupied with just one electron, or you can find them full being occupied by two electrons, as we'll see. There's s orbitals, p orbitals, d orbitals, there's also f orbitals that we'll definitely talk about in this course. But it turns out there's also g orbitals and h orbitals that you probably won't talk about in this class because we don't actually have any stable elements that have electrons in them normally.

But we will talk about the f orbitals, but it turns out the shapes of the f orbitals are more complex that I can't draw them for one. But you typically won't even find illustrations of them in your textbook. But if you Google them, you will find shapes for them. So it turns out where these shapes come from.

So it turns out they are mathematical, three-dimensional mathematical equations that we call... wave functions, and this is just the three-dimensional graph. So this is not two-dimensional.

Think of this as a three-dimensional sphere. We like to think of the p orbitals as being dumbbell-shaped, but again they're three-dimensional in space. And then the d orbitals are either going to be four-leaf clover-shaped, or they're going to have this kind of toroidal shape, we say. And it turns out that there's just a single s orbital.

There are three p orbitals, but there are five d orbitals, and then there are seven F orbitals. And if you see the trend, so again, 1s orbital, 3p orbitals, 5d orbitals, 7f orbitals, 9g orbitals if we cared, 11h orbitals if we cared. They count on the odd numbers. We'll see why when we get to quantum numbers, why exactly that is. But you should know it.

Well, for the d orbitals, I just got lazy and didn't draw them all. But four out of the five, turns out they get labeled. dxy, dxz, dyz, dx squared minus y squared.

Four out of the five look like a four-leaf clover. What's going to be different though is how they're oriented on the, you know, in space. So, dxy it turns out is in the xy plane, dxz in the xz plane, dyz is in the yz plane.

So, and then dx squared minus y squared it turns out is also in the xy plane, but whereas dxy is in between the x and y axes, dx squared minus y squared would be right on the x and y axes. And then dz squared here, so it turns out the big lobes here are going to be on the z axis. and then this lovely circular portion, the donut portion, if you will, is gonna be in the xy plane.

Cool, so I just wanted to distinguish that, but I wanted you to realize that even though I'm lazy, there are five d orbitals, even though I only drew two of them here. So, but these four lobes represent a single orbital, and again, you could hold a maximum of two electrons in an orbital like that. Okay, one thing you should know is that I'm giving you the most basic version of each of these orbitals.

So, The first time you'll ever hit the lowest shell number that actually has an s orbital is the first shell. The lowest shell number that has p orbitals is the second shell. The lowest shell number that has d orbitals is the third shell. But as you go higher and higher, it actually turns out that those same type of orbitals that you encounter again get a little more complex, a little more complicated. So like when you go to 2s, it's a bigger sphere, okay?

But it also turns out has what's called a node in it. And it's kind of like, you know, putting a softball inside a basketball. so to speak, and the air space in the middle is going to be a node where you don't find the electrons. And you could find them in the softball region in the middle, or you could find them in the outer region, but there's a hollow gap in the middle that you don't find them. It's really weird.

And they get more complex as you go up. As you go up to like 3s, you'd have two of those nodes. We call them radial nodes and stuff, and not super important at this level. So, but if you take a more advanced like physical chemistry class, or if you're taking maybe a majors class, they might go through something like that.

So, but the same thing happens with the p orbitals and d orbitals. If you go from 2p, if you go and talk about the 3p. these orbitals get bigger, but again, they have these radial nodes incorporated into their shapes and stuff. So I'm not going to say any more than what I just said about them. I just want you to realize that there are more complicated shapes when you get to the higher shell versions of these same types of orbitals.

All right. Now, one thing I did say, though, is that as you go from like 1s to 2s to 3s, they get bigger. And if you recall, then, that means the electrons that are going to occupy them get higher energy because, again, The closer an electron is to the nucleus, the lower its energy, as we learned a couple lessons ago.

So it turns out then that the lower the shell number, the smaller the orbitals, and the lower the energy of the electrons that are in those orbitals. And so back when we studied the Bohr model of the atom, Bohr thought electrons went around the nucleus in these two-dimensional circular orbits. What's really true we find out though is that it's actually these three-dimensional orbitals that electrons actually live, and they have a wave-like property that Bohr really didn't take into account.

Alright, so these wave functions in some way shape or form describe where an electron may be found. Okay, so in addition to knowing these shapes, now we've got to talk about these lovely quantum numbers. It turns out there are four of them.

There's n, there's l, there's m sub l, and there's m sub s. You should know the name of each of these quantum numbers, you should know what they tell you, and you should know what range of values are possible. So if we look at n first, so n is going to tell you, or n is called, I should say, the principal quantum number.

