#quantum numbers

Chemistry no 1

Quantum Numbers

Of course, I’m happy to share my knowledge about quantum numbers! 

Let’s start off with the question: what exactly are they?

Quantum Numbers: list of numbers that characterize an outer electron in an atom. Or in simpler words, the location of an electron within an atom! These numbers are represented in the form: .

There are four electronic quantum numbers that make up the “location”

Principle Quantum Number (n): refers to the electron shell or I like to call the energy level. This can be determined by the row that the atom is located in. For example, sodium is located in the third row so it’s principle quantum number will be n = 3.Note: n cannot equal 0 because there is no such thing as a zero energy level.

Orbital Angular Momentum Quantum Number (l): refers to the shape of the orbital (or subshell). In which, there are four* types of subshells - s, p, d, and f. Each of these subshells can be assigned a number where s = 0, p = 1, d = 2, and f = 3. To calculate the possible values for this quantum number take the formula: n-1. For example, fluorine is in the second row on the periodic table. So n = 2 and it’s possible l values are 1 and 0 because fluorine contains S and P orbitals. *In theory, there are infinite types of subshells but we only know of the four types that can be assigned to current atoms. (Correct me if I’m wrong!)  Tip: split up the periodic table in the types of subshells they represent. This will give you a better understanding of what l value to assign to a specific electron when you’re determining it’s “location”.

Magnetic Quantum Number (ml): refers to the number of orbitals and the orientation. In other words, where in the subshell is the electron located. To find this value take the formula: -l…0…+l. Going back to the fluorine example, let’s say the electron has an l = 1 so it’s ml = -1, 0, +1. Which makes sense because l = 1 indicates an P orbital and there are three orbitals in the x, y, z positions. So the electron can be located in either three of these S orbitals.

Electron Spin Quantum Number (ms): refers to the spin of the electron. This number is simple, the electron either spins +½ or -½ though I’m not positive if the negative or positive value goes first when assigning them.

House Address Analogy

Let me tell you a good analogy I heard from my professor when I took general chemistry. Think about quantum numbers like an address of someone’s house. Where n = country, l = state, ml = street, and ms = building number. Each number tells you how precise they are with the location of the house. So convert that into the address of an electron! 

Further Example: Find the quantum number of the 1st outer electron in a lithium atom. 

So lithium is located in the second row of the periodic table, n = 2. The electron we are focused on is in the first row, l = 0 because it’s the section for S orbitals (refer to the first diagram I posted about splitting sections of the periodic table into their orbital types). Now the ml value can only be 0 since an S orbital is only made up of a single sphere. Lastly, the ms value is where I get a little confused, so I’m going to go off the idea that -½ comes first. And there we have it! The quantum numbers for the 1st outer electron in a lithium atom: <2, 0, 0, -½>.

In case my explanation wasn’t fully comprehensible… I’m going to list some extra help and practice problems below.

There are 4 quantum numbers: the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (ml), and the spin quantum number (ms). These numbers can be used to identify electrons, and each quantum number increases in specificity.

The principal quantum number (n) identifies the shell/energy level of an electron. The lowest energy level possible is 1, which is closest to the nucleus. Every electron in the 1st energy level will have n = 1, every electron in the 2nd has n = 2, and so on. Larger values of n means an increase in energy and distance from the nucleus.

The angular momentum quantum number (l) gets closer in specificity and identifies the subshell. The subshells you need to know are s, p, d, and f. l = 0 corresponds to the s subshell, l = 1 corresponds to p, l = 2 is d, and l = 3 is f. The maximum l value for an electron depends on its principal quantum number: maximum l = n - 1. This should make sense, because the first energy level (n = 1) only has an s subshell (l = 0), so l = 1 is impossible. The 2nd energy level (n = 2) has s and p subshells, so p subshells (l = 1) have the highest angular momentum quantum number possible for that level.

The magnetic quantum number (ml)identifies the orbital. Each subshell has a few different orbitals. s has 1 orbital, p has 3, d has 5, and f has 7. Each of these orbitals has a different shape. The orbitals in each subshell are labeled on a range of -l to l (the angular momentum quantum number). For s: l = 0, so the only mlpossible is 0, which works because there is only 1 s orbital. For p: l = 1, so -1, 0, and 1 are possible, which works again because there are 3 different p orbitals. Don’t worry about whether the labels correspond to specific orbitals (the p subshell has px, py, and pz orbitals, but for Summer Chem it doesn’t matter what numbers you would give them). Usually, however, the first filled orbital is the one with the lowest ml.

The final quantum number is the spin quantum number (ms). This is the most specific quantum number, as it finally identifies a single electron. Every orbital can hold 2 electrons, but they must be spinning in opposite directions (the explanation is based in quantum physics so you don’t need to know that). These spins are defined as either ½ or -½. In any full orbital, one electron will be ½ and the other is -½. As before, don’t worry about which electron would be ½ and which would be -½; it doesn’t matter.

