okay so this would be the third and last lecture over chapter seven so in this chapter we get into more of what quantum mechanics really is okay so in the last lecture we made this point that heisenberg uncertainty principle so you cannot know both the position and the momentum or speed of an electron and i forgot to put this on the last one too right can't know them both simultaneously at the same time if you know one then you have no clue what the other really is um so but what we can do and what we can calculate however is the probability of where you could find an electron in a region of space using what's known as the schrodinger equation and we're not going to go through the math on how to calculate that because it's extremely complex math right if you have some calculus 1 some calculus 2 some calculus 3 some linear algebra some differential equations then you might know enough math to do the math involved in this work so we're obviously not going to do that what we're going to do instead and i'm perfectly okay with that is just look at nice pretty pictures okay so the probability of where an electron can be found in the space is what we call an orbital so probability means statistics basically right so statistically was the greatest shot of or chance of finding an electron at so if we plot that the probability versus the distance from the nucleus so for hydrogen then you get a picture that looks like this and so basically what's done then is to take ninety percent of this area take ninety percent of that region and call it an orbital so the maximum probability of finding hydrogen's one electron is 57 picometers from the nucleus of the atom okay so quantum numbers is um basically utilized it basically provides like an address for where you can find an electron at mm atom and so there's four quantum numbers so three of the quantum numbers uh n l m sub n l and m sub l are used to describe the different orbitals and the fourth quantum number m sub s is used to describe electrons in that orbital so basically they're just like addresses for uh an electron just like if you had a address for your house so you have a state you have a town or city and a street address and then a house address same kind of a similar concept for electrons is what these quantum numbers do is describe where that electron is at and what region of space that electron is in looks like okay so we're going to go through these quantum numbers in order n and then l and then m sub l and then m sub s and so if you haven't seen this before the material may not make much sense it's pretty abstract until we get through n l and m sub l then at that point we can start tying things together and hopefully it makes crystal clear sense at that point but as we go through it it may not make a whole lot of sense uh the material is not difficult once you wrap your mind around it it's really trivial stuff to to understand if you want to do the math behind it that's really complex but just to understand quantum numbers and the print pictures uh graphically involved is pretty simple but it's pretty abstract as we go through it okay so n is what's known as the principal quantum number so it can have values so for these quantum numbers you want to know what the possible values are for them so n can have values from one two three four any integer on up to infinity so it always starts at one and can run to infinity so any whole number greater than zero so this is the same n that we talked about from the bohr model uh so so n primarily describes an orbital size and its energy and the energy of the electron in that orbital so as n goes up right the size of the orbital goes up so we looked at the bohr model in the last lecture right so this is the nucleus of the atom and then you have shells and so n represents the different shells so as n goes up the size of the shell or orbital goes up and the energy of the electron goes up so where is the most stable place to put the the electron is in the first shell so if it's in the second or third shell then the electron is less stable so each quantum number has a letter and a number designation so the first shell so that so one two three four would be the number designation they have letter designations as well one is also called the casio two is called the l shell three the m shell for the n shell and then it's just following the alphabet so chemists usually don't use the letter designations for shells we usually just talk about or say first shell second shell third shell fourth shell right so that's all i'm going to expect you to know is just a numerical designation for shells okay so the l quantum number so it's what's known as the angular momentum quantum number so what are its possible values so we can have values from zero to n minus one and any integer in between so any whole number or integer in between so l primarily determines the shape of an orbital what an orbital looks like so for its number and letter designations so it's number designation right l zero one two three four uh if it's zero that's called an s orbital if l is one it's called a p orbital if l is two it's a d orbital three is an f orbital 4 is a g orbital this is something that you want to memorize as fast as possible because we use all both of these interchangeably all the time okay so again if l is zero that means it's an s orbital if l is one that means it's a p orbital that we're