A Closer Look at Shells, Subshells, and also OrbitalsSubshellsProblems

A complete of 4 quantum numbers are used to explain completely the activity and trajectories of each electron within an atom. The combicountry of all quantum numbers of all electrons in an atom is explained by a wave feature that adheres to the Schrödinger equation. Each electron in an atom has a distinctive set of quantum numbers; according to the Pauli Exclusion Principle, no 2 electrons deserve to share the same combicountry of four quantum numbers. Quantum numbers are essential bereason they deserve to be provided to recognize the electron configuration of an atom and the probable area of the atom"s electrons. Quantum numbers are also supplied to understand other attributes of atoms, such as ionization energy and the atomic radius.

You are watching: Explain why the quantum number set (2, 1, -2, -½) is not possible for an electron in an atom.

In atoms, tright here are a complete of four quantum numbers: the principal quantum number (n), the orbital angular momentum quantum number (l), the magnetic quantum number (ml), and also the electron spin quantum number (ms). The primary quantum number, (n), defines the power of an electron and the the majority of probable distance of the electron from the nucleus. In other words, it refers to the size of the orbital and also the power level an electron is put in. The number of subshells, or (l), defines the form of the orbital. It deserve to additionally be used to recognize the number of angular nodes. The magnetic quantum number, ml, defines the power levels in a subshell, and ms refers to the spin on the electron, which can either be up or dvery own.


The Principal Quantum Number ((n))

The principal quantum number, (n), designates the primary electron shell. Since n defines the a lot of probable distance of the electrons from the nucleus, the bigger the number n is, the farther the electron is from the nucleus, the bigger the size of the orbital, and also the bigger the atom is. n can be any type of positive integer founding at 1, as (n=1) designates the initially principal shell (the innermany shell). The first principal shell is also referred to as the ground state, or lowest power state. This explains why (n) can not be 0 or any type of negative integer, bereason there exists no atoms with zero or a negative amount of power levels/primary shells. When an electron is in an excited state or it gains energy, it might jump to the second principle shell, where (n=2). This is dubbed absorption because the electron is "absorbing" pholoads, or power. Known as emission, electrons deserve to likewise "emit" energy as they jump to lower principle shells, wright here n decreases by whole numbers. As the energy of the electron increases, so does the primary quantum number, e.g., n = 3 indicates the 3rd major shell, n = 4 indicates the fourth primary shell, and also 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?

Answer

Emission, because energy is shed by release 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 circulation. The variety of angular nodes is equal to the worth of the angular momentum quantum number (l). (For even more information around angular nodes, view Electronic Orbitals.) Each value of (l) indicates a particular s, p, d, f subshell (each distinctive in form.) The worth of (l) is dependent on the major quantum number (n). Unlike (n), the value of (l) have the right to be zero. It deserve to likewise be a positive integer, but it cannot be larger than one less than the primary quantum number ((n-1)):


Example (PageIndex3)

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

Answer

Due to the fact that (l) can be zero or a positive integer much less than ((n-1)), it can have actually a worth of 0, 1, 2, 3, 4, 5 or 6.


Example (PageIndex4)

If (l = 4), how many kind of angular nodes does the atom have?

Answer

The number of angular nodes is equal to the value of l, so the variety 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 within a subshell. Consequently, its worth relies on the orbital angular momentum quantum number (l). Given a particular (l), (m_l) is an interval varying from (–l) to (+l), so it deserve to be zero, an unfavorable integer, or a positive integer.


Example (PageIndex5)

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

Answer

Due to the fact that (m_l) should array 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 spin quantum number (m_s) does not depfinish on an additional quantum number. It designates the direction of the electron spin and might have actually a spin of +1/2, stood for by↑, or –1/2, represented by ↓. This indicates that once (m_s) is positive the electron has actually an upward spin, which deserve to be referred to as "spin up." When it is negative, the electron has a downward spin, so it is "spin dvery own." The definition of the electron spin quantum number is its determicountry of an atom"s capacity to generate a magnetic field or not. (Electron Spin.)


Example (PageIndex5)

List the possible combicountries of all four quantum numbers as soon as (n=2), (l=1), and (m_l=0).

Answer

The fourth quantum number is independent of the first 3, allowing the initially 3 quantum numbers of 2 electrons to be the very same. Because the spin deserve to be +1/2 or =1/2, tbelow are 2 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 a downward spin?

Answer

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


A Closer Look at Shells, Subshells, and Orbitals


Principal Shells

The worth of the primary quantum number n is the level of the major digital shell (principal level). All orbitals that have actually the very same n worth are in the exact same principal level. For example, all orbitals on the second major level have actually a principal quantum number of n=2. When the worth of n is higher, the number of primary digital shells is greater. This reasons a greater distance between the farthest electron and also the nucleus. As a result, the dimension of the atom and also its atomic radius rises.

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Since the atomic radius increases, the electrons are farther from the nucleus. Therefore it is easier for the atom to expel an electron bereason the nucleus does not have actually as solid a pull on it, and also the ionization energy decreases.



Subshells

The number of worths of the orbital angular number l can likewise be offered to determine the number of subshells in a principal electron shell:

When n = 1, l= 0 (l takes on one value and also thus there deserve to just be one subshell) When n = 2, l= 0, 1 (l takes on two values and also therefore tbelow are 2 possible subshells) When n = 3, l= 0, 1, 2 (l takes on 3 worths and therefore there are 3 feasible subshells)

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

Principal shell via n = 1 has one subshell Principal shell via n = 2 has two subshells Principal shell with n = 3 has three subshells

To recognize what kind of possible subshells n has, these 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

Therefore:

Principal shell via n = 1 has one s subshell (l = 0) Principal shell through n = 2 has one s subshell and one p subshell (l = 0, 1) Principal shell with n = 3 has actually one s subshell, one p subshell, and also one d subshell (l = 0, 1, 2)

We have the right to designate a primary quantum number, n, and a details subshell by combining the worth of n and also the name of the subshell (which have the right to be discovered making use of l). For instance, 3p refers to the third primary quantum number (n=3) and the p subshell (l=1).

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Orbitals

The variety of orbitals in a subshell is tantamount to the number of values the magnetic quantum number ml takes on. A beneficial equation to determine the number of orbitals in a subshell is 2l +1. This equation will not offer you the value of ml, however the number of feasible values that ml can take on in a specific orbital. For example, if l=1 and ml deserve to have actually values -1, 0, or +1, the value of 2l+1 will certainly be three and also tbelow will be three different orbitals. The names of the orbitals are called after the subshells they are uncovered 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 number below, we check out examples of 2 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 section of a jawbreaker around the circle. The layers are portraying the atoms angular nodes. To image a 3s orbital, imagine one more layer approximately the circle, and also so on and also so on. The p orbital is equivalent to the form of a dumbbell, through its orientation within a subshell depending on ml. The shape and orientation of an orbital counts on l and ml.

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To visualize and also organize the initially three quantum numbers, we can think of them as constituents of a home. In the complying with photo, the roof represents the primary quantum number n, each level represents a subshell l, and each room represents the different orbitals ml in each subshell. The s orbital, bereason the worth of ml deserve to just be 0, can only exist in one plane. The p orbital, yet, has actually three possible worths of ml and also so it has three feasible orientations of the orbitals, presented by Px, Py, and Pz. The pattern continues, via the d orbital containing 5 possible orbital orientations, and f has actually 7:

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Principle Quantum Number 4.jpgexplain why the quantum number set (2 1 -2 -½) is not possible for an electron in an atom.