A Closer Look in ~ Shells, Subshells, and OrbitalsSubshellsProblems

A total of four quantum numbers are provided to describe completely the movement and also trajectories of every electron in ~ an atom. The combination of all quantum numbers of all electrons in an atom is defined by a wave function that adheres to the Schrödinger equation. Every electron in an atom has actually a unique set of quantum numbers; follow to the Pauli exclusion Principle, no two electrons deserve to share the same combination of 4 quantum numbers. Quantum numbers are important because they deserve to be provided to determine the electron configuration of one atom and also the probable place of the atom"s electrons. Quantum numbers are likewise used to understand other attributes of atoms, such together ionization energy and also the atom radius.

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In atoms, there are a total of 4 quantum numbers: the principal quantum number (n), the orbital angular momentum quantum number (l), the magnetic quantum number (ml), and the electron rotate quantum number (ms). The major quantum number, (n), describes the power of an electron and the many probable street of the electron indigenous the nucleus. In various other words, it describes the dimension of the orbital and also the power level an electron is put in. The number of subshells, or (l), explains the shape of the orbital. The can additionally be used to identify the variety of angular nodes. The magnetic quantum number, ml, explains the energy levels in a subshell, and ms refers to the rotate on the electron, which have the right to either be up or down.

The primary Quantum Number ((n))

The principal quantum number, (n), designates the major electron shell. Because n describes the most probable street of the electrons from the nucleus, the larger the number n is, the furthermore the electron is indigenous the nucleus, the bigger the size of the orbital, and the bigger the atom is. n deserve to be any kind of positive integer starting at 1, together (n=1) designates the first principal shell (the innermost shell). The first principal covering is also called the ground state, or lowest power state. This explains why (n) deserve to not be 0 or any an unfavorable integer, since there exist no atoms through zero or a negative amount of power levels/principal shells. As soon as an electron is in one excited state or the gains energy, it may jump come the 2nd principle shell, whereby (n=2). This is called absorption because the electron is "absorbing" photons, or energy. Well-known as emission, electron can likewise "emit" power as they run to reduced principle shells, wherein n reduce by entirety numbers. As the energy of the electron increases, for this reason does the major quantum number, e.g., n = 3 suggests the third principal shell, n = 4 indicates the fourth principal shell, and so on.

Example (PageIndex1)

If n = 7, what is the major electron shell?

Example (PageIndex2)

If an electron jumped from energy level n = 5 to energy level n = 3, did absorption or emissions of a photon occur?


Emission, because energy is shed by relax of a photon.

The orbital Angular inert Quantum Number ((l))

The orbital angular momentum quantum number (l) determines the shape of one orbital, and therefore the angular distribution. The variety of angular nodes is same to the worth of the angular inert quantum number (l). (For an ext information around angular nodes, see digital Orbitals.) Each worth of (l) suggests a particular s, p, d, f subshell (each unique in shape.) The worth of (l) is dependency on the principal quantum number (n). Uneven (n), the worth of (l) deserve to be zero. The can likewise be a positive integer, however it can not be bigger than one much less than the principal quantum number ((n-1)):

Example (PageIndex3)

If (n = 7), what are the possible values the (l)?


Since (l) can be zero or a optimistic integer less than ((n-1)), it can have a worth of 0, 1, 2, 3, 4, 5 or 6.

Example (PageIndex4)

If (l = 4), how countless angular nodes walk the atom have?


The number of angular nodes is same to the value of l, therefore the variety of nodes is likewise 4.

The Magnetic Quantum Number ((m_l))

The magnetic quantum number (m_l) identify the number of orbitals and also their orientation within a subshell. Consequently, that is value relies on the orbit angular momentum quantum number (l). Provided a certain (l), (m_l) is one interval ranging from (–l) come (+l), so it deserve to be zero, a an unfavorable integer, or a optimistic integer.

Example (PageIndex5)

Example: If (n=3), and also (l=2), climate what are the possible values that (m_l)?


Since (m_l) must range from (–l) to (+l), climate (m_l) have the right to be: -2, -1, 0, 1, or 2.

The Electron rotate Quantum Number ((m_s))

Unlike (n), (l), and also (m_l), the electron turn quantum number (m_s) go not rely on one more quantum number. The designates the direction of the electron spin and also may have actually a rotate of +1/2, stood for by↑, or –1/2, represented by ↓. This method that when (m_s) is optimistic the electron has actually an increase spin, which deserve to be described as "spin up." as soon as it is negative, the electron has a bottom spin, so it is "spin down." The definition of the electron spin quantum number is its determination of one atom"s capability to generate a magnetic field or not. (Electron Spin.)

