# Atomic Quantum Numbers

by Jared Rovny

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00:02 Now that we have an idea of the basic structure of a Bohr atom.

00:05 We are ready to discuss of the electrons in those atoms which can be a little more intricate and complicated in the basic parts, the basic energy levels that we already introduced.

00:16 Just as a review and a starting point we have here an atom with the different electrons orbiting it.

00:22 And each electron is in some energy level.

00:24 Either the ground state or one of the excited states.

00:27 And again we number these levels, the energy levels starting with 1 and then going up to 2 and 3 and so on through the integers.

00:34 These electrons in fact don't have to orbit in a perfectly spherical round shape though.

00:40 So for a given energy level we could have different possible shapes to the orbit.

00:45 So for example, if we just picked the n = 2 energy level, this first excited state, we could have the perfectly spherical and rounder orbital which we already discussed.

00:55 And that's in the green circle here.

00:57 But we could also have sort of this lobed structure that you can see.

01:01 And we name these shapes with a new letter.

01:03 Instead of 'n' we have 'l.' And 'l' describes the shapes of these different orbitals.

01:08 The important thing about l, the different types of shapes an orbital can have, is that the higher energy level you are at, the more different kinds of shapes you can assume.

01:19 So for this n = 2 energy level that we have written here, there's only two possible shapes the electrons can have.

01:26 They can either be in a circular shape that you can see in the green here.

01:30 Or they can be in this lobed shape that we see in the blue.

01:33 We number the shapes of the energy levels in this same way that we number the energy levels themselves.

01:38 Except that 'l' will start at 0 instead of 1.

01:42 We have the l = 0 shape and that's always going to be this round spherical shape.

01:47 And then we have the l = 1 shape.

01:50 Which is this lobed structure.

01:51 And then we can keep counting up and up and up.

01:54 We have to stop our counting at some point because we can't have as many shapes as we want for a given energy level.

02:00 So for a given energy level 'n' the highest number of possible shapes we can have, we'll be n - 1.

02:07 So for example, we have l starting at 0 and then going to 1 for the n = 2 energy level just given by the equation we have for l here.

02:17 So for any energy level n you'll always start l at 0.

02:20 Start counting up to 0 to 1 to 2 and so on, until you arrive at the number right before the number of the energy level you are at.

02:29 So again for the n = 2 energy level, we start l at 0, which is always that round shape.

02:35 We go to 1 which it turns out is this nice lobe shape.

02:39 But then we stop, because we can't go any higher than n - 1.

02:42 So we two quantum numbers so far and that's what we are introducing here is different quantum numbers n and l.

02:49 The next quantum number just comes by looking at this lobed shape that we have and asking ourselves, Are there different possible orientations that could be in? So for example could we have these two lobes going into and out of the page rather than just being up and down.

03:02 Or may be side to side and left to right.

03:04 And of course the answer is yes.

03:06 We can have different shapes to our orbital to the particular shape that we have.

03:11 So these different orientations for a given shape are going to follow the same sort of convention.

03:16 We have a new name, a new lettering.

03:18 We call it 'm sub l' so m with a little l below it.

03:22 And then we give it a number which instead of starting at 1 and going up or starting at 0 and going up.

03:28 We'll instead start at a negative number and go up to the same positive number.

03:33 So for example let's take the exact case that we already had.

03:37 The n = 2 energy level.

03:39 The l = 1 orbital shape.

03:42 That's that lobed structure.

03:44 And now that we are going to start counting the different possible orientation.

03:47 Orientations that, that shape can have.

03:50 Just like with the energy levels having many possible shapes, every shape has many possible orientations.

03:57 And the number of orientations you have will depend on the type of shape that you have.

04:01 So for example for the very simple circular shape that we had with n = 1 and l = 0.

04:07 That circular shape cannot have different orientations.

04:10 Because no matter which way we turn a perfect sphere, it's always going to look like perfect sphere.

04:14 Whereas, when we have this lobed structure here, we suddenly acquired new different distinct orientations that those lobes structures could have.

04:23 So we can count those lobed structures different orientations by again using the equation that we have for m sub l here.