talking about if l is two that means it's a d orbital that we're talking about and so this describes the shape of an orbital so what is what does an s orbital look like well s orbitals are spherical so this is the two-dimensional thing that i'm riding on you have to keep in mind this is three dimensions really so it's a sphere it's not a circle it's a sphere so if an electron is in an s orbital then it's somewhere in that sphere right okay so what a p orbitals look like p orbitals are dumbbell shaped um so dumbbell shaped or if we want a nice prettier picture it looks like this on the right where you again you have to keep in mind that this is three dimensions so the electron is somewhere in that space if it's in a p orbital so it does not mean the electron does not mean the electron traces a path right does not mean it traces some figure eight path that's not what it means at all it just means the electron this is if you have the mathematical equation behind this and you plot it you get these regions of space that look like a dumbbell and it just means the electron is somewhere in there right where's the nucleus at the nucleus is here and of course the probability of finding the electron density in the nucleus is zero right because protons and neutrons are in the nucleus so the electron is somewhere in this top lobe or somewhere in this bottom lobe so sometimes um you see so the nucleus is there so sometimes you see p orbitals represented simply like this or sometimes you see them represented like this where they're shaded or sometimes you see them represented like it is here with a phase plus and minus so what does that do to so electrons have wave like properties so for a wave right you have a positive and a negative phase and so you could see to represent that p orbitals either represented shaded in unshaded lobe or a positive and a negative lobe trying to represent the wave like properties of electrons but again it doesn't mean the electron is traced on a path right it just means the electron is somewhere in that space okay so if n is so if n is one so if we're at the first shell then you l can only be zero so you can only have an s orbital at the first shell so this would be put these two together this would be what's called a one s orbital but if n is two so keep in mind l can be zero to n minus one so you always start at zero and then if n is two two minus one is one so l could also be one so at the second shell then you have what are known as two s orbitals right because if l is zero that means that's an s orbital or at the second shell l could be one and if l is one then that means that's a p orbital then at the second shell you have what are called two p orbitals so every time you go up a shell you add a new kind of orbital the first shell only has s the second shell has s and p orbitals so with the third shell if n is three then what are the possible values of l so l can be zero to n minus one so zero or one or three minus one would be two so l could be zero one or two and so if l is two then that means you have d orbitals at the third shell so here you would have three s orbitals right this would mean you have three p orbitals and these type of orbitals would be what are called 3d orbitals okay so third quantum number so it's probably looking a little abstract right now but we're going to tie it all together in a minute and hopefully it makes sense so the third after we get through this third quantum number so m sub l is the third quantum number that's what's known as the magnetic quantum number and so it primarily determines the number of orbitals of a given shape and their orientation in space so the possible values of m sub l can be anywhere from negative l to positive l so the value of m sub l m sub l depends on the value of l right the value of l depends on the value of n so if l is zero then m sub l can only be zero right because it goes from negative l to positive l if l is one so it can m sub l can be negative one or it can be zero or it can be positive one right all the way from negative l to positive l all of the integers in between so if l is two then m sub l can be negative two negative one zero one or positive two so some of the questions in this chapter would be that easy i tell you l is two what is m sub l and then you would just say well it's negative two negative one zero one or two so it may not make much sense what does that mean yet but if you can do that then you got some of the chapter down already if l is three what is m sub l it's negative three negative two negative one zero one two or three all the way from negative l to positive l okay so again so if l is zero then that's an s orbital right which is a sphere and all spheres only have one orientation in space but if l is one we set if l is one then that means this is a p orbital and p orbitals have dumbbell shape right but then m sub l tells you how many p orbitals there are there are three values of m sub l so there are three different p orbitals and they are orientated differently in space all right so if l is one m sub l could be negative one zero or one that's three values which means there are three different p orbitals so what do they look like they're orientated differently in space right so one so i have an xyz cartesian coordinate system here one can be orientated along the y-axis one is orientated along the x-axis so again they're just dumbbells but