A Closer Look in ~ Shells, Subshells, and OrbitalsSubshellsProblems

A full of four quantum numbers are provided to describe totally the movement and also trajectories of each electron within an atom. The mix of all quantum numbers of all electrons in one atom is described by a wave function that complies with the Schrödinger equation. Each electron in one atom has a unique set of quantum numbers; according to the Pauli exclusion Principle, no 2 electrons deserve to share the same combination of four quantum numbers. Quantum numbers are important due to the fact that they have the right to be offered to identify the electron construction of one atom and also the probable ar of the atom"s electrons. Quantum numbers are also used to know other qualities of atoms, such as ionization energy and also the atomic radius.

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In atoms, there are a complete of 4 quantum numbers: the primary quantum number (n), the orbital angular inert quantum number (l), the magnetic quantum number (ml), and the electron turn quantum number (ms). The primary quantum number, (n), defines the energy of an electron and also the most probable street of the electron native the nucleus. In other words, it describes the dimension of the orbital and also the energy level one electron is placed in. The number of subshells, or (l), describes the shape of the orbital. The can also be provided to recognize the variety of angular nodes. The magnetic quantum number, ml, describes the power levels in a subshell, and ms refers to the turn on the electron, which deserve to either be up or down.

The primary Quantum Number ((n))

The primary quantum number, (n), designates the major electron shell. Because n defines the many probable distance of the electrons from the nucleus, the bigger the number n is, the aside from that the electron is indigenous the nucleus, the larger the dimension of the orbital, and the bigger the atom is. n deserve to be any type of positive integer starting at 1, together (n=1) designates the an initial principal covering (the innermost shell). The first principal shell is likewise called the floor state, or lowest energy state. This defines why (n) can not it is in 0 or any an unfavorable integer, because there exist no atoms through zero or a negative amount of energy levels/principal shells. When an electron is in one excited state or the gains energy, it may jump come the 2nd principle shell, where (n=2). This is dubbed absorption due to the fact that the electron is "absorbing" photons, or energy. Recognized as emission, electrons can also "emit" energy as they run to reduced principle shells, where n decreases by entirety numbers. Together the energy of the electron increases, therefore does the primary quantum number, e.g., n = 3 indicates the 3rd principal shell, n = 4 suggests 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 absorb or emission of a photon occur?


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

The orbital Angular momentum Quantum Number ((l))

The orbit angular inert quantum number (l) determines the form of an orbital, and 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 about angular nodes, see digital Orbitals.) Each worth of (l) indicates a details s, p, d, f subshell (each distinct in shape.) The worth of (l) is dependency on the major quantum number (n). Unequal (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 primary quantum number ((n-1)):

Example (PageIndex3)

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


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

Example (PageIndex4)

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


The variety of angular nodes is equal to the worth of l, therefore the number of nodes is also 4.

The Magnetic Quantum Number ((m_l))

The magnetic quantum number (m_l) determines the variety of orbitals and their orientation in ~ a subshell. Consequently, that is value relies on the orbit angular momentum quantum number (l). Offered a particular (l), (m_l) is an interval ranging from (–l) come (+l), therefore it can be zero, a an adverse integer, or a hopeful integer.

Example (PageIndex5)

Example: If (n=3), and also (l=2), then what are the possible values that (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 spin quantum number (m_s) go not depend on one more quantum number. That designates the direction that the electron spin and also may have a spin of +1/2, represented by↑, or –1/2, stood for by ↓. This means that when (m_s) is optimistic the electron has actually an upward spin, which can be referred to as "spin up." when it is negative, the electron has actually a downward spin, so it is "spin down." The definition of the electron turn quantum number is its decision of one atom"s capacity to generate a magnetic field or not. (Electron Spin.)

Example (PageIndex5)

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


The 4th quantum number is live independence of the first three, enabling the very first three quantum number of 2 electrons to it is in the same. Because the spin have the right to be +1/2 or =1/2, there room 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 one electron through (m_s=1/2) have a bottom spin?


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

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

Principal Shells

The worth of the principal quantum number n is the level of the principal electronic shell (principal level). Every orbitals that have the exact same n value space in the same primary level. For example, all orbitals on the second principal level have actually a primary quantum number of n=2. As soon as the worth of n is higher, the number of principal electronic shells is greater. This causes a higher distance in between the farthest electron and the nucleus. As a result, the size of the atom and also its atom radius increases.


Because the atom radius increases, the electrons space farther indigenous the nucleus. For this reason it is easier for the atom come expel one electron due to the fact that the nucleus go not have actually as strong a pull on it, and the ionization power decreases.


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

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

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

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

To identify what kind of feasible subshells n has, this subshells have actually been assigned letter names. The worth 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 with n = 1 has actually one s subshell (l = 0) principal shell v n = 2 has actually one s subshell and one ns subshell (l = 0, 1) major shell through n = 3 has actually one s subshell, one ns subshell, and also one d subshell (l = 0, 1, 2)

We can designate a major quantum number, n, and also a particular subshell by combining the worth of n and the name of the subshell (which deserve to be found using l). Because that example, 3p describes the third principal quantum number (n=3) and the p subshell (l=1).

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The variety of orbitals in a subshell is equivalent to the variety of values the magnetic quantum number ml takes on. A beneficial equation to determine the variety of orbitals in a subshell is 2l +1. This equation will not offer you the value of ml, however the number of possible worths that ml can take top top in a specific orbital. For 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 names of the orbitals are named 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 that orbitals in designated subshell 1 3 5 7

In the figure below, we see instances of 2 orbitals: the p orbital (blue) and also the s orbital (red). The red s orbital is a 1s orbital. To photo a 2s orbital, imagine a layer similar to a cross ar of a jawbreaker around the circle. The class are portraying the atom angular nodes. To snapshot a 3s orbital, imagine another layer around the circle, and also so on and also so on. The p orbital is similar to the form of a dumbbell, with its orientation in ~ a subshell depending on ml. The shape and also orientation of an orbital depends on l and ml.


To visualize and also organize the first three quantum numbers, we deserve to think of them together constituents of a house. In the adhering to image, the roof represents the major quantum number n, every level represents a subshell l, and each room to represent the different orbitals ml in each subshell. The s orbital, since the value of ml deserve to only it is in 0, have the right to only exist in one plane. The ns orbital, however, has three possible values that ml and so it has three possible orientations of the orbitals, presented by Px, Py, and also Pz. The sample continues, through the d orbital containing 5 feasible orbital orientations, and also f has 7:

Principle Quantum Number 4.jpgn = 2 l = 1 ml = 1