Quantum numbers are fundamental concepts in quantum mechanics that play a crucial role in understanding the behavior of electrons in atoms. For students preparing for competitive exams such as JEE and NEET, a solid grasp of quantum numbers is essential for mastering topics related to atomic structure and electron configurations.
In this blog post, we will explore the different types of quantum numbers, their significance, and how they relate to various elements in the periodic table.
By solving the Schrödinger equation (HΨ = EΨ), we obtain a set of mathematical equations, called wave functions (Ψ), which describe the probability of finding electrons at certain energy levels within an atom.
A wave function for an electron in an atom is called an atomic orbital. This atomic orbital describes a region of space in which there is a high probability of finding the electron. Energy changes within an atom are the result of an electron changing from a wave pattern with one energy to a wave pattern with a different energy (usually accompanied by the absorption or emission of a
photon of light).
The set of numbers used to describe the position and energy of the electron in an atom are called quantum numbers.
Each electron in an atom is described by four different quantum numbers. The first three (n, l, ml) specify the particular orbital of interest, and the fourth (ms) specifies how many electrons can occupy that orbital.
Electronic quantum numbers (the quantum numbers describing electrons) can be defined as a group of numerical values which provide solutions that are acceptable by the Schrodinger wave equation for hydrogen atoms.
Types of quantum numbers
There are 4 types of quantum numbers, which are as follows:
- Principal quantum number
- Azimuthal quantum number
- Magnetic quantum number
- Spin quantum number
Representation and possible values
| Quantum number | Representation | Possible values |
| Principal Quantum Number | n | 1, 2, 3, 4…. |
| Azimuthal Quantum Number | l | 0 to (n-1) = 0,1,2,….(n-1) |
| Magnetic Quantum Number | m or ml | -l to +l = -l,…-1,0,+1,…,+l |
| Spin Quantum Number | s or ms | -1/2 or +1/2 |
Principal quantum number
- Principal quantum numbers are denoted by the symbol ‘n’. They designate the principal electron shell of the atom. Since the most probable distance between the nucleus and the electrons is described by it, a larger value of the principal quantum number implies a greater distance between the electron and the nucleus (which, in turn, implies a greater atomic size).
- It specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot).
- The value of the principal quantum number can be any integer with a positive value that is equal to or greater than one. The value n=1 denotes the innermost electron shell of an atom, which corresponds to the lowest energy state (or the ground state) of an electron.
- Thus, it can be understood that the principal quantum number, n, cannot have a negative value or be equal to zero because it is not possible for an atom to have a negative value or no value for a principal shell.
- When a given electron is infused with energy (excited state), it can be observed that the electron jumps from one principle shell to a higher shell, causing an increase in the value of n. Similarly, when electrons lose energy, they jump back into lower shells and the value of n also decreases.
- The increase in the value of n for an electron is called absorption, emphasizing the photons or energy being absorbed by the electron. Similarly, the decrease in the value of n for an electron is called emission, where the electrons emit their energy.
- The total number of orbitals for a given n value is n2.
Azimuthal quantum number
- The azimuthal (or orbital angular momentum) quantum number describes the shape of a given orbital. It is denoted by the symbol ‘l’ and its value is equal to the total number of angular nodes in the orbital.
- It specifies the shape of an orbital with a particular principal quantum number.
- A value of the azimuthal quantum number can indicate either an s, p, d, or f subshell which vary in shapes. This value depends on (and is capped by) the value of the principal quantum number, i.e. the value of the azimuthal quantum number ranges between 0 and (n-1).
- For example, if n =3, the azimuthal quantum number can take on the following values, i.e., 0,1, and 2. When l=0, the resulting subshell is an ‘s’ subshell. Similarly, when l=1 and l=2, the resulting subshells are ‘p’ and ‘d’ subshells (respectively). Therefore, when n=3, the three possible subshells are 3s, 3p, and 3d.
- In another example where the value of n is 5, the possible values of l are 0, 1, 2, 3, and 4. If l = 3, then there are a total of three angular nodes in the atom.
