Quantum Numbers describing Electronic Orbitals

An atom is composed of a nucleus containing neutrons và protons with electrons dispersed throughout the remaining space. Electrons, however, are not simply floating within the atom; instead, they are fixed within electronic orbitals. Electronic orbitals are regions within the atom in which electrons have the highest probability of being found.

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## Quantum Numbers describing Electronic Orbitals

There are multiple orbitals within an atom. Each has its own specific energy level và properties. Because each orbital is different, they are assigned specific quantum numbers: 1s, 2s, 2p 3s, 3p,4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. The numbers, (n=1,2,3, etc.) are called *principal quantum numbers *and can only be positive numbers. The letters (*s,p,d,f*) represent the *orbital angular momentum quantum number *(*ℓ*) và the orbital angular momentum quantum number may be 0 or a positive number, but can never be greater than n-1. Each letter is paired with a specific *ℓ* value:

An orbital is also described by its *magnetic quantum number *(m*ℓ*). The magnetic quantum number can range from *–ℓ* lớn +*ℓ*. This number indicates how many orbitals there are & thus how many electrons can reside in each atom.

Orbitals that have the same or identical energy levels are referred khổng lồ as **degenerate. **An example is the 2p orbital: 2px has the same energy level as 2py. This concept becomes more important when dealing with molecular orbitals. The **Pauli exclusion principle** states that no two electrons can have the same exact orbital configuration; in other words, the same quantum numbers. However, the electron can exist in spin up (ms = +1/2) or with spin down (ms = -1/2) configurations. This means that the s orbital can contain up to lớn two electrons, the p. Orbital can contain up lớn six electrons, the d orbital can contain up to lớn 10 electrons, và the f orbital can contain up lớn 14 electrons.

**s subshell**

**p subshell**

**d subshell**

**f subshell**Table 1: Breakdown & Properties of Subshells

ℓ = 0 | ℓ = 1 | ℓ = 2 | ℓ = 3 |

mℓ = 0 | mℓ= -1, 0, +1 | mℓ= -2, -1, 0, +1, +2 | mℓ= -3, -2, -1, 0, +1, +2, +3 |

One s orbital | Three p. Orbitals | Five d orbitals | Seven f orbitals |

2 s orbital electrons | 6 phường orbital electrons | 10 d orbital electrons | 14 f orbital electrons |

### Visualizing Electron Orbitals

As discussed in the previous section, the magnetic quantum number (ml) can range from –l lớn +l. The number of possible values is the number of lobes (orbitals) there are in the s, p, d, & f subshells. As shown in Table 1, the s subshell has one lobe, the phường subshell has three lobes, the d subshell has five lobes, and the f subshell has seven lobes. Each of these lobes is labeled differently and is named depending on which plane the lobe is resting in. If the lobe lies along the x plane, then it is labeled with an x, as in 2px. If the lobe lies along the xy plane, then it is labeled with a xy such as dxy. Electrons are found within the lobes. The plane (or planes) that the orbitals vì not fill are called nodes. These are regions in which there is a 0 probability density of finding electrons. For example, in the dyx orbital, there are nodes on planes xz & yz. This can be seen in Figure (PageIndex1).

api/deki/files/241973/180px-P3x.png?revision=2" />Figure (PageIndex2): Two orbitals. (left) The 3px orbital has one radial node & one angular node. (right) The 5dxz orbital has two radial nodes & two angular nodes. Images used with permission from WikipediaFor example, determine the nodes in the 3pz orbital, given that n = 3 and *ℓ *= 1 (because it is a p. Orbital). The total number of nodes present in this orbital is equal to lớn n-1. In this case, 3-1=2, so there are 2 total nodes. The quantum number *ℓ *determines the number of angular nodes; there is 1 angular node, specifically on the xy plane because this is a pz orbital. Because there is one node left, there must be one radial node. To sum up, the 3pz orbital has 2 nodes: 1 angular node and 1 radial node. This is demonstrated in Figure 2.

Another example is the 5dxy orbital. There are four nodes total (5-1=4) and there are two angular nodes (d orbital has a quantum number *ℓ*=2) on the xz & zy planes. This means there there must be two radial nodes. The number of radial & angular nodes can only be calculated if the principal quantum number, type of orbital (s,p,d,f), và the plane that the orbital is resting on (x,y,z, xy, etc.) are known.

## Electron Configuration within an Orbital

We can think of an atom lượt thích a hotel. The nucleus is the lobby where the protons & neutrons are, and in the floors above, we find the rooms (orbitals) with the electrons. The principal quantum number is the floor number, the subshell type lets us know what type of room it is (s being a closet, p being a single room, d having two adjoining rooms, and f being a suit with three rooms) , the magnetic quantum number lets us know how many beds there are in the room, và two electrons can sleep in one bed (this is because each has a different spin; -1/2 và 1/2). For example, on the first floor we have the s orbital. The s orbital is a closet và has one bed in it so the first floor can hold a total of two electrons. The second floor has the room styles s and p The s is a closet with one bed as we know & the p room is a single with three beds in it so the second floor can hold a total of 8 electrons.

Each orbital, as previously mentioned, has its own energy màn chơi associated to lớn it. The lowest energy màn chơi electron orbitals are filled first and if there are more electrons after the lowest energy cấp độ is filled, they move to the next orbital. The order of the electron orbital energy levels, starting from least to lớn greatest, is as follows: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p.

Since electrons all have the same charge, they stay as far away as possible because of repulsion. So, if there are xuất hiện orbitals in the same energy level, the electrons will fill each orbital singly before filling the orbital with two electrons. For example, the 2p shell has three p orbitals. If there are more electrons after the 1s, và 2s orbitals have been filled, each p orbital will be filled with one electron first before two electrons try to reside in the same phường orbital. This is known as **Hund"s rule**.

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