 Hi, I'm Zor. Welcome to Unisor Education. Today I would like to talk about how electrons are positioned within the atom. This lecture is part of the course called Physics for Teens, presented on Unisor.com. On this website, it's not the only course. There are other courses like for example Mass for Teens, which is prerequisite for this one. Physics without mathematics just cannot be studied at all in any good manner. Also, the site is completely free. There are no advertising, no financial strings attached. You don't even have to sign in if you don't want certain functionality which is related to signing in, like supervisory style of education. So, what else? Every lecture has a textual description. It's basically like a textbook. So, with every lecture you have notes, which is like a chapter of a textbook. All lectures obviously are kind of organized into a course, which means there are menus, there are submenus, etc. So, if you want to really study as a subject, physics as a subject, you just have to follow the menu and sequential order and it will be fine. If you found this particular lecture somewhere through a search engine or something like this on YouTube or wherever, I still suggest you to use the Unisor.com free website to access the educational material, because, as I was saying, it's organized into a course and there are certain other components like text and exams, for example, which are also part of this website. So, organized electrons within the atom. Now, first of all, this theory of how electrons are positioned within the atom is really complicated. The quantum theory, which was developed somewhere in the first half of the 20th century, brought a lot of information, actually, about this inner structure of the atom, especially electrons, their position. We can start from the planetary model of Rutherford and extended by Niels Bohr. That electrons are supposed to be on certain stationary orbits where they do not emit any electromagnetic oscillations and that's what makes them stable on the orbit. A slightly different, maybe, wording of this. Instead of the word orbit, it's preferable to use a determined shell. So, electrons, because it's a three-dimensional world, so orbit is kind of flat. So, to reflect this three-dimensional structure, we're using the term shell. And then, the further development, theoretical and experimental development of physics, especially the atoms physics, brought another concept, the concept of sub-shell. So, apparently, electrons which belong to a shell, they actually are slightly divided as far as, well, their radiuses are. So, it's still within the shell, but shell is kind of thick. It's not like a zero-thickness sphere. It really has certain widths, if you wish. And within that widths, you have sub-shells, also stationary, so that's what's important. Bohr was kind of right that there are stationary orbits, but it's not like one electron per orbit or something like this. No, electrons live in thick shells, which each shell divided into a certain number of sub-shells. Now, let me go further and again. There are certain theoretical formulations and they are related to quantum theories. And what's important is that physicists came up to a very kind of mathematically beautiful structure of the shells. There are theories and there are experiments which confirm it, so we basically take it as the truth as of right now. Okay, so what is this particular structure of the shell? Apparently, we have shells number one, which is the closest to the nucleus. Number two, number three, etc. So these are still the shells in the level of understanding how Bohr understood it. So this is the shell where a certain number of electrons live. And this is another shell a little bit higher from the nucleus, further from the nucleus and more energetic. And this is still higher and higher and higher. Now, okay, shells are fine. Now, the next step was, as I was saying, sub-shells. Now, apparently, and that's what actually physicists came up with, and again it looks mathematically quite beautiful, they're saying that shell number one contains only one sub-shell, number one. So this is shell, this is sub-shell. Shell number two contains two sub-shells. Number one and number two. Shell number three contains three sub-shells. Number one, number two, and number three, etc. Shell number n contains number one sub-shell. sub-shell number 2, number 2 sub-shell, number 3 sub-shell, et cetera, number N sub-shell. So the shell number N contains N sub-shells. So again, I can say, well, that's the result of theory and experiments which have been developed. I am not going to go into the details, number 1, because I myself do not really know all the details. It's really relatively complex theories which kind of was confirmed by experiments. Now, so sub-shells is a stationary orbit of a certain number of electrons. Now the question is how many electrons? Apparently, the sub-shell number of each shell has a fixed number of electrons, which depends on this number, a sub-shell number. So again, sub-shells have a certain number of electrons which depend only on the number of the sub-shell, not on the number of shell, which it belongs to, which means the first sub-shell of the nth shell is exactly the same number of electrons as the first sub-shell of the shell number 3 or the first sub-shell of the first number of the shell number 1. Now, the second sub-shell of any shell contains exactly the same number of electrons for each shell it belongs to. Now, what is that number of electrons? All right, if you have a sub-shell number m, then the number of electrons, maximum number of electrons, actually, it can hold is 4m minus 2. Again, the question is why? I should really mention the same thing. There are some theories, very advanced theories in quantum theory, which basically explain this in some way, and there are experiments which confirm it. Which means what? The shell number, sub-shell number 1 has two electrons, and here as well, and here as well, and here as well. Now, the second sub-shell, which is m is equal to 2, has 8 minus 2, that's 6 electrons, 6 electrons, in the shell number 2, in the shell number 3, and in any other shell it has 6 electrons. The third shell, which is 10, 10, 10, and the nth sub-shell has 2n, sorry, 4n, nth shell has 4n minus 2 electrons. It sounds like, you know, nice system, right? I mean, number of shells, number of sub-shells increasing as the shell number, as the shell is further and further, the number of sub-shells it has is increasing in exactly the same number. And the number of electrons is also increasing by 4, basically. 