Atomic Structure — Bohr's Model, Quantum Numbers for Class 11

What Is an Atom, Really?

Every bit of matter around you — your desk, the air, the ink on your textbook — is made of atoms. But for about 2000 years, the atom was just a philosophical idea. The real story of atomic structure begins when scientists started probing the atom with experiments, and the results kept breaking their models.

That’s the honest framing for this chapter: each model — Thomson’s, Rutherford’s, Bohr’s — was the best explanation available at the time. Each got replaced when new data arrived. Understanding why each model failed is just as important as knowing what the next one proposed. JEE and NEET questions regularly exploit the gaps between models.

By the end of this guide, we’ll work through quantum numbers, orbitals, and electronic configurations — the tools that actually let us predict how elements behave chemically.

Key Terms and Definitions

Atomic number (Z): Number of protons in the nucleus. Defines the element. Carbon is always Z = 6.

Mass number (A): Protons + neutrons. The integer closest to an element’s atomic mass.

Isotopes: Same Z, different A. $^{12}C$ and $^{14}C$ are both carbon — same chemistry, different nuclear mass.

Electromagnetic radiation: Energy that travels as oscillating electric and magnetic fields. Characterized by wavelength $\lambda$ (nm or m) and frequency $\nu$ (Hz).

Photon: A quantum (packet) of light energy. Energy of one photon = $h\nu$, where $h$ is Planck’s constant.

Spectrum: The distribution of radiation by wavelength. An emission spectrum shows bright lines; an absorption spectrum shows dark lines on a continuous background.

Orbital: A region of space where the probability of finding an electron is high (typically ≥90%). Not a fixed path — this distinction from Bohr’s “orbit” is tested directly.

The Models: From Pudding to Planets to Probability

Thomson’s Plum Pudding Model (1904)

Thomson discovered the electron in 1897. His model: a positively charged sphere with electrons embedded like plums in a pudding. Electrons could vibrate and emit radiation.

This model correctly explained that atoms are electrically neutral. It failed completely when Rutherford fired alpha particles at gold foil.

Rutherford’s Nuclear Model (1911)

The Geiger-Marsden experiment scattered alpha particles ($\alpha$, which are $^4_2He^{2+}$ nuclei) at a thin gold foil. Most passed straight through. A small fraction deflected at large angles. A few bounced nearly straight back.

Rutherford’s conclusion:

• The atom is mostly empty space.
• Almost all the mass and all positive charge is concentrated in a tiny nucleus.
• Electrons orbit the nucleus at a distance.

Why Rutherford’s model failed: Classical electrodynamics says a charged particle moving in a circle continuously radiates energy. An orbiting electron should spiral into the nucleus in about $10^{-8}$ seconds. Clearly, atoms are stable. Also, continuous radiation would produce a continuous spectrum — but hydrogen shows only discrete spectral lines. Rutherford’s model couldn’t explain either fact.

Bohr’s Model (1913)

Niels Bohr fixed Rutherford’s model by introducing two postulates:

1. Electrons orbit the nucleus only in certain allowed orbits (shells), where they do not radiate energy. These are called stationary states.
2. An electron emits or absorbs radiation only when it jumps between orbits. The energy of the photon equals the energy difference:

Allowed orbit condition: Angular momentum is quantized.

Energy of the $n$-th orbit in hydrogen:

Radius of the $n$-th orbit:

Velocity of electron in $n$-th orbit:

Spectral series of hydrogen: When electrons fall to a particular shell, a specific series results:

Limitations of Bohr’s model: It works only for hydrogen and hydrogen-like species (one electron). It fails for multi-electron atoms, can’t explain the splitting of spectral lines in magnetic fields (Zeeman effect), and contradicts Heisenberg’s uncertainty principle by assigning both definite radius and definite velocity to an electron.

Quantum Mechanical Model and Quantum Numbers

The modern model treats electrons as waves (de Broglie, 1924; Schrödinger, 1926). Instead of fixed orbits, we have orbitals — probability distributions described by wave functions $\psi$.

Four quantum numbers completely describe every electron in an atom.