And it's going to tell you what shell its electron is found in, what shell the orbital, I guess, really is in that the electron is found in. And in this case, you have like the first shell, the second shell, the third shell. And again, these are the same n values that correspond to the Bohr model of atom, like the first orbit, the second orbit, the third orbit, according to Mr. Bohr there.

So in this case, we're going to say that they give you the shell number. And if we look at the range of values that are possible, so in this case the lowest shell is shell number one, and then it turns out just like with Bohr's orbits could go up to the infinitieth orbit, so can the shells. The first shell all the way up to infinity. So you should know though that in the ground state we only have atoms with electrons up to like the seventh shell, but again we can verify that the additional shells do indeed exist. by looking at different electron transitions where electrons are actually promoted up into those higher energy orbitals.

Okay, so that's our principal quantum number. Next on the list is L here, and L is often referred to as our azimuthal quantum number. And your azimuthal quantum number is going to tell you the subshell. And you can kind of think of this as a code here.

So the subshell In this case could be S, P, D, or F, or again technically G or H, but you're never gonna see it. So S, P, D, or F, and it's a code here. So 0 is gonna tell you that it's an... let's put this in blue actually, just like it is on your study guide.

So when L equals 0, a value of 0, that tells you you're an S subshell. When L equals 1, it tells you you're in a P subshell. When L equals 2, it tells you you're in a D subshell.

And when L equals 3, it tells you you're in an F subshell. So these quantum numbers are going to describe, and I say you, but it's really describing where an electron is found in an atom. And so if I say L equals zero, you're supposed to know, oh, that electron is in an S orbital somewhere.

So if I say L equals one, you're supposed to know, oh, that electron's found in a P orbital somewhere. And we call it the P subshell because in any P subshell, there are actually three P orbitals, one on the X axis, one on the Y axis, one on the Z axis. And when L equals one, the electron that has that value of L equals one has an electron in one of those three orbitals and somewhere in the entire p subshell. All right when l equals 2 that electron is in the d subshell one of the five d orbitals and when l equals 3 that electron's in the f subshell one of the seven f orbitals. All right if we look at range of values here so we like to define the range of values for l in terms of n so and it turns out it can start from a minimum value of zero up to a maximum of n minus 1. So we'll see the profound implications of this here in a sec.

But in this case, notice the lowest value you could have is 0. We saw that. But the highest value it takes depends on what shell you're in. If you're in shell number 1, i.e. n equals 1, well, 1 minus 1 is 0, and 0 up to 0 means the only values it can take is 0. And so here from 0 to n minus 1, all the integer values in between, and I should have said that up here as well.

Your range for the shell number is any integer value from 1 to infinity. So 1, 2, 3, 4, but not like 3.476. So all integer values.

Cool. So if you're in the second shell, n equals 2. Well, 2 minus 1 is 1. And so it can take any integers from 0 up to 1. And so in the second shell, it could be 0 or it could be 1. And in the second shell, there are s and p orbitals only. In the first shell, there were only s orbitals because the only value l could take was 0. In the third shell, you've got 0 up to 3 minus 1, 0 up to 2, so 0, 1, and 2, and 0, 1, and 2 mean that in the third shell there are going to be S, P, and D subshells, i.e.

S, P, and D orbitals. And so every time you go to a higher shell, you get something new, it turns out. In the fourth shell, you've got L taking on values of 0 up to 4 minus 1, i.e. 0 to 3, so 0, 1, 2, and 3. Because in the fourth shell, there are S, P, D, and F subshells. Cool, and that's the way it works.

And technically, if you go up to the fifth shell, that's where you'd actually encounter the G orbitals, the G subshell. But again, no stable elements actually put any electrons in there, and we don't ever talk about them. So I just bring it up just so you can follow the pattern here.

So all you've got to know is up to F orbitals. Cool, but that's how the azimuthal quantum number works. And so... Moving on to m sub L, and m sub L is our magnetic quantum number.

So in this case it's going to tell you, it's going to actually identify a specific orbital, but oftentimes we say that it tells you the orientation in space. But it's going to specify an orbital, so it gives us a specific orbital. And so these mean the same thing technically.

Like if you look at the three different p orbitals, one's on the x-axis, one's on the y-axis, one's on the z-axis. So when we say that the magnetic quantum number m sub l here specifies the orientation in space, it's gonna tell you if a p orbital's on the x-axis, on the y-axis, or on the z-axis. In that way, it's specifying an individual orbital in that subshell.