2, 6, 10, next one will be 14, 18, etc. So this is the number, maximum number of electrons, which the sub-shell can hold. So now the next thing, okay, how about number of electrons per shell? So this total, well, this total is 2, this total is 8, this total is 18, which by the way, again, corresponds to number of electrons, as far as, well, before people came up with the sub-shell concept, etc., they do observe experimentally that 2, 8, 18, next one, I don't remember, I think it's 30, was the actually number of electrons per shell. But now this is some kind of a theoretical foundation, why do we have these numbers? Now, what are these numbers? If this is the first shell number one, and this is a shell number two, this is a shell number three, I think you can really have some kind of a, we can observe this particular thing. Shell number square double gives you this number. Shell number three, square would be 9, double 18, and same thing further. Now, so it looks like, it looks like shell number n has 2n square electrons, which is some of these. Now, is it true or not? Well, let's prove. Here we have a little bit of mass. So what I would like to prove is sum of 4m minus 2, where m is changing from 1 to capital N is equal to 2n square. That's what I would like to prove. And again, for n is equal to 1, I have 1 square, which is 1 times 2, 2. So for n is equal to 1, that's true. And then we will do it by induction. Remember, mathematical induction. Let's assume that for certain number n, this is true. And now let's prove it for n plus 1 using our assumption for n is equal to 2. So let's talk about a sum from 1 to n plus 1. This, what is it equal to? Well, it's sum from m equals to 1 to n, and then the next number, right? So from sum from m is equal to 1 to n of 4m minus 2 plus. Now, instead of m, we should substitute this as n plus 1. So it would be 4n plus 1 minus 2. This is the last n plus first member of this sum, right? So these are first n members. And n plus first member is just instead of m, I substitute it as n plus 1. Okay? Okay. Well, let's check it out. We have assumed that the formula is correct for n. So I can replace this equal to 2n square plus 4n plus 4 minus 2, which is plus 2, equals to 2 times n square plus 2n plus 1, which is 2 times n plus 1 square, which is exactly the formula we need for n is equal to n plus 1. So this is the proof by induction that this particular formula is correct. So for nth shell, we have 2n square electrons. Okay, so basically that's all I wanted to talk about today about how electrons can be, in theory, distributed among the shell, shells and subshells. Now, these are, well, this is the maximum number, obviously. It's not necessarily completely filled up. Now, how the electrons are filling shells and subshells would be a subject of the next lecture. But in theory, you just have to understand that these are capacities of the subshells and shells. So the capacity of each subshell is, as I was saying, 4m minus 2, where m is the sequence number of the subshell. And the total number of electrons which a shell, the entire shell can hold is this, 2n square. Now, about how we call these subshells, shells and subshells. Well, shells we call by number. Number 1 shell, number 2, number 3, etc. Now, subshells, well, physicists have decided to, instead of numbers, numbers were reserved for the shells. They were using letters. And in the beginning, the first subshells, which were actually discovered, they called with certain names. And the first letter from that name was used as the name of the subshell. So I don't remember the names themselves, but the first is s, for some reason, whatever it is. And then there are some other letters. I believe it's P, G, maybe F, something like this. After that, they decided, okay, we cannot really name them and use the first letter of the name. We'll just continue alphabetically. So after F, subshell, next was J, H, I, etc., so all along the Latin alphabet. So these are the first couple of letters. And then the next one are alphabetical. These are the subshell names, which physicists are using instead of I was using the subshell number 1, number 2, number 3. Now, why did they use the numbers to get to this formula? For M minus 2. Number of electrons that subshell number M can hold. So the first subshell would be 2, the next one would be 6, etc., etc. So, but you really should not use the subshell number. If you will read the textbook or something, you will see these letters instead of number. But basically, this is number 1, subshell number 1, this is subshell number 2, subshell number 3, number 4. I'm not sure actually the sequence might actually mix it up, but doesn't really matter. I mean, my purpose was to introduce you to a concept, a concept of a subshell. So that's what's important. So the shell contains certain number of subshells, named or numbered, doesn't really matter. And the subshells have this particular formula to give the maximum number of electrons, from which we can come up with this formula as total number of electrons per shell. Okay, so that's it for today. You probably have to read the notes for this lecture. I recommend you to read it on Unisor.com. You go to part called Atoms, and then you will have the next menu, and you will see the electronic structure of the atom, which contains this lecture. That's it. Thank you very much and good luck.