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The magnetic quantum number is one of the three quantum numbers. The magnetic quantum number was proposed by Arnold Somerfield to explain the Zeeman and Stark effects. In Spectroscopy, the splitting of spectral lines under the influence of a strong magnetic field is called the Zeeman effect. The splitting of the spectral line under the influence of a strong electric field is called the Stark effect.
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Keyterms: Quantum number, Atoms, Protons, Electron, Neutron, Magnetic field, Electric fiels, Spectroscopy, Stark effect, Zeeman effect, Energy
What are Quantum Numbers?
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Quantum numbers define the trajectory and movement of an electron within an atom. Besides this, the quantum numbers of every electron in an atom are found to be combined; it should obey the Schrodinger equation, a mathematical expression that describes the energy and position of the electron in space and time.
There are four numbers to describe the energy state of electrons:
- Principal Quantum Number: Expresses the energy levels. It is denoted by “n”
- Azimuthal Quantum Number: Also called Angular Momentum Quantum Number. It describes the subshell. It is denoted by “?”
(Shell is a grouping of electrons in an atom according to energies; A subshell is a pathway of electrons moves within the shell; Orbitals are the regions of space where electrons like to exist in an atom.)
- Magnetic Quantum Number: Expresses the orbital of a subshell. It is denoted by m? or m
- Spin Quantum Number: Expresses the spin. It is denoted by “ms” or “s”
Moreover, every electron has a set of four unique quantum numbers. According to Pauli’s exclusion principle, no two electrons are found to have the same combination of quantum numbers.
The video below explains this:
Magnetic Quantum Number Detailed Video Explanation:
Principal Quantum Number
This principal quantum number mainly gives the electron shell or energy level of an atom. Here, the value of ‘n’ starts from 1 and gradually increases to the shell containing the outermost electron of a particular atom.
Azimuthal Quantum Number
Also known as Angular or orbital quantum number, the Azimuthal Quantum number describes the electron’s unique quantum state. It can be defined as, the quantum number associated with the angular momentum of an atomic electron.
Spin Quantum Number
The spin quantum number describes the spin or intrinsic angular momentum of an electron within an orbital. It also allows a projection of the spin angular momentum (s) along a particular axis.
Let us now learn about Magnetic Quantum Number in detail.
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Magnetic Quantum Number: Definition
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It divides the sub-shells (named as s,p,d,f) into specific orbitals and positions the electron in one of them. It also defines the orientation of electrons in space of a given orbital of particular energy (n) and shape (I).
In each subshell, the number of orbitals is given by 2l+1, where l is the Azimuthal quantum number.
The value of the Magnetic quantum number ranges from – to + including0, as it depends on the Azimuthal Quantum number.
For instance, s subshell has only 1 orbital, the p subshell has 3 and d has 5, and so on.
How to find out Magnetic Quantum Number?
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Quantum numbers in general, especially the magnetic quantum number, can be derived while solving the Schrodinger equation.
However, limits need to be put in place to match the equations with the physical world. The magnetic quantum number can be derived from solving the Azimuthal equation of the Hydrogen Schrodinger equation.
What does Magnetic Quantum Number find out?
The magnetic quantum number primarily finds out the number of orbitals and the orientation of orbitals in a given subshell of an electron. Consequently, it relies on the orbital angular azimuthal quantum number, also known as the angular momentum quantum number.
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Things to Remember
- We use Quantum numbers to define the trajectory and movement of an electron within an atom.
- Quantum Number is a value that we use to describe the energy that is present in atoms and molecules.
- The four quantum numbers are – Principal quantum number, Azimuthal quantum number (also called angular/orbital quantum number), Magnetic quantum number, and Spin quantum number.
- Magnetic quantum number divides the sub-shells (named as s,p,d,f) into specific orbitals and positions the electron in one of them. It also defines the orientation of electrons in space of a given orbital of particular energy (n) and shape (I).
- The value of the Magnetic quantum number ranges from – to + including 0, as it depends on the Azimuthal Quantum number.
- The following is the table summarizing the quantum numbers, along with symbols.
| Name and Symbol | Meaning and Possible Values |
|---|---|
| Principal quantum number, n | Electron shell, n ≥ 1 |
| Azimuthal quantum number, l | Subshells (s=0, p=1, etc.) , (n-1) ≥ l ≥ 0 |
| Magnetic quantum number, ml | Total number and orientation of orbitals, l≥ml≥-l |
| Electron spin quantum number, ms | The direction of electron spin, ms = ±½ |
Sample Questions
Ques. An electron in an atom is in the n = 3 and ℓ = 1 quantum state. Identify the possible values of mℓ that it can have. (3 Marks)
Ans. 1) We recall the rule for identifying possible mℓ values: ranging from negative to positive values with 0 in between.
2) With ℓ = 1, we have the following mℓ values:
−1, 0, +1 [2(ℓ)+1= 2(1)+1=3 values]
Ques. Which of the following combinations of quantum numbers are allowed for an electron in a one-electron atom? (3 Marks)
(a) n = 4, ℓ = 2, mℓ = −1, ms = −1⁄2
(b) n = 6, ℓ = 2, mℓ = 1, ms = +1⁄2
(c) n = 1, ℓ = −1, mℓ = −2, ms = +1⁄2
(d) n = 6, ℓ = 0, mℓ = 1, ms = +1⁄2
Ans. 1) (a) and (b) are allowed. Let's look at why (c) and (d) are not allowed. First (c):
Rationale #1: ℓ values start at zero and go by integers up to n − 1. When n = 1, the only possible ℓ value is zero. A value of −1 is not allowed in this example.