04:31 m sub l starts at the negative of the l number, the shape and moves in integers up to the positive of the l number for the shape.

04:40 So again let's look at the example we have written.

04:43 We have the n = 2 energy level.

04:45 We have the l = 1 shape, the lobed shape.

04:48 So the m sub l, the orientations of that shape can go from -1 up to 1 in integer steps.

04:56 So we have the -1 m sub l shape.

04:59 The 0 m sub l and the 1 m sub l shape.

05:03 Meaning that the total number of orientations that we have for this particular l is 3, just for the l = 1 shape for the orbitals.

05:13 This is the three possibly more confusing to visualize types of quantum numbers.

05:18 There's only one more quantum number and that is called the "Quantum Spin." So this one does not play directly off of the different shapes and orientations that we have.

05:26 It instead just single number that can be + or - .

05:29 And it is simple as that.

05:31 So for a given electron you could give it all these 4 numbers.

05:35 For a particular electron I could ask you, What is it's energy? What shape is the orbital that is in? What orientation does that shape have? And then finally, What spin does you electron have? Is it spin up electron or spin down electron.

05:50 These 4 numbers are called the 4 Quantum Numbers or the atomic quantum numbers for a particular electron.

05:57 You might have heard of the Pauli Exclusion Principle.

06:00 This is something that is talked about a lot.

06:02 But now that we've introduced our 4 quantum numbers, we're able to talk about it very scientifically and accurately.

06:08 The Pauli Exclusion Principle says " that no two electrons can have the same 4 quantum numbers.

06:15 Because if two electrons have the exact same 4 quantum numbers they would in fact be identical in the sense of this atom.

06:21 So for example, we could have these two electrons that we have pictured here.

06:25 They might be both at the energy level 2, they might both also be in the l = 0, or these spherical shape.

06:32 And they might even also be in a same orientation m sub l = 0.

06:36 But if this were all the case, we would require that they have different spins from each other.

06:41 One would be spin up and one would be spin down.

06:44 And so this is how, the Pauli Exclusion Principle works.

06:47 You have to keep any electrons, any two electrons throughout your atom with 4 different quantum numbers.

06:53 Again even if 3 are the same like we have here, you would just change that last one, to make sure you have distinct electrons.

### About the Lecture

The lecture Atomic Quantum Numbers by Jared Rovny is from the course Electronic Structure.

### Included Quiz Questions

1. n ( = 1, 2, ... ), l ( = 0, 1, ..., n-1 ), m_l ( = -l, ...,0, ..., +l ), m_s ( = ±1 )
2. n ( = 1, 2, ... ), l ( = 1, ..., n ), m_l ( = -l, ...,0, ..., +l ), m_s ( = +1 )
3. n ( = 1, 2, ... ), h ( = 0, 1, ..., n-1 ), m ( = 0, ..., +h ), p ( = ±1 )
4. n ( = 1, 2, ... ), h ( = 0, 1, ..., n-1 ), m ( = 0, ..., +h ), p ( = -1 )
5. n ( = 0, 1, 2, ... ), l ( = 0, 1, ..., n ), m_l ( = -l, ...,0, ..., +l ), m_s ( = ±1 )
1. 2l + 1, m_l = -l, ..., 0, ..., +l
2. 2l, m_l = -l, ..., 0, ..., l -1
3. l + 1 , m_l = 0, ..., l
4. 2l - 1 , m_l = -l +1, ..., 0, ..., l -1
5. 2l + 1, m_l = 0, ..., 2l
1. n, l = 0, 1, ..., n-1
2. 2n, l = 1, 2, ..., 2n
3. 2n + 1, l = 1, 2, ..., 2n+1
4. [n/2], l = 0, 1, ..., [n/2] -1
5. n + 2, l = 0, 1, ..., n + 1
1. Two electrons cannot have the same 4 quantum numbers.
2. Two electrons cannot have the same spin.
3. Only one electron is allowed per energy level n.
4. At energy level n, each orbit shape which corresponds to a quantum number l, can only hold one electron.
5. The number of electrons and protons in an atom must be equal.

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