A Closer Look at Shells, Subshells, and also OrbitalsSubshellsProblems

A total of four quantum number are provided to describe fully the movement and trajectories of each electron in ~ an atom. The combination of all quantum number of all electrons in an atom is described by a wave duty that complies with the Schrödinger equation. Each electron in an atom has a unique collection of quantum numbers; follow to the Pauli exemption Principle, no 2 electrons can share the same combination of four quantum numbers. Quantum numbers room important since they can be provided to determine the electron configuration of an atom and also the probable location of the atom"s electrons. Quantum number are also used to know other features of atoms, such as ionization energy and also the atomic radius.

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In atoms, there are a full of four quantum numbers: the primary quantum number (n), the orbital angular momentum quantum number (l), the magnetic quantum number (ml), and also the electron turn quantum number (ms). The primary quantum number, (n), explains the energy of an electron and the most probable distance of the electron indigenous the nucleus. In various other words, it describes the dimension of the orbital and the power level one electron is put in. The number of subshells, or (l), explains the form of the orbital. The can likewise be supplied to recognize the variety of angular nodes. The magnetic quantum number, ml, defines the energy levels in a subshell, and ms describes the spin on the electron, which can either be up or down.

The principal Quantum Number ((n))

The principal quantum number, (n), designates the principal electron shell. Due to the fact that n explains the most probable street of the electron from the nucleus, the bigger the number n is, the aside from that the electron is native the nucleus, the bigger the dimension of the orbital, and the larger the atom is. n have the right to be any positive integer starting at 1, together (n=1) designates the very first principal shell (the innermost shell). The an initial principal shell is also called the soil state, or lowest energy state. This describes why (n) have the right to not be 0 or any negative integer, due to the fact that there exist no atoms v zero or a negative amount of power levels/principal shells. Once an electron is in an excited state or that gains energy, it might jump to the 2nd principle shell, where (n=2). This is dubbed absorption due to the fact that the electron is "absorbing" photons, or energy. Well-known as emission, electron can additionally "emit" power as they run to reduced principle shells, whereby n decreases by entirety numbers. Together the energy of the electron increases, for this reason does the major quantum number, e.g., n = 3 shows the third principal shell, n = 4 indicates the 4th principal shell, and also so on.

Example (PageIndex1)

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

Example (PageIndex2)

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


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

The orbit Angular inert Quantum Number ((l))

The orbital angular inert quantum number (l) identify the shape of an orbital, and therefore the angular distribution. The number of angular nodes is same to the value of the angular inert quantum number (l). (For an ext information about angular nodes, see electronic Orbitals.) Each value of (l) shows a specific s, p, d, f subshell (each distinctive in shape.) The worth of (l) is dependence on the principal quantum number (n). Uneven (n), the worth of (l) have the right to be zero. It can additionally be a optimistic integer, yet it can not be bigger than one less than the principal quantum number ((n-1)):

Example (PageIndex3)

If (n = 7), what room the feasible values that (l)?


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

Example (PageIndex4)

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


The number of angular nodes is equal to the value of l, therefore the number of nodes is additionally 4.

The Magnetic Quantum Number ((m_l))

The magnetic quantum number (m_l) identify the variety of orbitals and also their orientation in ~ a subshell. Consequently, that value depends on the orbit angular momentum quantum number (l). Provided a particular (l), (m_l) is one interval ranging from (–l) to (+l), so it can be zero, a an unfavorable integer, or a positive integer.

Example (PageIndex5)

Example: If (n=3), and also (l=2), then what room the possible values the (m_l)?


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

The Electron spin Quantum Number ((m_s))

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

Example (PageIndex5)

List the feasible combinations the all four quantum numbers when (n=2), (l=1), and also (m_l=0).


The 4th quantum number is independent of the an initial three, permitting the an initial three quantum number of 2 electrons to be the same. Due to the fact that the spin can be +1/2 or =1/2, there space 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 through (m_s=1/2) have a downward spin?


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

A Closer Look at Shells, Subshells, and Orbitals

Principal Shells

The worth of the principal quantum number n is the level of the principal digital shell (principal level). All orbitals that have actually the same n value are in the same major level. Because that example, every orbitals ~ above the second principal level have a major quantum variety of n=2. When the worth of n is higher, the variety of principal digital shells is greater. This reasons a higher distance in between the farthest electron and also the nucleus. As a result, the size of the atom and also its atomic radius increases.


Because the atomic radius increases, the electrons space farther from the nucleus. Hence it is easier for the atom to expel an electron since the nucleus walk not have as strong a pull on it, and the ionization power decreases.


The number of values that the orbital angular number together can additionally be supplied to recognize the variety of subshells in a primary electron shell:

as soon as n = 1, l= 0 (l bring away on one value and also thus there can only it is in one subshell) once n = 2, l= 0, 1 (l bring away on two values and also thus there room two possible subshells) once n = 3, l= 0, 1, 2 (l takes on 3 values and also thus there room three possible subshells)

After looking in ~ the instances above, we view that the worth of n is same to the variety of subshells in a principal digital shell:

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

To recognize what form of feasible subshells n has, this subshells have been assigned letter names. The worth of l identify the name of the subshell:

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


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

We have the right to designate a primary quantum number, n, and a specific subshell by combine the value of n and the surname of the subshell (which deserve to be found using l). Because that example, 3p describes the third principal quantum number (n=3) and also the ns subshell (l=1).

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The number of orbitals in a subshell is identical to the variety of values the magnetic quantum number ml bring away on. A useful equation to identify the variety of orbitals in a subshell is 2l +1. This equation will certainly not give you the worth of ml, yet the variety of possible values that ml can take top top in a particular orbital. Because that example, if l=1 and ml have the right to have values -1, 0, or +1, the worth of 2l+1 will certainly be three and there will be three different orbitals. The names of the orbitals are called after the subshells castle are discovered 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 figure below, we see examples of two orbitals: the p orbital (blue) and the s orbital (red). The red s orbit is a 1s orbital. To picture a 2s orbital, imagine a layer similar to a cross ar of a jawbreaker roughly the circle. The great are showing the atom angular nodes. To picture a 3s orbital, imagine another layer about the circle, and also so on and also so on. The ns orbital is comparable to the form of a dumbbell, through its orientation in ~ a subshell relying on ml. The shape and orientation of one orbital relies on l and ml.


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

Principle Quantum Number 4.jpgwhich quantum number determines each of the following?