Principal Quantum Number ($n$)

• Values: $1, 2, 3, \ldots$ (positive integers)
• Determines the energy and size of the orbital (shell).
• Higher $n$ → higher energy → larger orbital → electron farther from nucleus.
• Maximum electrons in shell $n$: $2n^2$

Azimuthal (Angular Momentum) Quantum Number ($l$)

• Values: $0, 1, 2, \ldots, (n-1)$
• Determines the shape of the orbital (subshell).

• Number of orbitals in a subshell: $2l + 1$

Magnetic Quantum Number ($m_l$)

• Values: $-l, \ldots, 0, \ldots, +l$ (total: $2l+1$ values)
• Determines the orientation of the orbital in space.
• A $p$ subshell ($l=1$) has $m_l = -1, 0, +1$ → three $p$ orbitals: $p_x, p_y, p_z$.

Spin Quantum Number ($m_s$)

• Values: $+\frac{1}{2}$ or $-\frac{1}{2}$ only (spin up or spin down).
• Intrinsic property of the electron — not derivable from classical physics.

Electronic Configuration: Filling Rules

Three rules determine how electrons fill orbitals.

Aufbau Principle: Fill lower energy orbitals first. The order follows increasing $(n + l)$ value. When $(n+l)$ is equal for two subshells, the one with lower $n$ fills first.

Energy order: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d \ldots$

Pauli Exclusion Principle: No two electrons in an atom can have all four quantum numbers identical. Consequence: maximum two electrons per orbital, and they must have opposite spins.

Hund’s Rule of Maximum Multiplicity: When filling orbitals of equal energy (degenerate orbitals), electrons occupy separate orbitals with parallel spins first. Only after each orbital has one electron does pairing begin.

This is why nitrogen ($1s^2 2s^2 2p^3$) has three unpaired electrons — one in each $2p$ orbital — making it paramagnetic.

Exceptions (JEE loves these):

Chromium (Z=24): Expected $[Ar] 3d^4 4s^2$. Actual: $[Ar] 3d^5 4s^1$

Copper (Z=29): Expected $[Ar] 3d^9 4s^2$. Actual: $[Ar] 3d^{10} 4s^1$

The reason: half-filled ($d^5$) and completely filled ($d^{10}$) configurations are extra stable due to symmetry and exchange energy.

Solved Examples

Example 1 — CBSE Level

Calculate the wavelength of radiation emitted when an electron in hydrogen jumps from $n=3$ to $n=2$.

Using the Rydberg formula:

Here $n_1 = 2$ (lower level, electron lands here), $n_2 = 3$:

This is the red line of the Balmer series — the most famous spectral line in hydrogen. It’s visible to the naked eye.

Example 2 — JEE Main Level

An electron in Li$^{2+}$ is in the second excited state. Find its energy and radius.

Second excited state means $n = 3$ (ground state is $n=1$, first excited is $n=2$, second excited is $n=3$).

For hydrogen-like species with $Z=3$:

Interesting: Li$^{2+}$ in $n=3$ has the same energy as hydrogen in $n=1$. This is not a coincidence — the condition is $Z/n =$ constant.

Example 3 — JEE Advanced Level

Write the quantum numbers for the 19th electron in the ground state of potassium (Z=19). Also determine whether this electron is in the same shell as the 18th electron.

Potassium configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^1$

The 19th electron is the single $4s$ electron.

Its quantum numbers: $n=4$, $l=0$, $m_l=0$, $m_s=+\frac{1}{2}$ (by convention)

The 18th electron completes the $3p$ subshell: $n=3$, $l=1$, $m_l=+1$, $m_s=+\frac{1}{2}$ (or $-\frac{1}{2}$)

The 19th electron is in $n=4$; the 18th is in $n=3$. Different shells — despite $4s$ filling before $3d$, it belongs to the fourth shell.

Exam-Specific Tips

CBSE Class 11

• Chapter 2 carries consistent weightage in board exams. Expect 1-2 questions in 1-mark section (define quantum numbers, state Pauli’s principle), one 3-mark derivation (Bohr radius or energy), and one 5-mark question (electronic configuration + magnetic properties).
• For the 5-marker, practice writing configurations in both orbital notation (box diagrams) and spdf notation.
• The Balmer series wavelength calculation and the ionisation energy of hydrogen ($13.6$ eV) are high-frequency board questions.

