A Closer Look at Shells, Subshells, and also OrbitalsSubshellsProblems

A full of four quantum numbers are supplied to describe fully the movement and also trajectories of each electron in ~ an atom. The mix of all quantum number of every electrons in one atom is defined by a wave duty that adheres to the Schrödinger equation. Every electron in one atom has actually a unique collection of quantum numbers; follow to the Pauli exemption Principle, no two electrons have the right to share the same combination of 4 quantum numbers. Quantum numbers room important due to the fact that they deserve to be used to identify the electron construction of an atom and the probable location of the atom"s electrons. Quantum number are also used to know other qualities of atoms, such as ionization energy and also the atom radius.

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In atoms, there room a complete of 4 quantum numbers: the major quantum number (n), the orbit angular momentum quantum number (l), the magnetic quantum number (ml), and also the electron turn quantum number (ms). The principal quantum number, (n), defines the power of an electron and the many probable distance of the electron indigenous the nucleus. In other words, it describes the dimension of the orbital and the power level one electron is put in. The variety of subshells, or (l), defines the form of the orbital. The can additionally be offered to identify the variety of angular nodes. The magnetic quantum number, ml, explains the energy levels in a subshell, and ms describes the rotate on the electron, which can either be up or down.

The major Quantum Number ((n))

The principal quantum number, (n), designates the primary electron shell. Due to the fact that n explains the many probable distance of the electrons from the nucleus, the bigger the number n is, the aside from that the electron is from the nucleus, the bigger the dimension of the orbital, and also the larger the atom is. n can be any kind of positive integer starting at 1, as (n=1) designates the very first principal shell (the innermost shell). The first principal shell is also called the ground state, or lowest energy state. This explains why (n) deserve to not it is in 0 or any negative integer, due to the fact that there exists no atoms v zero or a an adverse amount of energy levels/principal shells. As soon as an electron is in an excited state or the gains energy, it may jump to the second principle shell, where (n=2). This is referred to as absorption due to the fact that the electron is "absorbing" photons, or energy. Recognized as emission, electron can additionally "emit" power as they jump to lower principle shells, whereby n decreases by entirety numbers. Together the energy of the electron increases, so does the major quantum number, e.g., n = 3 suggests the 3rd principal shell, n = 4 shows the 4th principal shell, and also so on.

Example (PageIndex1)

If n = 7, what is the primary 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 orbit Angular momentum Quantum Number ((l))

The orbit angular momentum quantum number (l) identify the shape of an orbital, and also therefore the angular distribution. The number of angular nodes is equal to the worth of the angular inert quantum number (l). (For more information around angular nodes, see electronic Orbitals.) Each value of (l) suggests a certain s, p, d, f subshell (each distinctive in shape.) The worth of (l) is dependency on the primary quantum number (n). Unequal (n), the value of (l) can be zero. That can likewise be a optimistic integer, however it can not be bigger than one less than the principal quantum number ((n-1)):

Example (PageIndex3)

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


Since (l) have the right to be zero or a positive 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 does the atom have?


The variety of angular nodes is same to the worth of l, for this reason 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 in ~ a subshell. Consequently, the value counts on the orbital angular inert quantum number (l). Offered a particular (l), (m_l) is an interval varying from (–l) to (+l), therefore it have the right to be zero, a an unfavorable integer, or a positive integer.

Example (PageIndex5)

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


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

The Electron turn Quantum Number ((m_s))

Unlike (n), (l), and (m_l), the electron turn quantum number (m_s) walk not count on one more quantum number. That designates the direction of the electron spin and may have a spin of +1/2, represented by↑, or –1/2, represented by ↓. This method that once (m_s) is optimistic the electron has an increase spin, which deserve to be described as "spin up." once it is negative, the electron has actually a downward spin, so that is "spin down." The significance of the electron turn quantum number is its decision of one atom"s ability to generate a magnetic ar or not. (Electron Spin.)

Example (PageIndex5)

List the possible combinations the all 4 quantum numbers once (n=2), (l=1), and also (m_l=0).


The fourth quantum number is independent of the very first three, enabling the an initial three quantum number of two electrons to it is in 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 worth of (m_s) is positive, the electron is "spin up."

A Closer Look in ~ Shells, Subshells, and Orbitals

Principal Shells

The worth of the primary quantum number n is the level that the principal electronic shell (principal level). Every orbitals that have the exact same n value space in the same major level. Because that example, every orbitals on the 2nd principal level have a primary quantum number of n=2. Once the value of n is higher, the variety of principal digital shells is greater. This reasons a greater distance between the farthest electron and also the nucleus. Together a result, the size of the atom and also its atomic radius increases.


Because the atom radius increases, the electrons space farther indigenous the nucleus. Therefore it is simpler for the atom come expel one electron because the nucleus walk not have actually as strong a traction on it, and the ionization energy decreases.


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

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

After looking at the instances above, we view that the value of n is same to the number of subshells in a principal digital shell:

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

To determine what form of possible subshells n has, these subshells have been assigned letter names. The value of l determines the name 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 v n = 2 has actually one s subshell and one ns subshell (l = 0, 1) major shell v n = 3 has actually one s subshell, one p subshell, and one d subshell (l = 0, 1, 2)

We have the right to designate a primary quantum number, n, and a specific subshell by combining the worth of n and also 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 variety of orbitals in a subshell is identical to the variety of values the magnetic quantum number ml take away on. A helpful equation to recognize the number of orbitals in a subshell is 2l +1. This equation will certainly not provide you the worth of ml, however the number of possible worths that ml have the right to take top top in a particular orbital. Because that example, if l=1 and also ml have the right to have worths -1, 0, or +1, the worth of 2l+1 will certainly be three and there will certainly be three various orbitals. The name of the orbitals are called after the subshells lock 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 of 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 orbit is a 1s orbital. To picture a 2s orbital, imagine a layer comparable to a cross section of a jawbreaker around the circle. The great are showing the atoms angular nodes. To picture a 3s orbital, imagine an additional layer roughly the circle, and so on and also so on. The p orbital is similar to the form of a dumbbell, v its orientation in ~ a subshell depending on ml. The shape and orientation of an orbital counts on l and also ml.


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

Principle Quantum Number 4.jpgwhat are the possible values of ms for the orbital?