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# Quantum numbers for the first four shells

## Video transcript

now that we understand the four quantum numbers let's get some more practice using the quantum numbers and thinking about the first four shells the first four energy levels so we'll start with n is equal to one so when n is equal to one write the principal quantum number is equal to one we're talking about the first energy level or the first shell the angular momentum quantum number depends upon the principal quantum number and so when n is equal to one let's think about the allowed values for L so L goes from zero one and all the way up to n minus one well n minus one if n is equal to one and minus one one minus one is equal to zero and so that's the only allowed value for the angular momentum quantum number and we know L is equal to zero refers to an S orbital so we're talking about an S orbital here all right the next quantum number is the magnetic quantum number so ml so the magnetic quantum number depends on the angular momentum quantum number and it goes from negative L to positive L so if L is equal to zero we only get one value for the magnetic quantum number if L is equal to zero the only possible value we could get is zero and remember this tells us the orientation and so we have only one possible orientation we have only one S orbital all right so there's an S orbital there's only one of them here we know an S orbital is shaped like a sphere right so remember an orbital is the is the region where you're most likely to find an electron here so there's a total of one orbital in the first energy level and you could have gotten that by using N squared so when n is equal to one alright one squared is equal to one that's a total number of number of orbitals that you're going to find in the first energy level here let's move on to a number of electrons in the orbital so you can fit a maximum of two electrons in one orbital so the fourth quantum number right the fourth quantum number says the spin of an electron can either be it can be positive 1/2 or it can be negative 1/2 and so if you have two electrons in one orbital one electron has a spin a positive one half and one electron has a spin of negative one half and so if there's a maximum of two electrons in one orbital we have a maximum of two electrons in this 1s orbital in the first energy level and since that's the only orbital in this energy level or this shell that's also the total number of electrons in the shell all right so the total the total number of electrons is equal to two and you could have gotten that using 2n squared so if n is equal to one right so one squared is equal to one times two which is equal to two so that's the first energy level or the first shell let's move on to the second shell so this is where n is equal to two so if n is equal to two the principal quantum number is equal to two the angular momentum quantum number right what are the allowed values right well you start with zero and you go all the way up to n minus one so we start with 0 n minus 1 that'd be 2 minus 1 that'd be 1 so we go from 0 and then we go to 1 and then we have to stop so we have only two allowed values for the angular momentum quantum number right so if you have a n equal to 2 you get two allowed values here we already talked about what L is equal to zero means right L is equal to zero refers to an S orbital all right and there's 1s orbital so in the second energy level there's another S orbital this is different from the S orbital in the first energy level that we just talked about so there's another S orbital here it too is shaped like a sphere alright and what I drew here is is misleading I drew this as being a little bit smaller than the one before remember when n is equal to 2 you're further away your electron is on average further away from your nucleus here L is equal to 1 alright so if L is equal to 1 what are the allowed values for the magnetic quantum number so remember the magnetic quantum number is equal it goes from negative L to positive L so negative L would be negative 1 alright and then we include 0 and then we go to positive 1 so there are three possible values for the magnetic quantum number 1 2 3 the magnetic quantum number told us the orientation so there are three different orientations all right and we talked about L is equal to 1 is is referring to AP orbital which is shaped like a dumbbell so we have three different orientations we have three different P orbitals in the second energy level right one of them goes along the x-axis one of them the Y and one of them the Z so we talked about this in the previous video all right so a total of three P orbitals here so how many orbitals are there in the second energy level well we have one s orbital and three P orbitals so one plus three gives us four we could have done we could have done this math right and squared so 2 2 is N squared which gives us 4 all right let's do electrons now so 4 let's let's let's go back to the S orbital here remember there's one orbital so we can fit a maximum of two electrons in one orbital for the P orbitals we have three P orbitals so three if each P orbitals holding a maximum of two electrons three times two gives us six so we have a total of eight electrons in the second energy level so eight electrons and we can get that from two N squared again right because if n is equal to two all right square that you get four multiply that by two and you get eight all right let's go to the third energy level or the third shell here so when n is equal to three what are the allowed values for the angular momentum quantum number L so remember L goes from zero all the way up to n minus one so L goes from zero all