JEE Main

• Atomic structure has appeared in 1-2 questions every session for the past three years.
• Focus areas: spectral series identification, Bohr model calculations for hydrogen-like species, electronic configuration exceptions (Cr, Cu), and quantum number sets.
• 2024 trend: questions on the number of radial/angular nodes (radial nodes = $n - l - 1$; angular nodes = $l$; total nodes = $n - 1$).

NEET

• NEET tests more conceptual questions here — “which set of quantum numbers is not possible?”, “which element has maximum unpaired electrons?”, paramagnetic vs diamagnetic.
• Electronic configuration of transition metals in ionic form is a reliable NEET area.

Common Mistakes to Avoid

Practice Questions

Q1. How many radial nodes and angular nodes does a $3d$ orbital have?

Q2. Which of the following quantum number sets is NOT allowed: $(2, 1, 0, +\frac{1}{2})$, $(3, 3, 0, -\frac{1}{2})$, $(4, 2, -1, +\frac{1}{2})$?

Q3. Calculate the energy required to remove the electron from $He^+$ in its ground state. (Given: $E_1$ for H = $-13.6$ eV)

Q4. Write the electronic configuration of $Fe^{2+}$ (Z=26) and state whether it is paramagnetic or diamagnetic.

Q5. The wavelength of first line of Lyman series is $\lambda$. Find the wavelength of the second line.

Q6. How many spectral lines are possible when electrons from $n=5$ fall to $n=2$ in hydrogen?

Q7. Which element has the configuration $[Kr] 4d^{10} 5s^1$ and why?

Q8. An electron transitions from $n=4$ to $n=2$ in hydrogen. In what region of the electromagnetic spectrum does this radiation fall?

FAQs

Why does Bohr’s model work for hydrogen but not helium?

Bohr’s model treats each electron independently — it has no way to handle electron-electron repulsion. In helium, the two electrons repel each other, and this repulsion significantly affects energy levels. The quantum mechanical approach (Schrödinger equation) handles multi-electron systems through approximation methods. Even for helium, exact analytical solutions don’t exist.

What exactly is an orbital? Is it a path?

No. An orbital is a mathematical function ($\psi$) that describes the quantum state of an electron. $|\psi|^2$ gives the probability density of finding the electron at a given point. Chemists draw the boundary surface that encloses 90% of this probability — that’s the “shape” of an orbital. There is no path.

Why is the energy negative in Bohr’s formula?

By convention, an electron at infinity (completely separated from the nucleus) has zero energy. An electron bound to the atom has lower energy than infinity, so it’s negative. The more negative the energy, the more tightly bound the electron. For hydrogen, $E_1 = -13.6$ eV means you need to supply $+13.6$ eV to ionise it from the ground state.

What is the difference between a subshell and an orbital?

A subshell is a set of orbitals with the same $n$ and $l$. The $2p$ subshell (n=2, l=1) contains three orbitals ($2p_x$, $2p_y$, $2p_z$, corresponding to $m_l = -1, 0, +1$). Each orbital holds at most two electrons. So the $2p$ subshell holds up to 6 electrons.

Can two electrons in an atom have the same first three quantum numbers?

Yes — but they must differ in $m_s$. If $n$, $l$, $m_l$ are identical, the electrons are in the same orbital. Pauli’s exclusion principle then requires opposite spins: one with $m_s = +\frac{1}{2}$ and the other with $m_s = -\frac{1}{2}$.

Why does the $4s$ orbital fill before $3d$?

For neutral atoms with $Z$ around 19–20, the $4s$ orbital (with $n+l=4$) has slightly lower energy than $3d$ ($n+l=5$) due to penetration effects — the $4s$ electron has greater probability of being found close to the nucleus. This extra nuclear attraction lowers $4s$ energy below $3d$ in these atoms.

How many electrons can the $f$ subshell hold?

The $f$ subshell has $l=3$, giving $2l+1 = 7$ orbitals, each holding 2 electrons. Total: 14 electrons. This is why the $f$-block elements (lanthanides and actinides) span 14 elements in the periodic table.