they're orientated along different axis and the other is orientated along the z axis right so we would call this py we would call this px and we would call this pz so three orbitals orientated perpendicular to each other right this would be a nicer prettier picture from just a google search but you have to keep in mind they're all on the same cartesian coordinate system so that's three different p orbitals right this one this in this case they have z up and x that way and y coming in and out so p x p y p z three different p orbitals if you had one electron it could be in any of those p orbitals so we'll we'll see and i'll make the point stress the point later that any one orbital can have two electrons in it so since there's three p orbitals you could put six electrons in them two could be in the orbital that's in black two could be in the orbital that's in uh blue and two could be in the orbital that's in red right so you can have anywhere from one zero electrons and p orbitals to six maximum any any number in between okay so let's try to put this together if n is equal to one then l can only be equal to zero right because l can only be zero to n minus 1. and then if l is 0 m sub l can only be 0 because m sub l can only be negative l to positive l so with the first shell there is just the 1s orbital and this is so the number in front is n and the ladder is l and so you you have to keep them again you have to have this memorized you must memorize this l is zero means that's an s orbital l is one means that's a p orbital l is two means that's a d orbital okay so at the first shell you only have an s orbital if n is two so if you're at the second shell then l could be zero or l could be one because l is zero to n minus one and then if n l is zero m sub l could only be zero right or if l is one m sub l could be negative one zero or positive one so what does that mean so that means at the second shell there is a 2s orbital so this would be 2s and there are three p orbitals because of the three values of m sub l right so one of these would be 2py one of these would be 2pz and one of these would be 2px so at the second shell there are 4 orbitals total 2s 2px 2py 2pz so what's the difference between a 1s and or 2s orbital all s orbitals are spherical right so so that's 1s basically to us it's a little bit more complicated than this but for our purposes that's good enough so 2s is just bigger so if you went to the third shell and you had a 3s it's bigger right ok so the first shell schematic is you only have an s orbital so you have your nucleus and an s orbital right at the second shell you have your nucleus that would be your 1s orbital so if we consider the first and second shell and then you have a 2s orbital which is bigger and then at the second shell you also have p orbitals so you have a py right and then you have a px orbital so notice they or the start of the nucleus but there's no electron density in the nucleus so that would be p y and this was p that would be p x this was p y so this was p x and then you have pz right so you have pz so again i'm writing on a two dimensional board but i'm trying to illustrate three dimensions so what does the dotted line mean that means it's going behind the surface right so if you had x and you have y then you have z that's going in and out so one lobe is behind the board or one lobe is behind the ipad that i'm riding on and one mob is on the other face of the ipad coming out of the ipad so this would be pz and so the the solid wedge means it's coming out towards you right so we've gone from a simple model chapter two model where you have a nucleus and electron cloud to the bohr model where you have the nucleus and then you have shells right but the first shell is just simple it's just a sphere but the second shell is not this this simple as the bohr model illustrates right there then we've gone to the quantum mechanics model where the atom actually looks like that right the first shell you have a sphere at the second cell you have a sphere and you have these three p orbitals a low p y log p x and lobes and p y and p z direction and that's just the first two shells right and so every time you go up a shell you add a whole new kind of orbitals right after we go to the third shell and l could be two we did that right maybe not we're fixing to do that okay so if this was a hydrogen atom where there's just one electron where is that electron at well it wants to be closest to the nucleus what orbital was closest to the nucleus would be the 1s orbital it is the smallest and this is the most compact around the nucleus so it would be in the 1s orbital and then as you get to multi-electron atoms like sodium with 11 electrons were zt electron at and that's what chapter the next chapter is going to cover okay so if we if n is three then l can be zero one or two right because l again can be anywhere from zero to n minus one and so again we get this memorized as fast as possible so if l is zero then this is an s orbital and m sub l can only be zero so that would be the three s orbital right if l is one and then these are p orbitals then m sub l could be negative one zero or one if l is two then these are d orbitals according to this and m sub l can be negative two negative one zero one or two right so with the third shell s p and d orbitals exist how many p orbitals are there three p orbitals always come in sets of three