Q.How are the quantum numbers n, l and m_l arrived at? Explain the significance of these quantum numbers.
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Quantum Numbers
Why do we need quantum numbers?
Imagine trying to describe exactly where a student sits in a giant stadium. You'd need more than just "row 5" — you'd need the section, the row, the seat number, and maybe even which side of the aisle. An electron in an atom faces a similar problem. The atom is a tiny, crowded space, and each electron needs a unique "address" so that no two electrons occupy exactly the same spot (this is the Pauli exclusion principle, which we'll meet properly later).
Classical physics would let you pin down an electron's position and velocity exactly. Quantum mechanics says: no. Instead, we describe an electron's state using four numbers — the quantum numbers. They don't give you a precise orbit like a planet; they give you the electron's energy, its "shape" of probability, its orientation in space, and its intrinsic spin.
The four quantum numbers
1. Principal quantum number (n) — the shell
This is the simplest. n tells you the energy level and roughly how far the electron is from the nucleus. It can be any positive integer: n=1,2,3,…
- n=1 is the lowest energy, closest to the nucleus.
- As n increases, the electron has more energy and is, on average, farther away.
In the Bohr model (a useful first picture), n directly gives the orbit radius. In the full quantum model, n still determines the main energy, but the electron's position is spread out in a "cloud" — the orbital.
Example: For hydrogen, the ground state has n=1. The first excited state has n=2.
2. Azimuthal quantum number (l) — the subshell shape
For a given n, the electron can have different shapes of its probability cloud. l tells you which shape. It ranges from 0 to n−1.
- l=0: s orbital — spherical, like a fuzzy ball around the nucleus.
- l=1: p orbital — dumbbell-shaped, two lobes on opposite sides.
- l=2: d orbital — more complex, often four-lobed (clover shape).
- l=3: f orbital — even more intricate.
The letters s, p, d, f come from old spectroscopic terms: sharp, principal, diffuse, fundamental. You only need to remember the order.
Example: For n=2, l can be 0 or 1. So the second shell has a 2s subshell (spherical) and a 2p subshell (dumbbell). The 2p electrons have slightly higher energy than the 2s.
3. Magnetic quantum number (ml) — the orientation
If you have a dumbbell (a p orbital), which way does it point? Along the x-axis? y-axis? z-axis? ml tells you the orientation of the orbital in space. It ranges from −l to +l, including zero.
- For l=0 (s orbital): only ml=0 — one orientation (spherical, so no direction matters).
- For l=1 (p orbital): ml=−1,0,+1 — three orientations. These correspond to the px, py, and pz orbitals.
- For l=2 (d orbital): ml=−2,−1,0,+1,+2 — five orientations.
The names px, py, pz are just labels. In a free atom, all three p orbitals have the same energy. It's only when you apply a magnetic field (or put the atom in a molecule) that they become different.
4. Spin quantum number (ms) — the electron's spin
Even after you've fixed the orbital (shape and orientation), the electron itself has an intrinsic property called spin. It's not literally spinning like a top, but it behaves as if it has a tiny magnetic moment that can point in one of two directions.
ms can only be +21 or −21.
- Often called "spin up" and "spin down".
- Two electrons can share the same orbital (same n,l,ml) only if they have opposite spins.
This is the Pauli exclusion principle: No two electrons in an atom can have the same set of all four quantum numbers. That's why each orbital holds at most two electrons — one with spin up, one with spin down.
Putting it all together: the address
Let's take a concrete example: the electron configuration of carbon (Z=6). The first two electrons fill the 1s orbital:
- Electron 1: n=1,l=0,ml=0,ms=+21
- Electron 2: n=1,l=0,ml=0,ms=−21
Next two go into 2s:
- Electron 3: n=2,l=0,ml=0,ms=+21 …
These three quantum numbers are not chosen arbitrarily — they emerge as the mathematical requirement for a physically valid solution when the Schrödinger wave equation is solved for an atom, and each one describes a different physical property of the resulting orbital. …
Solving Schrödinger's wave equation for an atom requires three quantum numbers — n, l, ml — to fully specify each allowed orbital; each carries distinct physical significance about the orbital's size, shape, and orientation.
Origin: when Schrödinger's wave equation is solved for an atom (e.g. hydrogen), it does not have arbitrary solutions — only certain specific wavefunctions (ψ, called atomic orbitals) are mathematically permitted. Each permitted solution is uniquely labelled by a set of three quantum numbers that emerge naturally from the boundary conditions of the equation: the principal (n), azimuthal (l), and magnetic (ml) quantum numbers.
