JEE Mains Chemistry · Structure of Atom

Bohr Model: Radius, Energy and Velocity

In a one-electron atom or ion, Bohr's orbits have fixed radii, energies and speeds, and each scales with the orbit number n and the nuclear charge Z.

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Why this matters

Twenty-five PYQs, nineteen of them multiple choice, and five from 2026 — the largest page in the chapter. Seven scale an orbit radius with n²/Z; thirteen use the orbit energy, its kinetic and potential parts, the speed or the ionisation energy; five test what the Thomson, Rutherford and Bohr models could and could not explain. Three ideas cover the page.

Concept 1 of 3: Orbit radius and its scaling

Bohr fixed the electron's angular momentum at whole multiples of

h/2\pi

. That allows only certain orbits. Their radius grows as

n^2

, because outer orbits are much farther out. It shrinks as

Z

grows, because a larger nuclear charge pulls the electron in. Only one-electron species obey it: H,

\mathrm{He^+}

\mathrm{Li^{2+}}

\mathrm{Be^{3+}}

Definition

$mvr=\dfrac{nh}{2\pi}$ , $n=1,2,3,\ldots$
$r_n=a_0\dfrac{n^2}{Z}$ , with $a_0=52.9\ \mathrm{pm}=0.529\ \mathrm{\mathring{A}}$ .
Ratio of two orbits: $\dfrac{r_1}{r_2}=\dfrac{n_1^2/Z_1}{n_2^2/Z_2}$ .
Ground state is $n=1$ ; the first excited state is $n=2$ .

Bohr radius

r_n=52.9\,\frac{n^2}{Z}\ \mathrm{pm}

Worked example

An orbit of

\mathrm{He^+}

has radius 238.05 pm. Which orbit is it?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 2 · Q27Moderate

Example 1 · Structure of Atom · Bohr Model: Radius, Energy and Velocity

The Bohr radius of a hydrogen like species is 70.53 pm . The species and the stationary state(n) are respectively (Given: Hydrogen atom Bohr radius is 52.9 pm )

The radius goes as n², not n

The sixth orbit of H is

\dfrac{36}{16}=2.25

times the fourth, not

\dfrac{6}{4}

times. Square the orbit numbers before you divide.

First excited state is n = 2

"First excited" is one step above the ground state, so

n=2

. Using

n=1

n=3

is the common slip.

Drill 6 more on orbit radius and its scaling

Concept 2 of 3: Orbit energy, kinetic and potential energy, and speed

A bound electron has negative total energy. Zero is the free electron at rest far away. The energy grows with

Z^2

and falls with

n^2

, so inner orbits of heavier ions are much deeper. Kinetic energy is the size of the total energy, and potential energy is twice the total. The speed grows with

Z

and falls with

n

Definition

$E_n=-2.18\times10^{-18}\dfrac{Z^2}{n^2}\ \mathrm{J}=-13.6\dfrac{Z^2}{n^2}\ \mathrm{eV}$ .
$KE=-E_n$ , $PE=2E_n$ .
Ionisation energy from the ground state: $13.6Z^2\ \mathrm{eV}$ .
Energy to move between levels: $\Delta E=13.6Z^2\left(\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\right)\ \mathrm{eV}$ .
$v_n=2.18\times10^{6}\dfrac{Z}{n}\ \mathrm{m\,s^{-1}}$ .
Other scalings: frequency of revolution $\propto Z^2/n^3$ ; Coulomb force $\propto Z^3/n^4$ .

Bohr energy

E_n=-13.6\,\frac{Z^2}{n^2}\ \mathrm{eV}

Worked example

Find the energy needed to excite the electron of

\mathrm{He^+}

from

n=1

n=3

, in J and in eV.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 23 Jan 2026 Shift 1 · Q26Moderate

Example 2 · Structure of Atom · Bohr Model: Radius, Energy and Velocity

Which of the following statements regarding the energy of the stationary state is true in the following one-electron system ?

Every orbit energy is negative

A bound electron always has

E_n<0

. An option with a plus sign on an orbit energy is wrong before you do any arithmetic. Kinetic energy is the only positive one.

KE is minus E, PE is twice E

KE=-E_n

and

PE=2E_n

. So for

E=-3.4\ \mathrm{eV}

KE=+3.4\ \mathrm{eV}

and

PE=-6.8\ \mathrm{eV}

Drill 12 more on orbit energy, kinetic and potential energy, and speed

Concept 3 of 3: Thomson, Rutherford, Bohr and the quantum model

Each model fixed what the last one could not explain. Thomson spread the positive charge out. Rutherford's scattering put it in a tiny nucleus. Bohr's fixed orbits explained hydrogen's lines. The quantum model kept Bohr's stationary states and photons, but dropped the definite circular path.