because m sub l is always negative one zero or one three three numbers three orbitals it doesn't matter what the shell is how many d orbitals are there well how many m sub l's are there right l is two means d orbitals m sub l negative two negative one zero one or two is five values so there are five p d orbitals d orbitals always come in sets of five it doesn't matter what shell it is okay so again every time you go up a shell you add a whole new set of orbitals because there's another possible value of l okay so again at the first show you just have one s at the second shield sorry that's 2s you have 2s2px2y 2pz at the third shell you have 3s 3px3py 3pz plus these 5 orbitals so five d orbitals so what do d orbitals look like so let's sketch them so one of them looks like this so along the z axis you have what looks like a p orbital and then it has a doughnut around the middle that's what's called d z squared one of them looks like this it's in the uh let's get rid of this to make the less convoluted ones in the x y plane it's basically a cloverleaf in the x y plane but it's rotated 45 degrees off the axis that's what's called d x squared minus y squared so there's a third another one that's in the x y plane sorry i just messed up this one is called three d x y the it's another clove relief but this time it's orientated along the x and y axis that's what's called three d x squared minus y squared so one of them is a dumbbell with a doughnut around the middle the other four are clover leafs they're just orientated along different axises okay so one is in the x z plane so let me get rid of the x one is in the y z plane again it's a clover leaf but it's rotated 45 degrees off of the x z axis and that's what's called d y z and then the other one so let's get rid of y because it's in the x z plane it's another clove relief but it's in the x z plane so it's like two dumbbells squished together to make a clover leaf and what do you think that one's called well this one's called dxy 45 degrees off the x y axis and this is called y z 45 degrees off the y z axis this one's called x c dxz or if you google it you can get nice as prettier pictures right so they just have their xyz cartesian coordinate system defined differently right but there you have d z squared right you can see the dumbbell with a doughnut around the middle so the electron can be anywhere in that space okay so i just take this took this then and i filled in what these five t orbitals are called all right d a z squared d x y d y z d x z d x squared minus y squared so at the first shell how many orbitals are there there's one and each orbital can only have two electrons so you could put two electrons in that orbital at the second shell how many orbitals are there sorry there should be two 2s 2px 2py 2pz so there's four orbitals each could have two electrons so you could put 80 electrons there so you could put 2 in the 2s you could put 2 and 2px 2 and 2py 2 and 2pz at the third shell how many orbitals are there there's 3s that's one there's three p orbitals three px three py through pz so that's four and there's five d orbitals so five six seven eight nine total so there's nine orbitals how many electrons you could put 18 electrons in the third shell okay so we've seen these numbers before we've seen these numbers in the bohr model right first shell can have two electrons second shell eight electrons third shell 18 electrons so bohr figured out how many electrons each shell could hold but he didn't know why because he didn't understand quantum mechanics yet but quantum mechanics it illustrates why and especially once we have the fourth quantum number m sebastian it will all tied together okay so if we went to the fourth shell then l can be zero one two or three right so you go to another shell you add a whole new set of orbitals if l is three what does that mean uh that means again if we have this memorized where's that right there if l is three that means we have f orbitals as well okay um where are we at sorry yeah so if l is three then m sub l what are the possible values of m sub l then that's about to be negative three negative 2 negative 1 0 1 2 or 3 right so how many f orbitals are there there are seven you see seven values so there are seven f orbitals how many electrons can they hold well each can hold two so you can put 14 electrons there so we set up the first shell there's one orbital two electrons at the second shell so that would be the 4s orbital at the this yeah so we're n equals four so there's s there's p there's d and there's f there's four s at the fourth shell there's four p and there's three of them which could hold a total of six electrons there's four d orbitals right because of the five values of m sub l if l is equal to two so there's five of those right d orbitals always come in sets of five so that could hold uh 10 electrons and there's seven f orbitals and they get all 14 electrons so how many orbitals total 7 8 9 10 11 12 13 14 15 16 16 orbitals total at the fourth shell so how many electrons could that hold each can have two so 32 electrons right so each of these orbitals basically each of these m sub l values is represented in a different orbital that can have two electrons in it right so again fourth shell can hold 32 electrons okay so hopefully this is making sense so what all orbitals exist at the fifth shell well you have 5s 5p 5d 5f right so f orbitals