Principal quantum number (n):
- Takes positive integer values: n=1,2,3,…
- Determines the size and energy of the orbital/shell — larger n means the electron is on average farther from the nucleus and has higher energy.
- For hydrogen-like species, energy En∝−n21.
- The maximum number of electrons a shell of principal number n can hold is 2n2.
Azimuthal (angular momentum / subsidiary) quantum number (l):
- For a given n, l can take integer values from 0 to n−1.
- Determines the shape of the orbital and identifies the sub-shell: l=0,1,2,3 correspond to s,p,d,f sub-shells respectively.
- Determines the orbital angular momentum: L=l(l+1)ℏ.
Magnetic quantum number (ml):
- For a given l, ml can take integer values from −l to +l (including 0), giving 2l+1 possible values. …
Showing the 12 most recent of 34 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Which of the following sets of quantum number is not valid?(a) n = 5, l = 2, m = 0, s = -1/2(b) n = 1, l = 2, m = 0, s = +1/2(c) n = 5, l = 3, m = 2, s = +1/2(d) None of these
›Reveal solutionSolution
The rule to check is: for principal quantum number n, l can only take integer values from 0 to (n-1); m (magnetic quantum number) ranges from -l to +l; s (spin) is always +1/2 or -1/2.
Check each option against l <= n - 1:
- n = 5, l = 2, m = 0, s = -1/2: for n = 5, allowed l = 0,1,2,3,4. l = 2 is allowed. m ranges -2 to +2, so m = 0 is allowed. Valid.
- n = 1, l = 2, m = 0, s = +1/2: for n = 1, the ONLY allowed value of l is 0 (since l can be 0 to n-1 = 0). l = 2 is not permitted for n = 1. This set is INVALID. …
- CBSE 2026Set ANNUAL1 markMCQQ.In 7d orbitals, 7 denotes -(a) Principal quantum number(b) Azimuthal quantum number(c) Magnetic quantum number(d) Spin quantum number
›Reveal solutionSolution
In the orbital label 7d, the numeral '7' is the principal quantum number (n); the letter 'd' encodes the azimuthal quantum number (l = 2).
Atomic orbitals are labelled using a standard convention: (principal quantum number)(subshell letter), e.g. 1s, 2p, 3d, 7d. The principal quantum number n describes the shell (main energy level) and can be any positive integer 1, 2, 3, ... The subshell letters s, p, d, f correspond to azimuthal quantum number l = 0, 1, 2, 3 respectively, describing the shape of the orbital. The magnetic quantum number (m) then specifies orientation within a subshell, and the spin quantum number (s) …
- CBSE 2026Set ANNUAL1 markQ.For n = 1, the value of l is ______.
›Reveal solutionSolution
When n = 1, the only allowed value of the azimuthal quantum number l is 0.
The azimuthal quantum number l determines the shape of an orbital/subshell and is restricted by the rule l = 0, 1, 2, ..., (n-1) for a given shell n. For n = 1, the allowed values of l range from 0 to (1-1) = 0, …
- CBSE 2026Set ANNUAL1 markMCQQ.An orbital in which n = 4, l = 2 is expressed as :(a) 4p(b) 4d(c) 4f(d) 3d
›Reveal solutionSolution
An orbital with n = 4 and l = 2 is a 4d orbital.
Every orbital is labelled by its principal quantum number (n), which gives the shell, followed by a letter denoting the subshell fixed by the azimuthal quantum number (l): l = 0 → s, l = 1 → p, l = 2 → d, l = 3 → f. Here n = 4 fixes the shell number written first, and l = 2 fixes the subshell letter 'd'. Combining them gives the orbital designation 4d. (Note that option (d), …
- CBSE 2026Set ANNUAL1 markMCQQ.Match the correct pair: Principal quantum number(a) XeF4(b) Stock notation(c) Lassaigne's test(d) Size of orbital(e) Exothermic reaction
›Reveal solutionSolution
The principal quantum number (n) determines the size (and energy) of an orbital — larger n means a larger, higher-energy orbital.
The principal quantum number n (n = 1, 2, 3, ...) specifies the main energy shell an electron occupies. As n increases, the average distance of the electron from the nucleus increases, so the orbital becomes larger, and its energy also increases. This is why the …
- CBSE 2026Set ANNUAL1 markMCQQ.If the number of an orbit is expressed by n, then the total number of orbitals in it is(a) n(b) 2n(c) n^2(d) 2n^2
›Reveal solutionSolution
Number of orbitals in the nth shell = n^2.