- The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with value of l, i.e., s<p<d<f.
| Value of l | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Letter | s | p | d | f | g | h | i |
Magnetic Quantum number
- This quantum number describes the behaviour of electron in a magnetic field.
- It specifies the orientation in space of an orbital of a given energy (n) and shape (l).
- This number divides the subshell into individual orbitals which hold the electrons. There are (2l+1) orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on.
- This quantum number gives the number of orbitals in a given subshell. Its value is from -l to +l.
- If the value of l is given as 2, then the value of m is from -2 to +2, i.e., -2, -1, 0, +1, +2. It indicates that the number of orbitals in d-subshell are 5 called d-orbitals.
Spin Quantum Number
- The electron spin quantum number is independent of the values of n, l, and ml. The value of this number gives insight into the direction in which the electron is spinning, and is denoted by the symbol ms.
- It specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down).
- The Pauli exclusion principle states that no two electrons in the same atom can have identical values for all four of their quantum numbers. What this means is that no more than two electrons can occupy the same orbital, and that two electrons in the same orbital must have opposite spins.
- Because an electron spins, it creates a magnetic field, which can be oriented in one of two directions. For two electrons in the same orbital, the spins must be opposite to each other; the spins are said to be paired. These substances are not attracted to magnets and are said to be diamagnetic. Atoms with more electrons that spin in one direction than another contain unpaired electrons. These substances are weakly attracted to magnets and are said to be paramagnetic.
- The value of ms offers insight into the direction in which the electron is spinning. The possible values of the electron spin quantum number are +½ and -½.
- The positive value of ms implies an upward spin on the electron which is also called ‘spin up’ and is denoted by the symbol ↑. If ms has a negative value, the electron in question is said to have a downward spin, or a ‘spin down’, which is given by the symbol ↓.
- The value of the electron spin quantum number determines whether the atom in question has the ability to produce a magnetic field. The value of ms can be generalized to ±½.
Table: Allowed Quantum Numbers
| n | l | m | No. of Orbitals | Orbital Name | No. of Electrons |
|---|---|---|---|---|---|
| 1 | 0 | 0 | 1 | 1s | 2 |
| 2 | 0 1 | 0 -1, 0, +1 | 1 3 | 2s 2p | 2 6 |
| 3 | 0 1 2 | 0 -1, 0, +1 -2, -1, 0, +1, +2 | 1 3 5 | 3s 3p 3d | 2 6 10 |
| 4 | 0 1 2 3 | 0 -1, 0, +1 -2, -1, 0, +1, +2 -3, -2, -1, 0, +1, +2, +3 | 1 3 5 7 | 4s 4p 4d 4f | 2 6 10 14 |
| 5 | 0 1 2 3 4 | 0 -1, 0, +1 -2, -1, 0, +1, +2 -3, -2, -1, 0, +1, +2, +3 -4, -3, -2, -1, 0, +1, +2, +3, +4 | 1 3 5 7 9 | 5s 5p 5d 5f 5g | 2 6 10 14 18 |
Shapes of orbital
Some Important Questions
Question: In a multi electron atom which of the following orbital described by the three quantum number will have the same energy in the absence of magnetic and electric fields.
1. n=1, l=0, m=0
2. n=2, l=0, m=0
3. n=2, l=1, m=1
4. n=3, l=2, m=0
5. n=3, l=2, m=1
(a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5
Solution: We know, l=0 represents s-subshell, l=1 represents p-subshell, l=2 represents d-subshell and l=3 represents f-subshell.
For energy, consider the Aufbau’s principle. Greater the n+l value, greater the energy. If n+l value is same, then greater the n value, greater than energy.
Considering only n and l values, so
1. 1s-subshell
2. 2s-subshell
3. 2p-subshell
4. 3d-subshell
5. 3d-subshell
Points 4 and 5 have same energy because both belongs to same subshell, i.e., 3d.
Hence, option (d) is correct.
Question: What are the Possible Subshells when n = 4? How Many Orbitals are Contained by Each of these Subshells?
Solution: When n = 4, the possible l values are 0, 1, 2, and 3. This implies that the 4 possible subshells are the 4s, 4p, 4d, and 4f subshells.