exist starting at the fourth shell and then every shell after that we'll have f orbitals and then if we go to the fifth shell then there must be a new kind of orbital right because if l is five if n is five then l could go all the way up to four so that means you would have g orbitals as well so just following the alphabet after that you kind of have to memorize s and p but d on is just the alphabet d f g h so forth okay so then you would also have five g over those at the fist shield how many orbitals total in the fifth shell oh there's an easy way and a hard way to do this the harder way would be to work through this right all of the m sub l values or the easy way is this the fifth shell can hold 50 electrons how many orbitals it must be let's go up 50 electrons so there must be 25 orbitals right so there's one uh s there's three p's right these always come in sets of five f always come in sets of seven so how many g orbitals do you think there are well nine right there's a pattern one three five seven nine so this is because um so if l is equal to five m sub l is equal to negative five negative four negative three negative two negative one zero one two or three that's five values of m sub l so are nine values so that's nine g orbitals and if you add that up you get 25. okay so again every time you go up a shell you add a new kind of orbitals so we looked at s orbitals p orbitals we looked at d orbitals um there's always only one of the s there's always three of the p's there's always five of the d's there's always seven of the f's there's always nine g's so if we went to 11 orbitals or h orbitals there would be 11 of them right okay so we looked at s orbitals p orbitals d orbitals what the f orbitals look like and the answer is i have no i do i don't have these memorized i don't deal with that f4 orbitals very often so but you can google it so i googled it maybe it's in your textbook if not you can google it uh so that's what f orbitals look like seven of them much more complex so i don't know what f orbitals look like i don't expect you to know what f orbitals look like but i do expect you to know what s p d and f orbitals look like and i expect you to know the names of the different orbitals right what does dx squared minus y squared look like what does dz squared look like for example okay so this is another picture i just googled this is probably the best schematic of d orbitals i've ever seen so this has all of the d orbitals superimposed on each other you can see the what the yellow there is dz squared all right yeah okay okay so lastly um i meant to put this on the second lecture but i forgot so if we go back to the bohr model right quick so it's a simple model it's much too simple right quantum mechanics is what orbitals really look like and what shells really look like but the bohr model can be useful to illustrate some concepts in chemistry so if we just took hydrogen lithium sodium potassium for example all group one elements and we sketched the bohr model so hydrogen's got one electron so it would be in the first shell lithium has three electrons so two would be in the first shell and one would be in the second shell sodium's got eleven electrons so according to the bohr model the first can hold two the second kind of eight third and according to quantum mechanics then that third you only have one more to go so the third would have one electron or if you went to potassium right you could put two in the first eight in the second 18 in the third and then you have one on the fourth shell so if you notice there's something they all have in common so they're all group one group one elements what else did they have in common if you notice they have one electron they all have one electron in the outer shell or if we did this for fluorine chlorine bromine iodine well what do they have in common they're all halogens right um and if you drew their bore model so fluorine's got nine electrons so two in the first shell seven in the second shell outer shell chlorine 17 electrons two in this first shell eight in the second show seven in the third show in the outer shell um sorry this should be chlorine like copy and pasted and this should be bromine so eight two in the first shell eight in the second shell eighteen in the third shell and then seven in the outer shell so again what do they have in common if you look at their outer shell they all have the same number of electrons seven electrons in their outer shell and so why are elements in columns because elements and columns have different similar chemical properties right similar physical and chemical properties and what are chemical and physical properties due to uh well chemical properties are due to electrons in their outer shell and so it makes sense then that all of the elements in the same column same group have similar outer shell or electron configurations right also what group number is for the halogens in this is group seven right this is group one one electron in the outer shell seven electrons in the outer shell so this is where we can start tying things together in chemistry right elements in the same group have similar chemical and physical properties and one of the reasons behind that is because they have similar electron configurations in their outer shells which bohr model and quantum mechanics nicely illustrates that okay we will stop there and that is it for chapter seven you