Same thing with the d orbitals. They differ in their orientations in space. And so when m sub l...

specifies an orientation in space, it's actually specifying an orbital in whatever subshell you're in. And it turns out if we take a look at the range of values here, it goes from a minimum of negative L as an integer and all the integer values in between up to positive L. And so just like L is bounded by what value of n you have, well M sub L is going to be bounded by what value of L you have. So if you have L equals 0, L equaling 0 means you're in an S subshell, and in an S subshell there's just one orbital.

And so you only have to specify one orbital, and in going from negative 0 to positive 0, there's only 0. And there's only 0 because you only need to specify one orbital for an S subshell. Now if we've got a P subshell, in a P subshell L equals 1, and M sub L could therefore go from negative 1 up to positive 1. That means negative 1, 0, and positive 1. three different values because we have three different p orbitals that we need to specify their orientation in space. And so one of these corresponds to negative one, one of them corresponds to positive one, one of them corresponds to zero.

That's beyond the scope of this class though. you just have to know that the three possible values would be negative 1, 0, and 1 for the p orbitals. If we go on to d orbitals, for d orbital, L is equal to 2, which means that M sub L could take on a range of values from negative 2 to positive 2, which means negative 2, negative 1, 0, positive 1, positive 2, and there's five different values because there are 1, 2, 3, 4, 5 different d orbitals. So again, if L equals 3... L equaling 3 means F orbital.

There are 7 F orbitals. And that's going to allow us to get 7 different values now for M sub L. So from negative 3 to positive 3. So that means negative 3, negative 2, negative 1, 0, positive 1, positive 2, positive 3. 7 different values because each of those F orbitals is going to get a specific value of M sub L because it's identifying or specifying an individual orbital.

Cool. Now it turns out that these 3 quantum numbers, so we're kind of, you know, deriving them based on what they actually tell us, and they're kind of giving us an electron's address it turns out, but they actually come from the solutions of the Schrodinger equation. They fall right out of the Schrodinger equation, but this last one, m sub s, called the spin quantum number, does not come from the Schrodinger equation.

I've seen that tested on a timer too. It's the only one that doesn't come from the Schrodinger equation, but it turns out that an electron has a spin associated with it, and what that spin actually is, I can't... actually tell you, but it is a fundamental property of matter. So particles of matter, these individual subatomic particles, have spin associated with them much of the time, and this spin is weird.

Now it turns out that something with spin can interact with a magnetic field when it's moving, and usually we attribute that to charged particles and things of this sort in a weird way, and the idea is that if an electron was rotating around the nucleus in one direction, it would have a magnetic field associated with it, and if it was going around the nucleus in the opposite direction, it would actually have a magnetic field associated with it in the opposite direction. Well, it turns out that these don't even have to be rotating around the nucleus, and they don't really go around in nice circular orbits anyways, and even when they're not rotating around a nucleus, they still interact with a magnetic field and have a characteristic magnetic field associated with them that allows them to interact with a magnetic field, and so it turns out it... it's not that they're spinning around the nucleus in any way shape or form it's just some fundamental property that they have so we call it spin and based on the idea that maybe it's revolving around the nucleus spinning around the nucleus but it turns out it has nothing to do with that it's just a fundamental inherent property of electrons and it's not just electrons most particles have a spin associated with them okay it turns out that this spin is actually what we're going to represent here.

It takes on two possible It is either plus one half or minus one half. So we don't have this range of values. There's just two values possible. So it turns out there's only two types of spin. So, and you can kind of think of this.

There's only two types of charge in the universe. And we identified them as positive and negative charges. Now notice, that's just what somebody called them.

We could have called them left charges and right charges, or up charges and down charges, or blue charges and green charges, or we could have called them anything and just known they were opposites. So, well here, we've got these opposite spins, but it turns out they have actual numerical values associated with them that are indeed opposites. So, plus one half and minus one half, and we're going to find out that we're going to represent electrons here in a little bit with arrows like so. And we're going to have one arrow point up and one arrow point down, and these arrows represent electrons, and we represent the opposite spin quantum numbers for them by representing the arrows pointing in opposite directions. Now, we've got these lovely four quantum numbers here.

And in this case, we know their names, we know their symbols, we know what they tell us, and now we know the range of values they can take. And there's a few different kind of questions you might have to answer. Now, one thing we've got to give a prelude to the next lesson, and that's called the, we've got to talk about what's called the Pauli Exclusion Principle. And it essentially excludes any two electrons from having the same four quantum numbers. So...

every electron in an atom is described by some combination of four quantum numbers. And no two electrons in an atom can have exactly the same set of four quantum numbers. What that ultimately means, because again, the set of four quantum numbers ultimately describes an electron's address, it means that no two electrons can live in the same orbital and have the same spin, as it turns out.

