A Closer Look at Shells, Subshells, and OrbitalsSubshellsProblems

A complete of 4 quantum number are provided to describe completely the movement and also trajectories of each electron in ~ an atom. The combination of all quantum numbers of every electrons in one atom is described by a wave function that adheres to the Schrödinger equation. Every electron in an atom has a unique collection of quantum numbers; follow to the Pauli exemption Principle, no 2 electrons deserve to share the same combination of four quantum numbers. Quantum numbers room important because they can be supplied to identify the electron construction of one atom and the probable location of the atom"s electrons. Quantum numbers are likewise used to understand other attributes of atoms, such as ionization energy and also the atomic radius.

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In atoms, there are a total of 4 quantum numbers: the major quantum number (n), the orbital angular inert quantum number (l), the magnetic quantum number (ml), and the electron turn quantum number (ms). The major quantum number, \(n\), explains the power of one electron and the most probable distance of the electron from the nucleus. In various other words, it refers to the dimension of the orbital and the energy level one electron is placed in. The number of subshells, or \(l\), defines the form of the orbital. That can likewise be supplied to identify the number of angular nodes. The magnetic quantum number, ml, describes the power levels in a subshell, and also ms refers to the rotate on the electron, which have the right to either it is in up or down.

The major Quantum Number (\(n\))

The major quantum number, \(n\), designates the principal electron shell. Since n defines the most probable street of the electron 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 very first principal covering is also called the soil state, or lowest energy state. This explains why \(n\) deserve to not be 0 or any an unfavorable integer, since there exists no atoms v zero or a negative amount of power levels/principal shells. When an electron is in one excited state or the gains energy, it may jump come the second principle shell, whereby \(n=2\). This is referred to as absorption since the electron is "absorbing" photons, or energy. Known as emission, electrons can likewise "emit" power as they run to lower principle shells, whereby n decreases by whole numbers. Together the power of the electron increases, so does the primary quantum number, e.g., n = 3 shows the 3rd principal shell, n = 4 indicates the fourth principal shell, and also so on.


Example \(\PageIndex1\)

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

Example \(\PageIndex2\)

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


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

The orbital Angular momentum Quantum Number (\(l\))

The orbital angular momentum quantum number \(l\) determines the form of an orbital, and therefore the angular distribution. The number of angular nodes is same to the worth of the angular inert quantum number \(l\). (For more information around angular nodes, see digital Orbitals.) Each value of \(l\) indicates a specific s, p, d, f subshell (each unique in shape.) The value of \(l\) is dependency on the primary quantum number \(n\). Uneven \(n\), the worth of \(l\) have the right to be zero. It can also be a confident integer, but it cannot be larger than one much 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 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 same to the value of l, for this reason the variety 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 within a subshell. Consequently, that value depends on the orbit angular momentum quantum number \(l\). Offered a details \(l\), \(m_l\) is an interval varying from \(–l\) come \(+l\), therefore it have the right to be zero, a negative integer, or a positive integer.


Example \(\PageIndex5\)

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


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

The Electron rotate Quantum Number (\(m_s\))

Unlike \(n\), \(l\), and \(m_l\), the electron turn quantum number \(m_s\) does not depend on another quantum number. That designates the direction that the electron spin and also may have a spin of +1/2, represented by↑, or –1/2, represented by ↓. This means that once \(m_s\) is hopeful the electron has an increase spin, which have the right to be described as "spin up." as soon as it is negative, the electron has a downward spin, so that is "spin down." The definition of the electron rotate quantum number is its determination of one atom"s capability to generate a magnetic ar or not. (Electron Spin.)


Example \(\PageIndex5\)

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


The fourth quantum number is independent of the an initial three, allowing the first three quantum numbers of 2 electrons to it is in the same. Due to the fact that the spin have the right 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 through \(m_s=1/2\) 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 also Orbitals

Principal Shells

The value of the primary quantum number n is the level the the principal electronic shell (principal level). Every orbitals that have actually the same n value room in the same primary level. Because that example, every orbitals on the second principal level have a major quantum number of n=2. Once the value of n is higher, the number of principal electronic shells is greater. This causes a better distance in between the the furthest electron and the nucleus. Together a result, the size of the atom and also its atom radius increases.


Because the atom radius increases, the electrons room farther native the nucleus. Thus it is easier for the atom to expel one electron because the nucleus walk not have actually as strong a traction on it, and the ionization power decreases.


The variety of values the the orbit angular number l can also be offered to determine the number of subshells in a primary electron shell:

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

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

To determine what type of possible subshells n has, this subshells have been assigned letter names. The value 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


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

We deserve to designate a principal quantum number, n, and also a certain subshell by combine the worth of n and the surname of the subshell (which can be uncovered using l). For example, 3p refers to the third principal quantum number (n=3) and also the p subshell (l=1).

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The variety of orbitals in a subshell is identical to the number of values the magnetic quantum number ml take away on. A useful equation to identify 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 values that ml deserve to take ~ above in a particular orbital. Because that example, if l=1 and also ml deserve to have values -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 they 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, 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 photo a 2s orbital, imagine a layer comparable to a cross section of a jawbreaker around the circle. The class are showing the atom angular nodes. To snapshot a 3s orbital, imagine one more layer about the circle, and so on and also so on. The ns orbital is similar to the shape of a dumbbell, v its orientation in ~ a subshell depending upon ml. The shape and also orientation of an orbital depends on l and also ml.


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

Principle Quantum Number 4.jpgfor l=2 what are the possible values of ml?