6.4 Quantum Numbers
Chad's General Chemistry Videos
- 19.1 Oxidation Reduction Reactions and Oxidation States
- 19.2 Balancing Oxidation Reduction Reactions
- 19.3 Galvanic Cells
- 19.4 Standard Cell Potentials aka emf or Voltage
- 19.5 Nonstandard Cell Potentials the Nernst Equation
- 19.6 Reduction Potentials and the Relationship between Cell Potential, Delta G, and the Equilibrium Constant
- 19.7 Electrolytic Cells
- 19.8 Electrolysis Calculations
Cool we got to talk about quantum numbers, good times. So quantum numbers, a little bit of a funky concept, students often don't know what they're talking about and so we're not going to talk about where these numbers came from but we're gonna talk about how you remember them. I like to think of these numbers as an electron's address. An electron's address turns out these quantum numbers. We represent them with four letters: n, l, m sub l, m sub s. Notice we often represent numbers with letters. Right, like, pi is a letter but it represents a number and things of a sort. So in this case we call them quantum numbers and often throw students "but you call them numbers and then they are letters what's going on?" So, sorry it's just the way it is. These are the symbols that represent the four different quantum numbers. You're supposed to know their names, you're supposed to know what they tell you, as well as their range of values possible but again, they're kinda like an electron's address. So let's say we want to figure out my address, I know, and you didn't know where in the US I lived. Somehow magically. What might be the first question, if you want to narrow this down gradually? What might be the first question you ask me- "where you live?"- We'll narrow this down gradually. We're in the U.S. So what might be the first question? How do I narrow this down? What state you live in! and what if I wanted to be a little bit cryptic? Instead of saying Arizona, I said I live in state number 48. I called it a number. Is there a state number 48? How does state number 48 actually mean Arizona? Yeah 48th state in the Union. Something a little bit cryptic but I identify what state I live in using a number. That's the key here. And so then you went and asked "Well you know what city in Arizona do you live in?" And I said well I live in city number seven. So Mesa turns out. So and I have no idea if Mesa was the seventh city in Arizona I'm just making that up but maybe it was and so again I'm using a number to represent a city, a location. And then you might ask me my street and then you might ask me my house number and by the time you've asked me those four things you know exactly where I live, right? I know, creepy. So in this case the quantum numbers are an electron's address. Where in an atom an electron lives, these four things will tell you. So first thing I might want to know is just which shell. This is the first shell, second shell, third shell and fourth shell, and again there's an infinite number of shells. They just keep going up. so first question I might do to narrow this down is, what shell do you live in? That's what n actually means. n is the shell number. So we call it the principal quantum number- this is all on your handout there- and it tells me the shell number. Cool, so if you notice here n was the same n we had when we're doing electronic transitions here. And here's 1, 2, 3 and 4 and so on and so forth. Now l on the other hand. l is what we call the azimuthal quantum number. Not a word you will commonly hear. So, and this is going to tell me the type of sub shell, whether it's s, p, d or f. That's what l is ultimately going to tell me. So in this case we use it as a code. When l equals 0 that means s. When l equals 1 that means p. When l equals 2 that means d. And when l equals 3 that means f. it's a code so if i say l equals 2 you're like: d orbital or d subshell. If i say l equals 1 you're like p subshell. l equals 3 f subshell it's code. Cool. Then m sub l- on your handout there- this is your magnetic quantum number. We'll come back to that in a second, and m sub s we call your spin quantum number. Cool, if we look at the range of values- and this is the kicker here- that's possible for each of these, what is the lowest value n can have? Yep first shell. That's the lowest value. How high can it go? Infinity. So its range of values goes from 1 to infinity. Cool. Now what's the lowest possible value for l? Zero. What's the highest possible value? Well it turns out it goes higher than three but it goes up to n minus one. So here's the way this works, they define l based on what shell you're in. So the lowest value is always zero because every shell has an S orbital and when l equals zero that means you're in an S sub shell. Okay. But then they define it because, notice in your first shell that's all you have. In your second shell what kind of sub shells do you have? Two kinds s and p, right? And s means l equals 0. And p means l equals 1. So when you're in shell number 2; 0 and 1 are possible for l. When you're in shell number 3; 0, 1 and 2 are possible for l. When you're in shell number 4; 0, 1, 2 and 3 are possible and that's why they cap it off at n minus 1. If you're in the fourth shell you can get 0, 1, 2, and a maximum of 3 for the range. All integer values from 0 up to the maximum. So if you were in the second shell, what values can have ya have? Just zero and one. Second shell, 2-1 to be the max, that's one. Zero and one that's all you got. And in the second shell all you have is s and p: 0 and 1. Cool so, l gets across this direction and we've got 0, 1, 2 & 3 for the fs up there. Cool so, let's say I'm this electron right here, what's n? Yeah it's in the third shell, n is 3. What's l for a d subshell? It's 2. Great, so now I not only know what shell I'm in, I now know the sub shell as well. But now how do I narrow this down further? And what might I want to know now? Yeah, what box? And what do these boxes represent? They represent the orbitals. Orbitals. So there's five orbitals in the 3d subshell, so which orbital? That's what m sub l is ultimately going to tell us. Now it's a little cryptic here. They often tell us that m sub l tells us the orientation in space. That's another way of saying which orbital, because the orbitals are all different based on their orientations in space. Recall, we talked about the D orbital a second ago. Four of them look like four-leaf clovers, the only difference being their difference in the orientation space. And so this is just another way of saying that it tells us which orbital we're in. And the way this works: is this takes on integer values including zero from negative l to positive l. It's a little bit funky. So let's say I were in a p sub shell right here. Let's say it's the 2p, what's n? 2p, what's n? Yeah, what value of n? We're in the second shell so type 2. What's l for a p subshell. It's one, great. And if l equals 1 then what values could m sub l have here? negative 1, 0 and positive 1. 