So the first um quantum number we're going to look at again think of quantum numbers as like the address of an electron is something known as the principal quantum number. If we're going to look at this model of an atom kind of like an electron hotel, this would be the floor that the electron is located on. And so the principal quantum number we use the letter like a lowercase n as the principal quantum number gives you an idea of the overall size and energy of the orbital. So think of it like this. So here we have increasing energy and n can have any value any whole number value of 1 or greater 1 2 3 and so on and so forth. No nothing less than one no decimals or anything like this. And as that number increases, n= 1 is the lowest energy and then you have n= 2 and then n= 3 and so on and so forth. One thing to note is that as n increases the gap between n levels, so like between one and two or two and three that decreases. And something else important is the distance from the nucleus increases as n increases. Now if we kind of compare let's compare what uh n equals 1 to uh is you know might be this n= 2 is going to be bigger and then so on and so forth. So if you look at this diagram on the right, these represent like n= 1, n= 2, n= 3. Don't worry about this s yet. 1 s 2 s 3s. We will get to that. But I want to kind of highlight here that the colored regions you see there. So like this little blue right here, well maybe I'll do it in blue and not red. So this blue dot right there that represents the probability of finding an electron and then n= 2. So what this what that um red sphere is showing. So here's n= 1 and then the red part there is n equals 2. So notice how that energy um it's farther away from the nucleus and it's a much bigger probability. Now, one important thing I want to highlight that we didn't see with n equals 1. I'm going to zoom in. Look at this empty space right here. Well, the highlighter is not really letting me do it that close. But kind of look at that empty space around the blue, right between the blue and the red. That empty space is known as a node. And a node is a region where you will not find the electron at all. Definitely will not find it. And so if you look at this kind of uh this this uh xy graph down here, this little valley where it hits the y ais or I'm sorry the the x- axis that represents a probability of zero. So the probability of finding an electron there in that position in that node is zero. And if we go to n= 3. So again there's that node between 1 and 2. There's the node between 2 and 3. And so at n= 3 there are two nodes. Now, one of the cool things about the periodic table is that you can look at the table and instantly know what that highest energy level is for any atom on the periodic table. And all you got to do is look at the row. So for helium, I'm sorry, for hydrogen and helium, its highest energy level is one. There's no other energy levels. Starting with lithium, all of these atoms, the highest energy level is two and three and so on and so forth as you go down a column on the periodic table. Now, this will become uh a lot more helpful when we start putting all of the quantum numbers together so that you'll be able to describe any single electron inside of these atoms. And so again, just to to kind of review, n or the principal quantum number is about the overall size and energy of the orbital. Now the next one we're going to look at is known as the angular quantum number or we use this like a cursive lowercase L. Um this basically tells you the shape of the orbital. Again I want to highlight what an orbital is is it's a mathematical solution and that mathematical solution forms a shape that can be graphed on like an XYZ coordinate kind of forming a three-dimensional shape. Now the allowed values of L can be zero, 1, 2, and so on up until you get to n minus one. So for example, if n equals 1, remember that's the lowest energy level. That's the only value of n you can have. Well, what's uh what's n minus one? 1 - 1 is zero. So, the only value allowed for l is zero. If n equals 2, well then the allowed values for l or n minus one, which is 1, and anything less than that, and zero. What if n equals 3? So the highest L can be is n minus one which is 2 and anything below that 2 1 and zero and so on and so forth. Now to describe these shapes we don't what we're going to do is we're going to take those numbers for L and assign them a letter. Zero represents the letter S. One represents the letter P. Two represents the letter D. And if L is three, that represents the letter F. Don't need to worry about going any higher than that. Where the this class, that's outside the focus of this class. Now, the shape that corresponds to for S, that's a sphere. Now, I know I just drew a circle, but imagine a three-dimensional sphere. The shape for P is a dumbbell and it's oriented in three directions. you have up and down. So like on the Y ais, you have it oriented on the X axis and you have it oriented on the Z-axis coming out at you. So kind of imagine this dumbbell is coming out at you. So you have this X, Y, and Z axis. Now