NDA Physics · Modern Physics

Atomic Structure: Models, Shells, and Energy

The atom is a tiny dense nucleus (protons + neutrons) surrounded by electrons in fixed energy shells; a sequence of experiments — cathode rays, Rutherford scattering, Bohr's orbits — revealed this picture.

All Modern Physics notes NDA Physics strategy

Why this matters

This is the densest recall block in the chapter — six PYQs, all EASY. The NDA wants you to match each landmark experiment to what it discovered (Rutherford found the nucleus; cathode rays are electrons from the cathode; Bohr proposed stable orbits) and to recall fixed numbers (M-shell holds 18 electrons; hydrogen's ionisation energy is 13.6 eV). These are near-free marks if you have the table memorised.

Concept 1 of 4

Cathode rays — the discovery of the electron

Intuition

In a discharge tube at low pressure, a glowing stream travels from the negative electrode (cathode) to the positive one (anode). These cathode rays turned out to be fast-moving electrons — the discovery that the atom contains negatively charged particles. The exam loves to test their properties, especially the direction (cathode to anode) and what they actually are.

Definition

Cathode rays are streams of electrons emitted from the cathode (negative electrode) in a discharge tube.

They travel from cathode to anode (negative to positive electrode).
They are negatively charged, travel in straight lines, and are deflected by electric and magnetic fields.
Their discovery (J. J. Thomson) revealed the electron.

Property of cathode rays	Correct statement
What they are	A stream of electronsQ
Direction of travel	From cathode to anode (negative to positive) NDA 2019 — the FALSE statement was "cathode ray particles start from the anode and move towards the cathode." They go cathode to anode.
Charge	Negative
Path	Straight line; deflected by electric and magnetic fields

The most common trap reverses the direction: cathode rays start at the CATHODE, not the anode.

Practice this conceptself-check · 4 quick reps

Try it yourself

Which statement about cathode rays is NOT correct: (i) they are electrons, (ii) they travel from anode to cathode, (iii) they are negatively charged?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Cathode rays are a stream of which particles?
2.
In which direction do cathode rays travel?
3.
What is the charge on cathode rays?
4.
Discovery of cathode rays revealed which subatomic particle?

From the bank · past-year question

Example 1Modern PhysicsEASY

Which one of the following statements regarding cathode rays is

\textbf{\text{not}}

correct?

[Q87 · Sep · 2019]

Cathode to anode, never anode to cathode

The single most-tested cathode-ray trap is the direction. Rays start at the CATHODE (the negative electrode) and move to the anode. Any option saying "from anode to cathode" is the false statement.

Concept 2 of 4

Rutherford's alpha-scattering — the nucleus

Intuition

Rutherford fired alpha particles at a thin gold foil. Most passed straight through, but a tiny fraction bounced almost straight back. The only way to explain the rare big deflections was a tiny, dense, positively charged centre — the nucleus. This overturned the earlier "plum-pudding" model where charge was spread out evenly.

Definition

Rutherford's alpha-particle scattering experiment discovered the atomic nucleus: a tiny, dense, positively charged core that contains almost all the atom's mass, with electrons around it.

Most alpha particles passed undeflected (atom is mostly empty space).
A few bounced back sharply (a concentrated positive nucleus).

Observation	Conclusion
Most alpha particles pass straight through	Atom is mostly empty space
A few alpha particles deflect at large angles / rebound	A tiny, dense, positively charged nucleus exists NDA 2021 — Rutherford's alpha-scattering experiment discovered the atomic NUCLEUS.
Almost all mass concentrated centrally	Nucleus holds the protons (and neutrons)

Rutherford = nucleus. Do not confuse with Chadwick (neutron) or Thomson (electron).

Practice this conceptself-check · 4 quick reps

Try it yourself

Rutherford's alpha-scattering experiment was responsible for the discovery of which part of the atom?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Rutherford's alpha-scattering experiment discovered what?
2.
Most alpha particles passing straight through showed the atom is mostly what?
3.
Who discovered the neutron (not Rutherford)?
4.
What charge does the nucleus carry?

