Transcript for:
Quantum Numbers and Orbitals

quantum numbers and atomic orbital is 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 for 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 is 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 uh stable elements that have electrons in them normally so uh but we'll talk about the f orbitals but 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 four there turns out 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 one s orbital three p orbitals five d orbitals seven f orbitals nine g orbitals if we cared eleven h orbitals if we care they count on the odd numbers and we'll see why when we get to quantum errors why exactly that is so 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 labels d x y d x z d y z 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 uh in space so dx y it turns out is in the x y plane d x e and the x z plane d y z is in the y z plane so and then d x squared minus y squared it turns out is also in the x y plane but whereas d x y is in between the x and y axes d x squared minus y squared we'd write 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 going to be in the x y plane cool so i just want to distinguish that but i i want 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 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 ovals 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 in the 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 complicated to 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 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 all right 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 and 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 you know electron is found and what shell the orbital i guess really is in that the electron is found in uh 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 uh according to mr bohr there so uh 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 that in the ground state we only have uh atoms with electrons up to like the seventh shell but again we can verify that the additional shells do indeed exist by you know 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 asmuthal quantum number so and your azimuthal quantum number is going to tell you the subshell so and you kind of think of this as a code here so the sub shell in this case could be s p d or f or again technically g or h but you're never going to see it so s p d or f and it's a code here so zero is going to 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're an s subshell when l equals 1 it tells you you're in a p sub shell when l equals 2 it tells you you're in a d sub shell and when l equals three it tells you you're in an f sub shell so these quantum numbers are going to describe and i say u 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 is found in a p orbital some somewhere and we call it the p sub shell because in any p sub shell 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 two that electron is in the d subshell one of the five d orbitals and when l equals three that electrons in the f subshell one of the seven f orbitals all right uh 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 one so and we'll see the profound implications of this here in a sec so but in this case so notice the lowest value could have a zero we saw that but the highest value it takes depends on what shell you're in if you're in shell number one i.e n equals one well one minus 1 is 0 and 0 up to 0 means it's 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 one two three four but not like three point four seven six so all integer values cool so if you're in the second shell n equals two well two minus one is one and so it can take any integers from zero up to one and so in the second shell it could be zero or it could be one and in the second shell there are s and p orbitals only in the first shell they're only s orbitals because the only value l could take was zero in the third shell you've got zero up to three minus one zero up to two so zero one and two and zero one and two mean that in the third shell they're gonna be s p and d sub shells 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 zero up to four minus one ie zero to three so zero one two and three because in the fourth shell there are s p d and f sub shells cool and that's the way it works and technically if you go up to the fifth sub shell i'm sorry the fifth shell that's where you'd actually encounter the g orbitals the g sub shell 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 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 and 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 ovals 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 going to tell you if a orb is on the x axis on the y axis or on the z axis that 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 zero l equaling zero means you're in an s sub shell and an s sub shell there's just one orbital and so you only have to specify one orbital and then going from negative zero to positive zero there's only zero and there's only zero because you only need to specify one orbital for an s subshell now if we've got a p subshell and a p sub shell l equals one and m sub l could therefore go from negative one up to positive one 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 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 one zero and one for the p orbitals if we go on to d orbitals for d orbital l is equal to two which means that m sub l could take on a range of values from negative two to positive two which means negative two negative one zero positive one positive two and there's five different values because there are one two three four five different d orbitals so again if l equals three l equaling three means f orbital there are seven f orbitals and that's gonna allow us to get seven different values now for m sub l so from negative three to positive three 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 three 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 time or two it's the only one that doesn't come from the schrodinger equation so 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 so and usually we attribute that to charged particles and things of a sort in a weird way and 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 the magnetic field and so it turns out 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 values it is either plus one half or minus one 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 gonna represent electrons here in a little bit with arrows like so and we're gonna 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 um 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 gotta give a prelude to the next lesson that's called the we got to talk about what's called the polyexclusion principle and it's 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 this 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 seven 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 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 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 one half or minus one 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 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 sub shell you remember l tells you the code zero uh 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 sub shell that that electron is in l is going to equal zero and then you recall that m sub l takes on values from negative l to positive l so in this case from negative zero to positive zero means the only possible value is zero cool and then finally m sub s well again we said that when you have two electrons in the same orbital we draw them as one spin up one spin down so and in this case when you only draw one of them though 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's 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 an uh 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 poly exclusion principle tells us that is not possible and it's not possible because paulie said so polly 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 than energy than the s the d's are higher than energy the p which are higher energy than the s and then f are higher energy than d or higher energy than p or 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 of hydrogen it turns out that all the subshells 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 it's only true for one electron system same thing here 3s3p3d 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 all right 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 the 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 3. and it's in a p sub shell and again l tells you the subshell and it the p sub shell corresponds to l equals 1. 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 so we can't actually get it specific here we can only say that it's either negative one zero or plus one it's one of those three it can't be negative two 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 uh 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 four is that possible yes but what if i told you it was n equals negative four well again n only takes on a range of values from one 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 let'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 4 l could be 0 1 2 or 3 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 2 and again it could have been 0 1 2 or 3 but no higher than 3. 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 so again outside that range of negative 2 to positive 2 but once again if i go in and say how about negative one that works it could have been negative two negative one zero plus one or plus two 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 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 any negative numbers are 0 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 it uh 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 and check out my general chemistry master course i'll leave a link in the description a free trial is available happy studying