notice that for S there's only one orbital and what we call the S subshell. There are three orbitals in the P subshell. Now the D um orbitals kind of look like a fourleaf clover. And I'm not the best at drawing that, but you got four of those. And then there's one more that kind of looks like a dumbbell with this ring around it. Kind of weird. Again, we're dealing with quantum stuff. Quantum stuff is weird. And so you have a total of five orbitals. The four fourleaf clovers and then the one dumbbell ring. in the D subshell and the F subshell. Um, I'm not even going to try to draw them. Just know that there's like seven of them. There's seven F orbitals in an F subshell. Again, I've said this before. I'll I'll keep on saying it. An orbital is just simply a mathematical solution of the shortinger equation of where you can find an electron. So this kind of shows you the shapes of the orbitals better than what I can draw. So notice here's like the s orbital again. There's one of them. Okay? So in an in an S subshell there's only one orbital and these represent the p orbitals. Notice there's three of them. And I want to highlight this also that the orbitals in a subshell so like in the p subshell are all degenerate. That's a term I've used before. They're all equal in energy. And then here's our D subshell. So you got five of them. So they all kind of, you know, you got four of them that look like a sort of like a fourleaf clover. And you got this weird one right here. And then F. Got seven of them. Don't worry about having to draw any f orbitals, but you should know what sp and d look like. So, you should be able to kind of uh draw or recognize um those orbitals. Now, just like I did with the um the uh principal quantum number and how we were able to easily find that using a periodic table, we're going to do the same thing here. So, how does this angular quantum number look on a periodic table? Okay. Well, these first two rows here, and we're going to include helium in this, are what we call S block elements. I'll tell you why we call them that in a little bit. It'll be, like I said, most of this will become clear once we put it all together. And then these here that I'm highlighting in orange, we call them PB block elements. And then these I'm highlighting here in purple are called DB block elements. And then these I'm highlighting in red down here are flock elements. So start kind of and you don't necessarily need to commit it to memory but like start you know thinking about how these um we can use the periodic table to look at these trends both now or for what we talked about first with the principal quantum number being on the rows and then being able to organize the periodic table into these blocks for S P D and F. Now, the next quantum number we're going to look at is the mag uh magnetic quantum number. And this refers to the orientation of an orbital. We're primarily going to use this um as to find how many orbitals there are in a subshell. I've highlighted this, but now I'm going to put it to a specific quantum number. So, m subl is allowed to have values of negative l to 0 to positive l. Now, let's do an example here. If n equals 4, what are the possible values of l? So go back a couple of pages. L can be zero, 1, 2, or 3. And again, just to review, zero meant S, one meant P, two meant D, and three meant F. Now if L is zero, that means M subL has only one value, zero. Well, how many s orbitals were there in the s subshell? Go back a couple pages when we when I was showing you the uh the diagrams there. There's only one orbital. So that the number of values m subl can take tells you how many orbitals there are in that subshell. If l is one, what are the possible values of m subl? Well, negative 1, 0, and positive 1, which is three orbitals. If L is 2, M subl can be -2, -1, 0, + 1, + 2. That's five values. So, there are five orbitals in the D subshell. And then finally with uh if L is 3 m subl can be all the way from -31 0 + 1 + 2 and + 3 which is a total of seven orbitals. All right. And then finally, the spin quantum number or m subs identifies a specific electron in an orbital. Now, m subs can only have one of two values. It's either + one/2 or minus one/2. And I know this is called a spin quantum number. It really doesn't relate to something like spinning around in a circle or spinning like a top. But what this um alludes to is something um important is that in an orbital or I should say only two electrons at most can be in a single orbital. And this leads to what's known as the polyexclusion principle, which is no two electrons can have exactly the same quantum number. A qu set of quantum numbers. So for example, if we take helium, helium has two electrons. Now how do I know that? Well, helium has a um atomic number of two. two protons. So it if it's neutral it has two electrons. Now those two electrons have a different address. one of them. So if we look at where helium is on the periodic table, n equals 1 and then that means l can only be zero. m subl can only be zero. And then for one of the electrons in helium it will be plus 1/2. The other electron