From the bank · past-year question

Example 2Modern PhysicsEASY

Rutherford's alpha-particle

(\alpha)

scattering experiment was responsible for the discovery of which one of the following?

[Q120 · Sep · 2021]

Rutherford found the nucleus, not the neutron

Rutherford's scattering experiment discovered the NUCLEUS. The NEUTRON was discovered later by James Chadwick. Distractors swap these two.

Concept 3 of 4

Bohr's model — electrons in stable orbits without radiating

Intuition

Classical physics said an orbiting electron should continuously radiate energy and spiral into the nucleus — but atoms are stable. Niels Bohr fixed this by postulating that electrons can occupy only certain special orbits in which they do NOT radiate energy. An electron only emits or absorbs energy when it jumps between these allowed orbits.

Definition

Bohr's model of the atom postulates that:

Electrons revolve in certain stable (allowed) orbits without emitting radiation.
Each orbit has a fixed energy; energy is quantised.
Energy is emitted or absorbed only when an electron jumps between orbits (emits a photon dropping down, absorbs one going up).

Bohr energy levels of hydrogen

E_n = -\dfrac{13.6}{n^2}\ \text{eV}

E_nenergy of the n-th orbit (eV)
norbit number (1, 2, 3, ...)
13.6 eVmagnitude of the ground-state (n=1) energy of hydrogen

Energy is quantised: an electron can sit only on a rung, never between them. Ground state of hydrogen is -13.6 eV, so its ionisation energy is 13.6 eV.

Worked example

Which scientist proposed that an electron in an atom can revolve in certain stable orbits without emitting radiant energy?

Classical theory predicted a radiating, spiralling electron — but atoms are stable.
Niels Bohr resolved this by postulating special stable orbits in which the electron does not radiate.

Answer:Niels Bohr.

Practice this conceptself-check · 4 quick reps

Try it yourself

Using

E_n = -13.6/n^2

eV, find the energy of hydrogen's second orbit (n = 2).

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Who proposed electrons revolve in stable orbits without radiating?
2.
In Bohr's model, when does an electron emit a photon?
3.
Is energy in Bohr's atom continuous or quantised?
4.
Ground-state energy of the hydrogen atom?

From the bank · past-year question

Example 3Modern PhysicsEASY

Which one of the following scientists mentioned that an electron in an atom could revolve in certain stable orbits without the emission of radiant energy ?

[Q130 · Apr · 2024]

Bohr's electrons radiate only when they JUMP

An electron sitting in an allowed orbit does NOT radiate. Radiation (a photon) appears only during a transition between orbits — emitted when dropping down, absorbed when moving up.

Concept 4 of 4

Electron shells and ionisation energy — the fixed numbers

Intuition

Electrons fill shells labelled K, L, M, N (n = 1, 2, 3, 4). Each shell holds a maximum of

2n^2

electrons. Ionisation energy is the energy needed to pull the outermost electron completely free of the atom. For hydrogen this is exactly the depth of the ground state, 13.6 eV. These are pure recall facts the NDA repeats.

Definition

Key fixed numbers for atomic structure:

Maximum electrons in a shell = $2n^2$ : K (n=1) holds 2, L (n=2) holds 8, M (n=3) holds 18, N (n=4) holds 32.
Ionisation energy of hydrogen (ground state) = 13.6 eV — the energy to remove its single electron.
Energy stored in the bonds (links) between atoms is chemical energy.

Quantity	Value	How to get it
K-shell (n=1) capacity	2 electrons	$2n^2 = 2(1)^2 = 2$
L-shell (n=2) capacity	8 electrons	$2n^2 = 2(2)^2 = 8$
M-shell (n=3) capacity	18 electrons	$2n^2 = 2(3)^2 = 18$ NDA 2021 — the M-shell holds a maximum of 18 electrons.
N-shell (n=4) capacity	32 electrons	$2n^2 = 2(4)^2 = 32$
Hydrogen ionisation energy	13.6 eV	Depth of the n=1 ground stateQ
Energy in atomic bonds	Chemical energy	Stored in the links between atomsQ

Shell capacity is 2n². Hydrogen ionisation energy 13.6 eV is the single most-repeated number in the chapter.