the way to n minus one so L is equal to zero L is equal to one and L is equal to 2 because 3 minus 1 is equal to 2 so if we have n is equal to 3 all right have three possible values for L 0 1 and 2 we already talked about what L is equal to 0 means right L is equal to 0 is an S orbital right and there's one of them L is equal to 1 is AP orbital and with 3 allowed values for the magnetic quantum number right we're going to have 3 P orbitals in the third energy level so let's focus in on Elle's equal to 2 so when L is equal to 2 what are the allowed values for the magnetic quantum number well those go from negative L to positive L so if L is equal to 2 let me use a different color here so if L is equal to - all right we could go negative 2 negative 1 0 1 & 2 so that's a total of 1 2 3 4 5 5 values so 5 different orientations for this orbital and when L is equal to 2 we call this a D orbital so 5 different orientations so 5 D orbitals so I'm going to write a 5 here so the total number of orbitals that we have in this energy level all right that'd be 1 + 3 + 5 and so that's 9 which we could have gotten from 3 squared right so the total number of orbital is equal to n squared if n is equal to 3 3 squared gives us 9 alright how many electrons can we fit in each one of those orbitals well let me once again use different colors so the S orbital we have we have one of those so we could fit a maximum of 2 electrons into that S orbital for the P orbitals in the third energy level we have three of them so each one of those orbitals could fit a maximum of 2 so 3 times 2 gives us 6 and then we go to the D orbitals all right so five D orbitals each one can hold a maximum of two electrons so 5 times 2 gives us 10 so what are the what's a total number of electrons that we can fit in this third shell here so that'd be 2 + 6 + 10 that's 18 all right and again we could have used this little formula over here so - N squared so if n is equal to 3 square that that's 9 times 2 this gives us 18 all right let's do let's do one one more and before I do that we already talked about the shape of an S orbital we talked about the shape of the p orbital when you get into things like D orbitals you start to get a little bit complicated and it's a little bit tricky for me to draw so I'm not going to attempt to draw all the D orbitals in the 5 different orientations so let's just move on to n is equal to 4 all right and so let's go ahead and rewrote our Eero what we were going for here all right so we're going to go for n so we're going to pretend is equal to 4 now and is equal to 4 what are the allowed values for L so the zero all the way up to n minus one so L is equal to zero l is equal to 1 l is equal to two and then n minus 1 that's 4 minus 1 that's equal to 3 so that's the last about that's the last allowed value for L the angular momentum quantum number and once again if you have 4 here you get 4 allowed values all right when L is equal to 0 we said that's an S orbital and when L is equal to 1 we said that's a p orbital when L is equal to 2 we call that a D orbital and when L is equal to 3 we're going to call this an F orbital and s P D and F come from old nomenclature used in atomic spectroscopy and so it's it's kind of its kind of not it's not really used anymore but like the S I think used to stand for sharp and so that's where that's where your your letters come from here we we use them to think about the orbitals and the different shapes here so an S orbital shaped like a sphere we know we have one of those P orbitals shaped like a dumbbell we know we have three of those D orbitals we just talked about the fact that we have five of them all right so five different possible values here for the magnetic quantum number so we have five D orbitals in the fourth energy level and then finally F right so an F orbital when L is equal to 3 what are the allowed values for the 4 for the magnetic quantum number so M L so what are the allowed values well they go from negative L to positive L all right so if L is equal to 3 let me use a different color if L is equal to 3 all right we get negative 3 negative 2 negative 1 0 1 2 & 3 so how many different how many different orbitals are we talking about now that'd be 1 2 3 4 5 6 7 so 7 F orbitals all right in the fourth energy level all right maximum number of electrons in the orbital right we have only one let me use me what color should I use here let's use magenta so I have 1s orbital right maximum two electrons so two electrons here three P orbitals two times three is six five D orbitals in the fourth shell or the fourth energy level so five times two gives us ten and then we just talked about F orbitals which are which would be way too difficult for me to draw so you can get some nice nice nice pictures of F orbitals online or in your textbook here so if we have seven F orbitals all right seven times two gives us a total of fourteen so we could have a maximum of 14 electrons in the in the F orbitals all right what's that total right so we add all those up so two plus six plus ten plus 14 all right that's 32 so there's 32 electrons in the fourth shell in the fourth energy level and once again we could have used our formula here so 2n squared so when n is equal to four all right we square that and we get 16 multiply that by two and we get 32 and so hopefully this gives you this gives you some experience playing with the kalam numbers so you have to be this is a very useful exercise so just sit down and think about think about how the quam numbers depend on each other right so L L depends on N and M L write the magnetic wall number depends on depends on this and so it it's going to allow you to understand understand the periodic table and electron configurations