Definition

Thomson: positive charge spread evenly, electrons embedded in it. It predicts only small deflections of $\alpha$ -particles.
Rutherford: a tiny, dense, positive nucleus. It explains large-angle scattering. It cannot explain why the atom is stable or why its spectrum has lines.
Bohr: fixed orbits with $mvr=nh/2\pi$ . It works only for one-electron species. It fails for the Zeeman (magnetic) and Stark (electric) splitting of lines, and a definite orbit breaks the uncertainty principle.
Quantum model: the electron is a wave, described by an orbital (a probability region). It keeps stationary states and $\Delta E=h\nu$ .

Model	Picture of the atom	What it explained	Where it failed
Thomson	Uniform sphere of positive charge with electrons embedded	The atom is neutral overall	Predicts only small deflections, so cannot explain large-angle scatteringQ If Thomson were right, α-particles would pass through gold foil with only small deflections.
Rutherford	Tiny dense positive nucleus with electrons around it	Large-angle scattering; a few α-particles bounce back	An orbiting electron should radiate and spiral in; no line spectrum
Bohr	Electrons in fixed circular orbits, mvr = nh/2π	Stability and line spectrum of H, He⁺, Li²⁺	Many-electron atoms (even Li⁺), Zeeman and Stark splitting; a definite path breaks the uncertainty principleQ Li⁺ has two electrons, so Bohr's theory does not apply to it. Li²⁺ has one.
Quantum mechanical	Electron as a wave; orbitals are regions of probability ψ²	All atoms, including many-electron ones	Keeps stationary states and ΔE = hν, but drops the definite orbit

The one Bohr postulate the quantum model rejects: the electron moves in a definite circular orbit.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 3 Apr 2025 · Q26Moderate

Example 3 · Structure of Atom · Bohr Model: Radius, Energy and Velocity

Which of the following postulate of Bohr's model of hydrogen atom is not in agreement with quantum mechanical model of an atom?

Li⁺ is not a one-electron ion

Count the electrons:

\mathrm{Li^{+}}

has two. Only

\mathrm{H}

\mathrm{He^+}

\mathrm{Li^{2+}}

\mathrm{Be^{3+}}

and so on are hydrogen-like.

Drill 4 more on thomson, rutherford, bohr and the quantum model

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 (2)

Orbit radius and its scaling
Bohr radius
$r_n=52.9\,\frac{n^2}{Z}\ \mathrm{pm}$
Orbit energy, kinetic and potential energy, and speed
Bohr energy
$E_n=-13.6\,\frac{Z^2}{n^2}\ \mathrm{eV}$

Reference tables (1)

Thomson, Rutherford, Bohr and the quantum model4 rows

Model	Picture of the atom	What it explained	Where it failed
Thomson	Uniform sphere of positive charge with electrons embedded	The atom is neutral overall	Predicts only small deflections, so cannot explain large-angle scatteringQ If Thomson were right, α-particles would pass through gold foil with only small deflections.
Rutherford	Tiny dense positive nucleus with electrons around it	Large-angle scattering; a few α-particles bounce back	An orbiting electron should radiate and spiral in; no line spectrum
Bohr	Electrons in fixed circular orbits, mvr = nh/2π	Stability and line spectrum of H, He⁺, Li²⁺	Many-electron atoms (even Li⁺), Zeeman and Stark splitting; a definite path breaks the uncertainty principleQ Li⁺ has two electrons, so Bohr's theory does not apply to it. Li²⁺ has one.
Quantum mechanical	Electron as a wave; orbitals are regions of probability ψ²	All atoms, including many-electron ones	Keeps stationary states and ΔE = hν, but drops the definite orbit

The one Bohr postulate the quantum model rejects: the electron moves in a definite circular orbit.

Watch out for (5)

The radius goes as n², not n
→ Orbit radius and its scaling
First excited state is n = 2
→ Orbit radius and its scaling
Every orbit energy is negative
→ Orbit energy, kinetic and potential energy, and speed
KE is minus E, PE is twice E
→ Orbit energy, kinetic and potential energy, and speed
Li⁺ is not a one-electron ion
→ Thomson, Rutherford, Bohr and the quantum model

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