will have almost all of the same quantum numbers except its spin number will be minus one/2. Now the way we kind of diagram that is let's imagine you know we're just going to say 1s here. The way we diagram a electrons having like a different spin plus 1/2 or minus 1/2 is we typically use an up arrow like a half arrow for plus one/2. So plus one half will be the up arrow and then we use a half down arrow or minus one/2. You're going to kind of see that notation a whole lot um in the uh in the coming lectures. Now let's kind of put this all together. So we said the row tells you the principal quantum number or I should say the highest energy level and then we organize them into blocks. So these first two columns we said were like our S block. Now I want you to notice something here. So this was our S block. How many orbitals are in an S subshell? I'm going kind of pause there. I want you to think about it. One orbital because that's what an S subshell is is a one orbital. How many electrons can I put in um at most in in in an orbital? I can put two at most. So look at this here. If I can include including helium in this one electron, two electrons, one electron, two electrons, one electron, two electrons. And I can do that all the way down those two columns. I know it sounds a little confusing now, but once we put all four of these things together, then it'll become a lot more clear. We look at the P block. So there in red. Remember how many orbitals are in a p block? Three orbitals. Which means how many electrons can the P block hold total? That means six total electrons. Now, they're always not going to be filled. But look at how many elements as you go across a the row there in red. How many elements are there in the P block? Just pick one row. Oh, one electron. two three electrons four electrons five six and you can do that for each subsequent row. Now do this for the d subshell. So, I'm not going to, you know, count them all, but notice in the D subshell there's 10 across. And notice in the um F block there's 14 across. All right. So, again, we're using the periodic table here to kind of help us out with these u these quantum numbers. And what we're going to go into next and in the next lecture is how we're going to put all these quantum numbers together. And we're going to use the periodic table to um find what's called the electron configuration or how atoms I'm sorry, how electrons are arranged in an atom. How do we put them in particular orbitals? What order do we fill them up? But first thing we're going to do, let's just do some practice and then that will be it for this lecture. Okay. So in this uh first problem, identify the subshell in which electrons with the following quantum numbers are found. So at a n= 1 I'm sorry n= 3 and l= 1. Now first of all what does l mean? What subshell does that correspond to? That corresponds to the p subshell. So if I want to identify specifically that subshell, I'm going to say it is in the 3p subshell. Three refers to N. The P refers to L. Now for B, N= 5, L= 3. Okay. Well, what letter did L I'm sorry, if L is three, what letter does that correspond to? That's the F subshell. So, the specific subshell that this is located in is five. Again, five is for the energy level and F represents L. and then C n= 2 and L= 0. Remember 0 represents S. And so the specific subshell, if an electron were in this subshell, we would say it's in the 2s subshell or the 2s orbital. Okay. Now in the second problem, we're going to just complete this table. Now we'll just identify here again. N is the principal quantum number. L tells you the subshell SPD or F. We're going to put a specific number for that. So 0, 1, 2, or 3. M subL degeneracy. That's just kind of another way of saying like how many orbitals are there equal in energy. So if the orbital is labeled as 4f, that four tells you what n is. Now the way we can get l is by looking at the letter of the orbital. The letter is f. That corresponds to l= 3. Now if L is three, we'll just kind of review that means that these are the possible values of M subl3 -21 0 1 2 and 3. That's a total number of seven values or seven orbitals in that subshell. Now, if you look at the next row, we have to figure out what the orbital is. Well, n is four and l is one. What letter does um one correspond to that corresponds to p? Now if L is one the possible values of M subL are -1 0 and + one. The P subshell has three orbitals that are equal in um equal in energy. Now the next row is a little tricky because I mean we can get the energy level seven that's n but how do we know what letter it is? We don't we're not given L but we are given the number of orbitals and there's only one subshell that has that many orbitals that's the F subshell. So we would say 7 F and the number for L to represent F is three. And then finally on the last row, the number five there gives us N. D helps us figure out what L is because a value of two represents the D subshell. And if L is 2, the possible values of M subL are all the way from -2 to positive2. That's five possible orbitals. That's five orbitals in the D subshell.