A Closer Look in ~ Shells, Subshells, and also OrbitalsSubshellsProblems

A full of 4 quantum numbers are offered to describe completely the movement and also trajectories of each electron in ~ an atom. The combination of all quantum number of every electrons in an atom is described by a wave duty that complies with the Schrödinger equation. Each electron in one atom has actually a unique collection of quantum numbers; follow to the Pauli exclusion Principle, no two electrons deserve to share the same mix of four quantum numbers. Quantum numbers space important due to the fact that they can be supplied to determine the electron construction of one atom and the probable location of the atom"s electrons. Quantum number are likewise used to recognize other qualities of atoms, such together ionization energy and the atom radius.

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In atoms, there space a complete of four quantum numbers: the primary quantum number (n), the orbit angular momentum quantum number (l), the magnetic quantum number (ml), and also the electron turn quantum number (ms). The major quantum number, (n), explains the power of an electron and also the most probable street of the electron from the nucleus. In other words, it describes the size of the orbital and also the energy level one electron is inserted in. The number of subshells, or (l), defines the form of the orbital. It can also be offered to determine the variety of angular nodes. The magnetic quantum number, ml, explains the power levels in a subshell, and ms describes the turn on the electron, which can either it is in up or down.

The major Quantum Number ((n))

The principal quantum number, (n), designates the principal electron shell. Since n defines the most probable street of the electron from the nucleus, the larger the number n is, the farther the electron is indigenous the nucleus, the bigger the dimension of the orbital, and also the larger the atom is. n can be any type of positive integer starting at 1, together (n=1) designates the an initial principal shell (the innermost shell). The an initial principal shell is also called the soil state, or lowest power state. This defines why (n) can not it is in 0 or any an adverse integer, since there exist no atoms v zero or a an unfavorable amount of energy levels/principal shells. As soon as an electron is in an excited state or the gains energy, it may jump come the second principle shell, where (n=2). This is referred to as absorption since the electron is "absorbing" photons, or energy. Recognized as emission, electrons can likewise "emit" energy as they run to reduced principle shells, where n to reduce by totality numbers. Together the energy of the electron increases, for this reason does the major quantum number, e.g., n = 3 indicates the third principal shell, n = 4 indicates the 4th principal shell, and so on.

Example (PageIndex1)

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

Example (PageIndex2)

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


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

The orbital Angular momentum Quantum Number ((l))

The orbit angular inert quantum number (l) determines the form of an orbital, and also therefore the angular distribution. The number of angular nodes is equal to the worth of the angular momentum quantum number (l). (For an ext information around angular nodes, see digital Orbitals.) Each value of (l) shows a certain s, p, d, f subshell (each distinctive in shape.) The value of (l) is dependence on the major quantum number (n). Unlike (n), the value of (l) can be zero. That can also be a hopeful integer, however it cannot be bigger than one much less than the major quantum number ((n-1)):

Example (PageIndex3)

If (n = 7), what space the feasible values of (l)?


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

Example (PageIndex4)

If (l = 4), how many angular nodes go the atom have?


The variety of angular nodes is same to the value of l, so the variety of nodes is additionally 4.

The Magnetic Quantum Number ((m_l))

The magnetic quantum number (m_l) determines the variety of orbitals and also their orientation within a subshell. Consequently, its value counts on the orbit angular inert quantum number (l). Given a details (l), (m_l) is one interval ranging from (–l) to (+l), for this reason it deserve to be zero, a negative integer, or a positive integer.

Example (PageIndex5)

Example: If (n=3), and (l=2), then what are the feasible values the (m_l)?


Since (m_l) must selection from (–l) to (+l), then (m_l) deserve to be: -2, -1, 0, 1, or 2.

The Electron spin Quantum Number ((m_s))

Unlike (n), (l), and (m_l), the electron turn quantum number (m_s) does not count on an additional quantum number. It designates the direction the the electron spin and also may have actually a rotate of +1/2, represented by↑, or –1/2, represented by ↓. This means that once (m_s) is confident the electron has actually an upward spin, which can be referred to as "spin up." once it is negative, the electron has a bottom spin, so the is "spin down." The meaning of the electron turn quantum number is its determination of an atom"s capacity to generate a magnetic field or not. (Electron Spin.)

Example (PageIndex5)

List the feasible combinations of all 4 quantum numbers as soon as (n=2), (l=1), and also (m_l=0).


The fourth quantum number is elevation of the an initial three, allowing the first three quantum number of two electrons to be the same. Due to the fact that the spin deserve to 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 actually a downward spin?


No, if the value 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 the same n value are in the same primary level. For example, every orbitals on the 2nd principal level have actually a principal quantum number of n=2. When the value of n is higher, the number of principal digital shells is greater. This reasons a greater distance in between the the furthest electron and the nucleus. As a result, the size of the atom and also its atomic radius increases.


Because the atomic radius increases, the electrons room farther native the nucleus. Hence it is much easier for the atom come expel an electron because the nucleus go not have as strong a pull on it, and also the ionization energy decreases.


The number of values of the orbit angular number l can additionally be supplied to recognize the variety of subshells in a major electron shell:

when n = 1, l= 0 (l bring away on one value and also thus there have the right to only it is in one subshell) once n = 2, l= 0, 1 (l take away on 2 values and also thus there are two feasible subshells) once n = 3, l= 0, 1, 2 (l takes on 3 values and thus there are three feasible subshells)

After looking at the examples above, we check out that the worth of n is same to the number of subshells in a principal electronic shell:

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

To determine what form of possible subshells n has, these subshells have actually been assigned letter names. The worth of l determines the surname of the subshell:

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


major shell v n = 1 has one s subshell (l = 0) principal shell through n = 2 has actually one s subshell and one p subshell (l = 0, 1) primary shell with 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 a certain subshell by combining the value of n and the surname of the subshell (which have the right to be uncovered using l). For example, 3p describes the 3rd 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 number of values the magnetic quantum number ml takes on. A advantageous equation to determine the variety of orbitals in a subshell is 2l +1. This equation will not offer you the value of ml, yet the number of possible values that ml deserve to take ~ above in a certain orbital. Because that example, if l=1 and ml have the right to have worths -1, 0, or +1, the worth of 2l+1 will be three and there will certainly be three different orbitals. The name of the orbitals are named after the subshells castle 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 the orbitals in designated subshell 1 3 5 7

In the figure below, us see examples of 2 orbitals: the p orbital (blue) and the s orbit (red). The red s orbital is a 1s orbital. To snapshot a 2s orbital, imagine a layer similar to a cross ar of a jawbreaker roughly the circle. The great are depicting the atoms angular nodes. To photo a 3s orbital, imagine one more layer around the circle, and so on and so on. The p orbital is comparable to the form of a dumbbell, through its orientation within a subshell depending upon ml. The shape and orientation of an orbital counts on l and also ml.


To visualize and organize the an initial three quantum numbers, we can think the them together constituents that a house. In the complying with image, the roof to represent the principal quantum number n, each level represents a subshell l, and also each room represents the various orbitals ml in every subshell. The s orbital, since the worth of ml have the right to only be 0, have the right to only exist in one plane. The ns orbital, however, has actually three possible values the ml and also so it has three feasible orientations that the orbitals, displayed by Px, Py, and also Pz. The sample continues, with the d orbital containing 5 possible orbital orientations, and also f has 7:

Principle Quantum Number 4.jpghow many electrons are there in the 2nd principal energy level (n = 2) of a phosphorus atom?