3 different values because I would have 3 different orbitals that all needed a value, to identify them and now I can identify each of them with a number in the same way that I could identify a state or a city with a number. Now I put negative 1, 0 and positive 1 here, notice this is random. It could be 0, positive 1, negative 1. Negative 1 positive 1, 0 -it's a random order. Here I just picked a convenient order. So notice we never would actually assign these on a diagram. Cool. Good, so in the f it's negative 3, negative 2, negative 1, 0, 1, 2 & 3 but again in any order. But when you're in the f orbitals where l equals 3, you've got from negative 3 to positive 3. 7 different values because you have 7 different orbitals. Nope not at all. You're never going to worry about which of these has which value of m sub l, not at all. Cool. Finally your spin quantum number. Turns out when we draw- when we start filling diagrams in here, right? We'll draw electrons in like this, right? And this is just a way of diagramming out opposite spins. Turns out your spin either has a value of positive 1/2 or negative 1/2 those are your only options. Their way of saying we have two opposites: is an up arrow and a down arrow. It's all it represents. Up doesn't mean positive and down doesn't mean negative it's arbitrary here as well. We never tell you which one's positive and which one's negative, but we do know that if you have two electrons in the same orbital one's plus and one's minus. They can't be the same that would be a violation of the Pauli exclusion principle which we'll talk about in just a little bit. Cool. You need to know the symbol for each one of these, their names what they tell you and their range of values, cool? We start filling in electrons in an atom as long as we're in the ground state- the most stable configuration- where do we start filling electrons? Yeah, you start down here. Why do you start here? Because it's the first shell? Why do we start there? Because it's the lowest energy. The universe likes things low in energy, so you start there. Who told us to start low energy and start filling as we go up? Mr. Aufbau. It's the Aufbau principle. You start filling in lowest energy and work your way up. So after 1s would come-what do I fill in next?-2s. And where do I go from there? 2p. Well now I got a problem all three of these orbitals are equal in energy- might be what somebody- good friend of yours- calls your ex-boyfriend? we call them degenerate. When you have equal energy orbitals they're degenerate. That's our special word for this. Cool so, these are degenerate when you have degenerate orbitals they all get one before you ever pair any of them up. Whose rule is that? That's Hund's rule. The other part of Hund's rule is that when you start putting them in they all get the same spin: they're all spin up or all spin down-it's arbitrary. You don't have to fill these in left to right. You could put, you know, one here first and it could be spin up or spin down but once you've put that there and it's spin down then the other ones you fill in also have to be spin down, they all match. This is part of Hund's rule. But once they each have one, now what? Now you can start pairing them up. Nope not at all, I don't have to pair them up in the same order the first ones went in at all. They're all equal energy, it's kind of arbitrary. Cool, now where do I go? Good, 3s. So what if I went to 4s instead? That would be bad. What would that be a violation of? The other one, the name you can't- no. Aufbau principle says you go from low energy to high- you don't skip over the lower ones before you go to a higher one. That would be a violation of the Aufbau principle- you should know what that looks like. Okay let's not do that. That would be bad. That might be an example of what we call the excited state. Definitely a violation to the Aufbau principle. What if- that's not blue not that it matters but I like blue electrons yes OCD indeed. *whispers about yellow* All right, so what would this be a violation of? That's- Violation of Hund's rule. In degenerate orbitals everybody gets one before you start pairing up so I can't do that either. And finally-so let's say we not violate Hund's rule and then we do this. That is a violation of the Pauli exclusion principle. The Pauli exclusion principle says no two electrons in an atom can have the same four quantum numbers. If you notice that means no two electrons can be in the same orbital and have the same spin. Because if they're in the same orbital then they're definitely in the same shell, they're definitely in the same subshell and they're in the same orbital. So all three of these are the same, which means they'd better have a different spin because all four numbers can't be the same. That's a violation of the Pauli exclusion principle. So, now you know what a violation of the Aufbau principle looks like, of Hund's rule looks like and of the Pauli exclusion principle looks like. You should recognize them all- makes a great question. Cool.
In this lesson you will learn:
-The name, meaning, and range of possible values for the 4 quantum numbers
1. The Principal quantum number (n)
2. The Azimuthal quantum number (l)
3. The Magnetic quantum number (ml)
4. The Spin quantum number (ms)
-A description/explanation of the Pauli Exclusion Principle
The following table summarizes the name, meaning, and range of possible values for each of the quantum numbers.
|l||Azimuthal||Subshell (Type of Orbital)||[0...(n-1)]|
|ml||Magnetic||Orientation in Space||[-l...+l]|
|ms||Spin||Spin Up or Spin Down||+1/2 or -1/2|
Pauli Exclusion Principle
The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers. Ultimately, this means that you can't have two electrons in the same orbital with the same spin.