Rationale #2: ℓ values cannot be negative. Ever.
2) Let's have a look at (d):
The n, ℓ combination of 6, 0 is allowed. mℓ values range from −ℓ to +ℓ. When ℓ = 0, the only possible mℓ value is 0. Since mℓ is incorrect, (d) is the set that is not allowed.
Ques. In an atom, which has 2K, 8L,18M, and 2N electrons in the ground state. What is the total number of electrons having magnetic quantum number, m = 0 ? (4 Marks)
Ans. Given are 2K, 8L,18M and 2N electrons in the ground state which means,
2K = 1s2, 8L = 2s2 2p6, 18M = 3s2 3p6 3d10, 2N = 4s2
For, s, p and d, m=0 with spins +1/2 i.e., 2 electrons
Hence we now find the electrons for each orbital,
2K = 1s2 = 2 electrons
8L = 2s2 2p6 = 4 electrons
18M = 3s2 3p6 3d10 = 6 electrons
2N = 4s2 = 2 electrons
Total = 14 electrons
Therefore. the total number of electrons having magnetic quantum number, m = 0 is 14
Ques. For the quantum number ℓ values below, how many possible values are there for the quantum number mℓ ? (4 Marks)
(a) 5; (b) 3; (c) 2; (d) 1
Ans. The rule for m? is that, the values of ℓ spans from negative to positive with 0 in between.
We use this formula to find m? values for a given ℓ : 2ℓ + 1.
(a) For ℓ = 5, the mℓ values −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, a total of 11 values.
(b) 7 values of mℓ resulting from 2(3) + 1 = 7
(c) 2ℓ + 1 gives 5 values. The enumeration is −2, −1, 0, 1, 2.
(d) 3 values (−1, 0, 1) or 2(1) + 1 = 3
Ques. Identify the subshell/shell that each of the following sets. (5 Marks)
(a) n = 2, ℓ = 1, mℓ = 1, ms = +1⁄2
(b) n = 3, ℓ = 2, mℓ = 2, ms = +1⁄2
(c) n = 4, ℓ = 1, mℓ = −1, ms = −1⁄2
(d) n = 4, ℓ = 3, mℓ = 3, ms = −1⁄2
(e) n = 5, ℓ = 0, mℓ = 0, ms = +1⁄2
Ans: Finding out the solution involves looking at the n, ℓ pairings. The mℓ and ms have no role in answering this question.
The values of n tell us the first part of the answer. The values of ℓ tell us the second half of the answer. Follow this guide:
ℓ → 0 1 2 3 4
Subshell → s p d f g
(a) 2, 1 ---> 2p
(b) 3, 2 ---> 3d
(c) 4, 1 ---> 4p
(d) 4, 3 ---> 4f
(e) 5, 0 ---> 5s
Ques. An orbital has n = 4 and mℓ = −1. What are the possible values of ℓ for this orbital? (5 Marks)
Ans. All possible values of ℓ range from 0 to n − 1 by integers, so: 0, 1, 2, 3 for all possible ℓ values.
We look for cases when mℓ = −1
When ℓ = 0, then mℓ = 0. [2(ℓ)+1= 2(0)+1=1 value]
If ℓ = 1, then mℓ = −1, because the range of mℓ values is from −ℓ to 0 to +ℓ by integers. So, with ℓ = 1, we have −1, 0, +1 for our mℓ values .
[2(ℓ)+1= 2(1)+1=3 values]
When ℓ = 2, we have mℓ values of −2, −1, 0 +1, +2. [2(ℓ)+1= 2(2)+1=5 values]
The third ℓ is 3. It generates m? values of −3, −2, −1, 0, +1, +2, +3
[2(ℓ)+1= 2(3)+1=7 values]
So, our answer is 1, 2, 3 for the ℓ values. The only one eliminated is 0.
Ques. What are the possible values of n and mℓ for an electron in a 5d orbital? Write the n, ℓ, mℓ for each of the orbitals in the 5d subshell. (5 Marks)
Ans. 1) Find out the value for n:
The use of a 5d subshell provides the answer.
The 5 in 5d is the value of n, which the question wants.
Other examples:
The value of n in the 4p orbital is 4.
The value for n in the 2s orbital is 2.
2) Find out the values for mℓ:
We need to find out what ℓ value belongs to d orbitals.
ℓ=2 (s orbitals have ℓ = 0, p orbitals have ℓ = 1, d has ℓ = 2 and f has an ℓ of 3.)
We now apply the rule for m? and find out the mℓ values for the 5d orbital:
−2, −1, 0, 1, 2 [2(ℓ)+1= 2(2)+1=5 values]
3) The five n, ℓ, mℓ sets are as follows:
- 5, 2, −2
- 5, 2, −1
- 5, 2, 0
- 5, 2, 1
- 5, 2, 2
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