Now, what do I mean by these being like an electron's address? Well, think about Think about this. Let's say I gave you my address and I said that I live in state number 48. And so I didn't tell you what state I lived in.

I just assigned a number value to my state. Well, if you've heard me say it in other videos, I live in Arizona. And the reason I call it state 48, that was a code because Arizona was the 48th state to be made a state in the union, if you will. So that's where that code comes from.

And then maybe I tell you, hey, I live in city number 7. So, and again, you might... start doing some research and find out, you know, what was the seventh city in Arizona? And I have no idea if this is correct, but I live in Tempe.

So we'd be like, oh, the seventh city in Arizona was, you know, Tempe. And I'm just giving you a code. And so I'm like, I live in state 48. I live in city number seven.

And so, and then I might tell you, I live in street 104. And then we might do some research and find out what was the 104th street to be paved in the city of Tempe or something like that. And then I might finally give you a house number. And so instead of giving you my address in normal terms like house number, street, city, state, I'm giving you numerical values as a code for this.

Well we're doing the same thing here. When I give you a set of four quantum numbers, it's a code for where an electron lives. And we'll take a look at how this works here.

All right, so here's a common diagram we use to represent the energy location for the different subshells and orbitals in an atom. And the lowest energy orbital is a 1s, so this box represents the orbital. And then we're going to draw arrows inside these boxes to represent the electrons.

So we might put an electron like here, and then when the next one turns out we'd have to draw would be spin down. So it turns out one spin up, one spin down, and again that corresponds to the opposite values of the spin quantum number as we'll see. And based on where an electron is found somewhere in this diagram, we can assign values to n, l, m sub l, and m sub s.

What you'll find is that most of the time you can assign a specific value to n and l, But you might have a range of values that are possible for m sub l, and you might not be able to specify which it is. And then m sub s, you're probably just going to know it's either plus 1 half or minus 1 half, and good chance you might not know which is which. All right, so let's say we talk about one of these electrons. It doesn't matter which one.

Let's just erase one of them and say we're talking about one of those 1s electrons. So that electron, and we might say, what is its set of quantum numbers? Well, it is in shell number one here, and that one is the principal quantum number. And so here n would equal one. So and then l, since we're in an s subshell, you remember, l tells you the code.

l equals zero means s, l equals one means p, l equals two means d, l equals three means f. So in this case, because it's an s subshell that that electron is in, l is going to equal 0. And then you recall that m sub l takes on values from negative l to positive l. So in this case, from negative 0 to positive 0 means the only possible value is 0. Cool. And then finally, m sub s.

Well again, we said that when you have two electrons in the same orbital, we would draw them as one spin up, one spin down. So, and in this case, when you only draw one of them, though, and it doesn't matter if it's spin up or spin down, we actually don't specify that, you know, spin up means plus one half and spin down means minus one half. There's no specificity there at all. We just know we have one of them.

This is either the plus one half spin or the minus one half spin. We don't know which. And so all we can say here is that it's either plus one half or minus one half, and we can't get any more specific to that. Now, what I could tell you is this.

If I drew both electrons in this particular orbital, And if I told you the one on the left is the minus one half electron, well, then you'd know that the one on the right would have to be the plus one half electron. And the key is they have to have opposite spins because if they had the same spin, if they were both plus one half or both minus one half, well, they're both in shell number one. They're both in the S orbital. They're both in the same S orbital. And if they had the same spin, then they would have the same four quantum numbers.

And again, the Pauli exclusion principle tells us that that is not possible. And it's not possible because Pauli said so, Pauli was just describing nature. It just turns out that nature doesn't put two electrons in the same orbital with the same spin, it's just the way it works.

Cool. So that was that electron. So let's say we put an electron somewhere else in this lovely atom.

And so a couple things you realize about the orbitals here is that we're going increasing energy as we go up. So notice shell one is lower than shell two, shell two is lower than shell three, and shell three is lower than shell number four of electrons. But also note that, you know, when you get to the higher shells, you got like S's and P's. The P's are higher in energy than the S.

The D's are higher than energy than the P, which are higher energy than the S. And then F are higher energy than D, are higher energy than P, are higher energy than S, with one exception. So for any system that only has one electron, it turns out that the way this is represented is not true. And the most notable atom or ion that only has one electron is hydrogen.

And for something with only one electron, most notably, again, the only neutral species is hydrogen, It turns out that all the sub-shells and orbitals within a single shell are going to be equal. So instead of having the P higher than S, they would all be at the same level, all the same energy. And that's only true for one electron system.