Practice this conceptself-check · 4 quick reps

Try it yourself

What is the maximum number of electrons that the M-shell can hold?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Maximum electrons in the M-shell?
2.
Ionisation energy of the hydrogen atom in its ground state?
3.
Energy stored in the links/bonds between atoms is called what?
4.
Maximum electrons in the L-shell?

From the bank · past-year question

Example 4Modern PhysicsEASY

What is the maximum number of electrons in the M-Shell?

[Q121 · Sep · 2021]

Shell capacity is 2n², not a fixed 8

The "octet" of 8 is only the L-shell. The general rule is

2n^2

: M (n=3) holds 18, N (n=4) holds 32. Do not cap every shell at 8.

Hydrogen ionisation energy = 13.6 eV (positive energy IN)

The ground-state energy is

-13.6

eV; the ionisation energy (the energy you must supply to free the electron) is

+13.6

eV. The exam answer is 13.6 eV.

Drill 2 more on electron shells and ionisation energy — the fixed numbers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Bohr's model — electrons in stable orbits without radiating
Bohr energy levels of hydrogen
$E_n = -\dfrac{13.6}{n^2}\ \text{eV}$

Reference tables (3)

Cathode rays — the discovery of the electron4 rows

Property of cathode rays	Correct statement
What they are	A stream of electronsQ
Direction of travel	From cathode to anode (negative to positive) NDA 2019 — the FALSE statement was "cathode ray particles start from the anode and move towards the cathode." They go cathode to anode.
Charge	Negative
Path	Straight line; deflected by electric and magnetic fields

The most common trap reverses the direction: cathode rays start at the CATHODE, not the anode.

Rutherford's alpha-scattering — the nucleus3 rows

Observation	Conclusion
Most alpha particles pass straight through	Atom is mostly empty space
A few alpha particles deflect at large angles / rebound	A tiny, dense, positively charged nucleus exists NDA 2021 — Rutherford's alpha-scattering experiment discovered the atomic NUCLEUS.
Almost all mass concentrated centrally	Nucleus holds the protons (and neutrons)

Rutherford = nucleus. Do not confuse with Chadwick (neutron) or Thomson (electron).

Electron shells and ionisation energy — the fixed numbers6 rows

Quantity	Value	How to get it
K-shell (n=1) capacity	2 electrons	$2n^2 = 2(1)^2 = 2$
L-shell (n=2) capacity	8 electrons	$2n^2 = 2(2)^2 = 8$
M-shell (n=3) capacity	18 electrons	$2n^2 = 2(3)^2 = 18$ NDA 2021 — the M-shell holds a maximum of 18 electrons.
N-shell (n=4) capacity	32 electrons	$2n^2 = 2(4)^2 = 32$
Hydrogen ionisation energy	13.6 eV	Depth of the n=1 ground stateQ
Energy in atomic bonds	Chemical energy	Stored in the links between atomsQ

Shell capacity is 2n². Hydrogen ionisation energy 13.6 eV is the single most-repeated number in the chapter.

Watch out for (5)

Cathode to anode, never anode to cathode
→ Cathode rays — the discovery of the electron
Rutherford found the nucleus, not the neutron
→ Rutherford's alpha-scattering — the nucleus
Bohr's electrons radiate only when they JUMP
→ Bohr's model — electrons in stable orbits without radiating
Shell capacity is 2n², not a fixed 8
→ Electron shells and ionisation energy — the fixed numbers
Hydrogen ionisation energy = 13.6 eV (positive energy IN)
→ Electron shells and ionisation energy — the fixed numbers

Mastery check — 2 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Modern PhysicsEASY

Which one of the following energy is stored in the links between the atoms?

[Q57 · Apr · 2019]

Example 2Modern PhysicsEASY

The ionization energy of hydrogen atom in the ground state is

[Q79 · Sep · 2017]

Drill every past-year question on this subtopic

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