Each shell n has subshells with l = 0 to (n−1). The number of orbitals in a subshell is (2l+1). Summing over all subshells gives 1 + 3 + 5 + ... = n^2 orbi …
- CBSE 2025Set ANNUAL1 markMCQQ.Which of the following is not possible?(a) n = 3, l = 0, m = 0(b) n = 3, l = 0, m = -1(c) n = 2, l = 0, m = -1(d) n = 2, l = 1, m = 0
›Reveal solutionSolution
When l = 0 (an s-orbital), m can only be 0 — any set with l = 0 and m ≠ 0 is not a physically allowed combination.
Rule: for a given l, m may only take the (2l+1) integer values from -l to +l. So if l = 0, the only allowed value of m is 0.
Checking the options:
- (a) n=3, l=0, m=0 — allowed (m=0 is the only value when l=0).
- (b) n=3, l=0, m=-1 — l=0 forces m=0, so m=-1 is not allowed here. …
- CBSE 2025Set ANNUAL1 markMCQQ.Maximum number of electrons in 4d orbital is(a) 14(b) 10(c) 8(d) 6
›Reveal solutionSolution
A d subshell always has 5 orbitals; with 2 electrons per orbital (Pauli exclusion), the maximum electron capacity of any d subshell (3d, 4d, 5d...) is 10.
For a given subshell with azimuthal quantum number l, the number of orbitals is (2l + 1), and the maximum electrons that subshell can hold is 2(2l + 1).
For a d subshell, l = 2, so:
Number of orbitals = 2(2) + 1 = 5
Maximum electrons = 2 x 5 = 10
…
- CBSE 2025Set ANNUAL1 markMCQQ.Which of the following set of quantum numbers is not possible? (columns: n, l, m, s)(a) n=4, l=3, m=2, s=-1/2(b) n=3, l=0, m=0, s=+1/2(c) n=2, l=2, m=-1, s=+1/2(d) n=2, l=1, m=-1, s=-1/2
›Reveal solutionSolution
The quantum number rules are: l = 0 to (n-1), m = -l to +l, and s = +1/2 or -1/2. The set n=2, l=2 violates l <= n-1, so it is impossible.
Check each option against the rules governing quantum numbers:
- n (principal quantum number) can be any positive integer.
- l (azimuthal/angular momentum quantum number) can take values from 0 to (n-1).
- m (magnetic quantum number) can take values from -l to +l.
- s (spin quantum number) is always +1/2 or -1/2.
- n=4, l=3, m=2, s=-1/2: for n=4, l can be 0,1,2,3 — l=3 is allowed (4f); m ranges -3 to +3, so m=2 is allowed. Valid.
- n=3, l=0, m=0, s=+1/2: for n=3, l=0 is allowed (3s); m must be 0 when l=0. Valid. …
- CBSE 2025Set hz1 markMCQQ.Select the correct one: The number of radial nodes in 3S orbital are:(a) 2(b) 0(c) 1(d) 3
›Reveal solutionSolution
Radial nodes = n - l - 1; for 3s, n=3 and l=0, giving 2 radial nodes.
A radial node is a spherical surface around the nucleus where the probability of finding the electron is zero. The number of radial nodes in any atomic orbital is given by:
Radial nodes = n - l - 1
For the 3s orbital: principal quantum number n = 3, azimuthal quantum number l = 0 (s-orbital).
Radial nodes = 3 - 0 - 1 = 2.
…
- CBSE 2025Set ANNUAL1 markMCQQ.Principal quantum number shows:(a) Orbit(b) Orbital(c) Shape of orbital(d) Orientation of electron
›Reveal solutionSolution
The principal quantum number n denotes the shell (orbit) number and the electron's distance from the nucleus.
Quantum numbers describe the state of an electron in an atom:
- Principal quantum number (n) = shell/orbit, decides size and energy.
- Azimuthal quantum number (l) = sub-shell, decides shape of orbital.
- Magnetic quantum number (m) = orientation of orbital.
- Spin quantum number (s) = spin of electron. …
- CBSE 2024Set ANNUAL1 markMCQQ.2p orbitals have the values(a) n = 2, l = 0(b) n = 2, l = 1(c) n = 1, l = 2(d) n = 1, l = 0
›Reveal solutionSolution
The orbital name '2p' directly encodes the quantum numbers: the digit is n, and the letter p corresponds to azimuthal quantum number l = 1.
For any orbital labelled n(letter):
- The number is the principal quantum number, n.
- The letter denotes the sub-shell via the azimuthal quantum number l, where s -> l = 0, p -> l = 1, d -> l = 2, f -> l = 3.
For 2p: n = 2 (second shell) and the letter 'p' means l = 1.
…
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