If you haven’t already been bored or are still confused by the explanation, here’s a full example.

Let’s consider fluorine. The full electron configuration is 1s2 2s2 2p5. n = 2 refers to the 7 electrons found in the 2nd shell/energy level of the atom (2s2 2p5). l = 1 would refer to the p subshell (also the maximum possible value; max l = n - 1). We have narrowed our group of electrons down to 5 (2p5). ml= -1 refers to one of the orbitals in the p subshell, for example px. Each orbital can hold 2 electrons, and the only unfilled orbital would be the last one (ml= 1). With ml= -1, our group is narrowed down to 2 electrons. To differentiate between the two, we use ms: with ms = ½, we’ve finally gotten to a single electron. Our final quantum numbers are 2, 1, -1, ½. Every electron can be labelled this way, and each electron has a different set of numbers (the Pauli exclusion principle).

Since I just got done writing about electron configuration and energy levels of atomic orbitals, I’ve been thinking about why the electrons arrange themselves the way they do. 

To recap, electrons surround an atomic nucleus in ordered “shells”.  The idea is that each shell represents a discrete energy level.  It’s easiest to occupy the lowest level, but because electrons are attracted to the positively charged nucleus, they’ll settle for the higher levels if the lower ones are already full.  This bottom to top arrangement was postulated in the 1920′s as the Aufbau Principle, which states: The orbitals of lower energy are filled in first with the electrons and only then the orbitals of high energy are filled.

However, larger atoms with greater numbers of electrons manage to defy this principle.  For example, manganese has 25 protons in its nucleus, and so it attracts 25 electrons to surround it.  If the Aufbau principle applied, the electron configuration of manganese ought to be 1s22s22p63s2 3p63d7 . The first two electrons would fill up the 1s subshell, which is the lowest energy level.  Then two more would head for the 2s subshell.  Then the 2p subshell would fill up, and it can hold six more electrons.  Then the third shell would start to fill.  Two electrons in the 3s subshell, then six more in the 3p subshell, and that just leaves seven more, which can all fit in the 3d subshell with room to spare.  Right?

Wrong.  The actual electron configuration of manganese is 1s22s22p63s2 3p64s23d5.  For some reason the electrons skip the 3d subshell in favor of the 4s subshell, which ought to have a higher energy level because, you know, 4 is bigger than 3.  Once the 4s subshell is full, then the electrons will start filling up the 3d subshell. 

The Madelung Rule tries to cover this situation with a mathematical relationship.   The electron shells are identified by the principal quantum number, n=1, 2, 3,...  The subshells are defined by the azimuthal quantum number, ℓ=0, 1, 2, 3...   The letters used to identify subshells represent specific values for ℓ.    When ℓ=0, it’s an s subshell.  When ℓ=1, it’s a p subshell.  When ℓ=2, its a d subshell. 

What the Madelung Rule says is that you actually have to take the sum of n and ℓ together to determine which energy level fills up next.  And if we crunch the numbers for the first several subshells, we find:

1s:  n+ℓ = 1+0 = 1

2s: n+ℓ = 2+0 = 2

2p: n+ℓ = 2+1 = 3

3s: n+ℓ = 3+0 = 3

3p: n+ℓ = 3+1 = 4

3d: n+ℓ = 3+2 = 5

4s: n+ℓ = 4+0 = 4

4p: n+ℓ = 4+1 = 5

So 4s has a lower n+ℓ value than 3d, even though it’s part of a higher shell.  According to the Madelung Rule, that’s why manganese’s electron configuration is the way it is. 

It strikes me, however, that in the event of a tie, as with 2p and 3s, the subshell with the lower value of n gets priority.  When I think about it, it makes sense, because even if 2p and 3s cost the same amount of energy to occupy, there are other benefits to consider.  Let’s consider an oxygen atom for this scenario. 

The actual electron configuration for oxygen is 1s22s22p4.  But if the Madelung rule applies, you could argue that some of the 2p electrons could just as easily go in 3s.  So we could suggest a configuration of 1s22s22p23s2.   I mean, why not, right? 

The problem with doing that is that you make it tougher for the oxygen atom to completely fill its outermost electron shell.  If the configuration is 1s22s22p4, then the outermost shell is the second one, which has six out of eight slots filled.  If an oxygen atom can grab two more electrons (by forming a water molecule with two hydrogen atoms, for example), then the outermost shell would be full, and the oxygen atom would become even more stable. 

But a configuration of 1s22s22p23s2 shoots that idea all to hell.   Now the oxygen atom has to lose two electrons from shell #3, only to seek out four more electrons to fill up shell #2.  It’s just too big a hassle, so even though both configurations might cost the same energy to achieve, one is simply more practical than the other.  I don’t know if there’s a quantum mechanical justification for my reasoning, but I bet there is.