Same thing here, 3S, 3P, 3D. For hydrogen, they're all equal energy. But for any multi-electron system, they split into different energies.

And we'll talk about that at a much later date, why that is and stuff like that. So just something you should realize. Alright, so let's say we've got an electron now, and let's put this electron right... here. And so now the questions again are what are the possible, you know, what is the four, or the possible sets of four quantum numbers that that electron could have?

Well in this case it's in the 3p, so that means it's in shell number three, so n is going to equal three, and it's in a p subshell, and again l tells you the subshell, and the p subshell corresponds to l equals one. Cool, now here's the deal. We got to go to m sub L, and m sub L takes on a range of values from negative L to positive L, so in this case negative 1 to positive 1, so either negative 1, 0, or positive 1. And on a diagram like this, we don't specify.

It could be any of those three, and so we can't actually get it specific here. We can only say that it's either negative 1, 0, or plus 1. It's one of those three. It can't be negative 2, it can't be positive 3, but it's got to be one of these three specific values. but can't get any more specific than that. And then finally m sub s, same thing as before, it's either plus one half or it's minus one half and we can't get any more specific than that either.

And so you might get a question on the test that says, you know, multiple choice, let's say, and which of the following is a possible set of quantum numbers for that electron right there or for an electron in a 3p orbital? Well, any answer that doesn't have n equals three is wrong. Any answer that doesn't have l equals one is wrong.

But notice for M sub L, there are three possible correct answers here. Either negative one, zero, or positive one. And then for M sub S, there are two possible correct answers, plus one half and minus one half.

And you've got to find the combination of four that are all within this range. So with three different options here and two different options here, there would actually be six possible correct answers they could include on your answer sheet. Obviously, they're only going to include one of those options as a correct answer.

Cool. The other way you could see quantum numbers showing up is they might just start giving you different combinations of quantum numbers and asking which of the following is even possible. So which of the following is a possible set of quantum numbers? So let's say we start messing with this a little bit.

And so I first tell you that n equals 4. Is that possible? Yes. But what if I told you it was n equals negative 4?

Well again, n only takes on a range of values from 1 at the lowest up to infinity. it can't be a negative number. So that whatever else we fill in for L, m sub L and m sub s, this would not be a possible set of quantum numbers. But as long as I just said n equals plain old four, positive four, okay, so far so good.

Now, what values could L have here? Well, in this case, you got you remember that L takes on a range of values from zero at the minimum to a maximum of n minus one, in this case to a maximum of three. And so in this case, if I put It's not, let's just say four again. Well, then again, no matter what I fill in for m sub l and m sub s, you'd know that this is not a possible set of quantum numbers because with n equals four, l could be zero, one, two, or three, but it couldn't be anything greater than that. So that wouldn't be possible anymore.

Okay, so let's say I put one of the possible numbers. Let's say I put l equals two. And again, it could have been zero, one, two, or three, but no higher than three.

So this is possible though. Okay, so then we move on to m sub l. And in this case, m sub l depends on the value of l in the same way that l depended on the value of n. And for m sub l it takes on a range of values from negative l to positive l, those integer values in between. So negative 2, negative 1, 0, plus 1, plus 2. So if here again I put minus 4, well that's outside the range of possible values.

And again no matter what I fill in for m sub s, this would not be a possible set of quantum numbers for anything. Again, outside that range of negative 2 to positive 2. But once again, if I go in and say, how about negative 1, that works. It could have been negative 2, negative 1, 0, plus 1, or plus 2. Any of those would have been possible. And then finally, for m sub s.

So there's only two values possible. And so if I say, hey, what if it's plus one? Well, that's not one of the possible values. Again, the values for m sub s are either plus one half or minus one half. And this is usually the first place I look.

And so usually on a multiple choice question, I'll start with m sub s. And any answer that doesn't have plus one half or minus one half, I just get rid of them. Then I go look at n. And usually that's not where they're going to get you. But if I have any negative numbers or zero for n, that's out.

It's got to be any positive integer. And then I go to l and I compare l to n. And I say, does this go from zero to a maximum of n minus one? And then I go and compare l to m sub l. And does m sub l range from negative l?

to positive L. And slowly but surely you're going to find out that you're going to eliminate all the wrong choices and be left with one that actually fits this criteria. Cool, but that's essentially what you need to understand regarding quantum numbers up to this stage of the game. And this will start to make even more sense when we start looking at electron configurations in the next lesson. Now if you found this lesson helpful, would you give me a thumbs up?

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Transcript for:Quantum Numbers and Atomic Orbitals Overview

Transcript for:
Quantum Numbers and Atomic Orbitals Overview