- The 4s subshell contains 1 orbital and can hold up to 2 electrons.
- The 4p subshell contains 3 orbitals and can hold up to 6 electrons.
- The 4d subshell contains 5 orbitals and can hold up to 10 electrons.
- The 4f subshell has 7 orbitals and can hold up to 14 electrons.
Thus, a total of 4 subshells are possible for n = 4.
Question: What are the Possible ml values for l = 4?
Solution: Since the value of the magnetic quantum number ranges from -l to l, the possible values of ml when l = 4 are: -4, -3, -2, -1, 0, 1, 2, 3, and 4.
Question: Which energy level has the least energy?
Solution: There is a single 1s orbital that can accommodate 2 electrons at the lowest energy level, the one nearest to the atomic core. There are four orbitals at the next energy level; a 2s, 2p1, 2p2 and a 2p3. Each of these orbitals can carry 2 electrons, so we can find a total of 8 electrons at this energy level.
Quantum Number Assignment
1. The principle quantum number represents:
(a) Shape of an orbital (b) Number of electrons in an orbit
(c) Distance of an electron from the nucleus (d) Orientation of orbitals in space
2. Which set of quantum numbers is not consistent with the quantum mechanical theory? Give reason.
(a) n = 2, l = 1, m = 1, s = +1/2 (b) n = 4, l = 3, m = 2, s = -1/2
(c) n = 3, l = 2, m = 3, s = +1/2 (d) n = 4, l = 3, m = 3, s = +1/2
3. Which of the following sets of quantum numbers is/are incorrect? Give reason.
(a) n = 3, l = 3, m = 0, s = +1/2 (b) n = 3, l = 2, m = 2, s = -1/2
(c) n = 3, l = 1, m = 2, s = -1/2 (d) n = 3, l = 0, m = 0, s = +1/2
4. What is the subshell designation for the following?
(a) n = 2, l = 1
(b) n = 3, l = 1
(c) n = 3, l = 2
(d) n = 1, l = 1; Is the designation correct? If not mention reason for the same.
(e) n = 4, l = 3
(f) n = 4, l = 2
5. What subshells are possible in n = 4 energy levels?
6. What are the possible values of m, when the value for l is 3?
7. The maximum number of orbitals which can be present in a subshell can be represented by:
(a) 2n + 1 (b) 2l + 1 (c) 2m + 3 (d) 4l + 2
8. Give all possible value of l, m and s for the N-shell.
9. How many orbitals can have the following quantum numbers, n = 3, l = 1, m = 0?
(a) 4 (b) 2 (c) 1 (d) 3
10. What is the maximum number of electrons, which can have following quantum numbers, n = 3, l = 1, m = -1?
(a) 2 (b) 6 (c) 10 (d) 4
11. The maximum number of electrons which can be present in a subshell can be represented by:
(a) 2l + 1 (b) 2n2 (c) 4l + 2 (d) 4l – 2
12. Which of the following quantum numbers governs the spatial orientation of an atomic orbital?
(a) Magnetic quantum number (b) Spin quantum number
(c) Azimuthal quantum number (d) Principal quantum number
13. The maximum number of electrons with l = 3 is:
(a) 14 (b) 2 (c) 10 (d) 6
14. What are the possible ml values for l = 4?
15. Principal, azimuthal and magnetic quantum numbers are respectively related to:
(a) Size, shape and orientation (b) Shape, size and orientation
(c) Size, orientation and shape (d) None of the above
16. Two electrons occupying the same orbital, can be distinguished by:
(a) Principal quantum number (b) Azimuthal quantum number
(c) Magnetic quantum number (d) Spin quantum number
17. Is the given set of quantum number permissible? If no, why?
n = 4, l = 3, m = -2, s = +1
18. How many orbital can fit in the orbital for which n = 3 and l = 1?
(a) 2 (b) 6 (c) 14 (d) 10
19. How many orbital can fit in the orbital for which n = 3, l = 1, m = -1?
(a) 2 (b) 6 (c) 14 (d) 10
20. What is the orbital angular momentum for value of l = 3?
21. What is the spin angular momentum for the value of s = ½?
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