Example (PageIndex5)

List the possible combinations the all four quantum numbers as soon as (n=2), (l=1), and (m_l=0).


The fourth quantum number is live independence of the an initial three, enabling the an initial three quantum numbers of 2 electrons to be the same. Because the spin can be +1/2 or =1/2, there are two combinations:

(n=2), (l=1), (m_l =0), (m_s=+1/2) (n=2), (l=1), (m_l=0), (m_s=-1/2)

Example (PageIndex6)

Can an electron with (m_s=1/2) have a downward spin?


No, if the worth of (m_s) is positive, the electron is "spin up."

A Closer Look at Shells, Subshells, and also Orbitals

Principal Shells

The value of the major quantum number n is the level that the principal electronic shell (principal level). All orbitals that have actually the exact same n value room in the same major level. Because that example, all orbitals ~ above the 2nd principal level have a principal quantum number of n=2. As soon as the value of n is higher, the number of principal electronic shells is greater. This reasons a better distance in between the farthest electron and also the nucleus. As a result, the dimension of the atom and its atomic radius increases.


Because the atom radius increases, the electrons are farther from the nucleus. Therefore it is easier for the atom come expel an electron because the nucleus walk not have as solid a traction on it, and the ionization energy decreases.


The variety of values the the orbital angular number l can also be supplied to identify the number of subshells in a primary electron shell:

when n = 1, l= 0 (l takes on one value and also thus there deserve to only it is in one subshell) once n = 2, l= 0, 1 (l take away on 2 values and thus there room two possible subshells) when n = 3, l= 0, 1, 2 (l takes on three values and also thus there space three feasible subshells)

After looking in ~ the instances above, we see that the worth of n is equal to the variety of subshells in a principal electronic shell:

major shell through n = 1 has actually one subshell principal shell v n = 2 has two subshells primary shell with n = 3 has actually three subshells

To determine what type of possible subshells n has, this subshells have been assigned letter names. The value of l identify the surname of the subshell:

name of Subshell value of (l)
s subshell 0
p subshell 1
d subshell 2
f subshell 3


principal shell with n = 1 has one s subshell (l = 0) major shell with n = 2 has actually one s subshell and one p subshell (l = 0, 1) major shell through n = 3 has actually one s subshell, one ns subshell, and one d subshell (l = 0, 1, 2)

We deserve to designate a primary quantum number, n, and also a certain subshell by combining the worth of n and also the name of the subshell (which can be discovered using l). Because that example, 3p refers to the 3rd principal quantum number (n=3) and the ns subshell (l=1).

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The number of orbitals in a subshell is indistinguishable to the variety of values the magnetic quantum number ml takes on. A valuable equation to identify the number of orbitals in a subshell is 2l +1. This equation will not provide you the worth of ml, however the variety of possible values that ml can take ~ above in a particular orbital. Because that example, if l=1 and ml can have worths -1, 0, or +1, the value of 2l+1 will certainly be three and also there will be three different orbitals. The surname of the orbitals are called after the subshells they are found in:

s orbitalsp orbitalsd orbitalsf orbitals
l 0 1 2 3
ml 0 -1, 0, +1 -2, -1, 0, +1, +2 -3, -2, -1, 0, +1, +2, +3
Number that orbitals in designated subshell 1 3 5 7

In the number below, we see examples of 2 orbitals: the ns orbital (blue) and the s orbit (red). The red s orbit is a 1s orbital. To picture a 2s orbital, imagine a layer comparable to a cross ar of a jawbreaker about the circle. The class are illustrating the atoms angular nodes. To photo a 3s orbital, imagine an additional layer approximately the circle, and so on and so on. The p orbital is comparable to the shape of a dumbbell, with its orientation within a subshell relying on ml. The shape and also orientation of an orbital relies on l and ml.


To visualize and also organize the very first three quantum numbers, we can think that them as constituents the a house. In the complying with image, the roof to represent the major quantum number n, every level represents a subshell l, and also each room represents the different orbitals ml in every subshell. The s orbital, due to the fact that the worth of ml deserve to only be 0, can only exist in one plane. The p orbital, however, has actually three possible values that ml and so it has actually three feasible orientations the the orbitals, presented by Px, Py, and Pz. The sample continues, through the d orbit containing 5 possible orbital orientations, and f has 7:

Principle Quantum Number 4.jpghow many subshells are in n=3