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

A total of 4 quantum numbers are offered to describe completely the movement and also trajectories of each electron in ~ an atom. The mix of all quantum numbers of all electrons in an atom is defined by a wave duty that adheres to the Schrödinger equation. Each electron in one atom has actually a unique set of quantum numbers; follow to the Pauli exclusion Principle, no two electrons have the right to share the same mix of four quantum numbers. Quantum numbers room important since they have the right to be provided to recognize the electron construction of an atom and the probable location of the atom"s electrons. Quantum number are additionally used to recognize other characteristics of atoms, such together ionization energy and the atomic radius.

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In atoms, there are a complete of 4 quantum numbers: the major quantum number (n), the orbit angular inert 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 also the most probable street of the electron native the nucleus. In other words, it refers to the size of the orbital and the energy level an electron is put in. The number of subshells, or (l), explains the form of the orbital. It can additionally be provided to identify the number of angular nodes. The magnetic quantum number, ml, describes the energy levels in a subshell, and ms describes the rotate on the electron, which deserve to either be up or down.

The principal Quantum Number ((n))

The primary quantum number, (n), designates the principal electron shell. Due to the fact that n defines the many probable distance of the electrons from the nucleus, the larger the number n is, the farther the electron is native the nucleus, the larger the dimension of the orbital, and also the larger the atom is. n deserve to be any kind of positive integer starting at 1, together (n=1) designates the an initial principal covering (the innermost shell). The first principal covering is likewise called the ground state, or lowest energy state. This defines why (n) have the right to not be 0 or any negative integer, due to the fact that there exist no atoms with zero or a an unfavorable amount of power levels/principal shells. When an electron is in an excited state or that gains energy, it might jump come the 2nd principle shell, whereby (n=2). This is called absorption because the electron is "absorbing" photons, or energy. Known as emission, electron can additionally "emit" power as they jump to lower principle shells, where n to reduce by entirety numbers. As the energy of the electron increases, so does the principal quantum number, e.g., n = 3 indicates the 3rd principal shell, n = 4 suggests the fourth principal shell, and also so on.

Example (PageIndex1)

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

Example (PageIndex2)

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


Emission, due to the fact that energy is shed by release of a photon.

The orbital Angular inert Quantum Number ((l))

The orbit angular inert quantum number (l) identify the form of one orbital, and also therefore the angular distribution. The number of angular nodes is equal to the value of the angular momentum quantum number (l). (For an ext information around angular nodes, see digital Orbitals.) Each worth of (l) indicates a particular s, p, d, f subshell (each unique in shape.) The worth of (l) is dependent on the principal quantum number (n). Uneven (n), the worth of (l) deserve to be zero. That can additionally be a hopeful integer, but it cannot be larger than one much less than the primary quantum number ((n-1)):

Example (PageIndex3)

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


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

Example (PageIndex4)

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


The variety of angular nodes is same to the worth of l, for this reason the number of nodes is likewise 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, the value relies on the orbit angular inert quantum number (l). Provided a certain (l), (m_l) is an interval ranging from (–l) to (+l), therefore it have the right to be zero, a an unfavorable integer, or a hopeful integer.

Example (PageIndex5)

Example: If (n=3), and (l=2), climate what space the feasible values of (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 rotate Quantum Number ((m_s))

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

Example (PageIndex5)

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


The 4th quantum number is elevation of the very first three, permitting the first three quantum numbers of 2 electrons to be the same. Since the spin can 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 an electron v (m_s=1/2) have actually a bottom spin?


No, if the worth 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 primary quantum number n is the level that the principal digital shell (principal level). All orbitals that have the exact same n value space in the same major level. Because that example, all orbitals ~ above the 2nd principal level have actually a primary quantum number of n=2. Once the value of n is higher, the variety of principal electronic shells is greater. This causes a higher distance between the the furthest electron and the nucleus. As a result, the size of the atom and its atomic radius increases.


Because the atom radius increases, the electrons room farther native the nucleus. Therefore it is easier for the atom come expel one electron since the nucleus go not have as strong a traction on it, and also the ionization power decreases.


The number of values that the orbital angular number l can additionally be offered to determine the number of subshells in a primary electron shell:

once n = 1, l= 0 (l bring away on one value and thus there deserve to only be one subshell) when n = 2, l= 0, 1 (l take away on two values and thus there are two feasible subshells) when n = 3, l= 0, 1, 2 (l bring away on three values and also thus there room three feasible subshells)

After looking at the examples above, we see that the worth of n is same to the variety of subshells in a principal digital shell:

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

To identify what kind of possible subshells n has, these subshells have been assigned letter names. The worth of l identify the surname of the subshell:

surname 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) major shell through n = 2 has one s subshell and one ns 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 have the right to designate a major quantum number, n, and a specific subshell by combine the value of n and also the name of the subshell (which have the right to be uncovered using l). For 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 indistinguishable to the variety of values the magnetic quantum number ml bring away on. A helpful equation to recognize 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 have the right to take top top in a certain orbital. Because that example, if l=1 and ml have the right to have values -1, 0, or +1, the value of 2l+1 will be three and also there will certainly be three various orbitals. The names 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 number below, us see examples of two orbitals: the p orbital (blue) and also the s orbital (red). The red s orbital is a 1s orbital. To picture a 2s orbital, imagine a layer comparable to a cross ar of a jawbreaker approximately the circle. The layers are depicting the atom angular nodes. To snapshot a 3s orbital, imagine an additional layer approximately the circle, and so on and so on. The ns orbital is comparable to the shape of a dumbbell, with 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 have the right to think the them as constituents of a house. In the complying with image, the roof represents the major quantum number n, each level represents a subshell l, and each room to represent the different orbitals ml in every subshell. The s orbital, because the value of ml have the right to only it is in 0, deserve to only exist in one plane. The ns orbital, however, has actually three feasible values of ml and so it has actually three possible orientations of the orbitals, presented by Px, Py, and Pz. The sample continues, v the d orbital containing 5 possible orbital orientations, and also f has actually 7:

Principle Quantum Number 4.jpghow many different values of l are possible for an electron with principal quantum number n = 4?