Bohr's Atomic Model

In 1913, Danish physicist Niels Bohr proposed a new model of the atom that combined Rutherford's nuclear model with the emerging idea of quantized energy.

Rutherford had shown that atoms contain a tiny, positively charged nucleus surrounded by electrons. However, classical physics could not explain why those electrons remained stable or why atoms produced distinctive line spectra.

Bohr proposed a major new idea:

Electrons can occupy only certain allowed energy states.

These discrete states are called:

quantized energy levels

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From Rutherford to Bohr

Rutherford's model described the atom as:

• a tiny positive nucleus
• electrons surrounding the nucleus
• mostly empty space

This successfully explained alpha-particle scattering.

However, Rutherford's model had a serious problem.

According to classical electromagnetism, an electron moving in a circular path is accelerating.

An accelerating charged particle should emit electromagnetic radiation.

Therefore, classical physics predicted:

electron radiates energy → electron loses energy → orbit shrinks → electron falls into nucleus

Real atoms do not behave this way.

Bohr's model was designed partly to solve this:

atomic stability problem

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Bohr's Main Ideas

Bohr proposed several important ideas.

Electrons occupy allowed states

An electron cannot have just any energy.

It can occupy only certain:

discrete energy levels

Stable states do not continuously radiate

An electron remaining in an allowed state does not continuously lose energy as predicted by the classical model.

Electrons can change energy levels

An electron can move from one allowed state to another.

Energy is exchanged during transitions

When an electron changes energy level, electromagnetic radiation can be:

• absorbed
• emitted

The photon energy equals the difference between the two atomic energy levels.

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Quantized Orbits

In the original Bohr model, electrons were pictured moving in specific allowed circular orbits around the nucleus.

Each orbit corresponded to a particular:

energy

The allowed states were labelled using the principal quantum number:

n = 1, 2, 3, 4, ...

The lowest energy state is:

n = 1

This is called the:

ground state

Higher values of n represent higher-energy states.

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Energy Levels Are Discrete

Imagine a staircase.

You can stand on:

• step 1
• step 2
• step 3

but not halfway between two steps.

Bohr suggested that atomic energy behaves similarly.

An electron can occupy:

E₁, E₂, E₃, E₄ ...

but cannot have an arbitrary energy between allowed levels.

This is what is meant by:

quantization

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Energy Levels Are Not Equally Spaced

The allowed energy levels become closer together as n increases.

For hydrogen:

Eₙ = −13.6/n² eV

where:

• Eₙ = energy of level n
• n = principal quantum number
• eV = electron volt

For example:

Notice that the energy levels become closer together as n increases.

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Why the Energies Are Negative

The zero-energy reference represents an electron completely separated from the atom.

A bound electron has:

negative energy

Energy must be added to remove it from the atom.

At:

E = 0

the electron is no longer bound.

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Ground State

The lowest possible energy level is:

n = 1

For hydrogen:

E₁ = −13.6 eV

This is the atom's:

ground state

The electron cannot move to a lower energy state because no lower allowed state exists.

This helps explain why the hydrogen atom is stable.

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Excited States

If an electron absorbs energy, it can move to a higher energy level.

This produces an:

excited state

For example:

n = 1 → n = 2

or:

n = 2 → n = 4

The electron does not gradually move through every possible intermediate energy.

It changes from one allowed energy state to another.

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Absorption

For an electron to move to a higher energy level, it must absorb exactly the required energy.

For example:

lower energy → absorb photon → higher energy

The required photon energy is:

E_photon = E_high − E_low

or:

ΔE = hf

where:

• ΔE = difference between energy levels
• h = Planck's constant
• f = photon frequency

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Emission

An electron in an excited state can move to a lower energy level.

When it does:

higher energy → lower energy + photon

The energy of the emitted photon is:

E_photon = E_high − E_low

Therefore:

ΔE = hf

The atom cannot emit arbitrary photon energies because only certain energy differences are available.

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Photon Energy

The energy of a photon is:

E = hf

where:

• E = photon energy
• h = 6.63 × 10⁻³⁴ J·s
• f = frequency

Because:

c = fλ

photon energy can also be written:

E = hc/λ

Therefore:

larger energy transition → higher frequency → shorter wavelength

and:

smaller energy transition → lower frequency → longer wavelength

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Worked Example: Energy Between Levels

For hydrogen:

n = 1:

E₁ = −13.6 eV

n = 2:

E₂ = −3.40 eV

How much energy must be absorbed for the transition:

n = 1 → n = 2?

Calculate:

ΔE = E₂ − E₁

ΔE = −3.40 − (−13.6)

ΔE = 10.2 eV

Therefore, the atom must absorb a photon with energy:

10.2 eV

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Worked Example: Emission

Suppose an electron falls from:

n = 3 → n = 2

For hydrogen:

E₃ = −1.51 eV

E₂ = −3.40 eV

Energy released:

ΔE = E₃ − E₂

ΔE = −1.51 − (−3.40)

ΔE = 1.89 eV

A photon with energy approximately:

1.89 eV

is emitted.

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Converting Electron Volts to Joules

Atomic energies are often measured in:

electron volts (eV)

The conversion is:

1 eV = 1.60 × 10⁻¹⁹ J

For the previous transition:

E = 1.89 eV

E = 1.89 × 1.60 × 10⁻¹⁹

E ≈ 3.02 × 10⁻¹⁹ J

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Finding the Photon Frequency

Using:

E = hf

Rearrange:

f = E/h

For:

E = 3.02 × 10⁻¹⁹ J

f = (3.02 × 10⁻¹⁹)/(6.63 × 10⁻³⁴)

f ≈ 4.56 × 10¹⁴ Hz

This frequency corresponds to visible light.

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Finding the Wavelength

Use:

c = fλ

Therefore:

λ = c/f

λ = (3.00 × 10⁸)/(4.56 × 10¹⁴)

λ ≈ 6.58 × 10⁻⁷ m

or:

658 nm

This is approximately the red hydrogen spectral line associated with the n = 3 → n = 2 transition.

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Atomic Spectra

One of Bohr's greatest successes was explaining the:

line spectrum of hydrogen

When hydrogen gas is excited, it emits light at specific wavelengths rather than producing every possible wavelength.

The visible hydrogen spectrum contains several characteristic lines.

These lines occur because hydrogen electrons can make only certain:

energy transitions

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Why the Spectrum Is Not Continuous

Suppose atomic energy could take any value.

Electrons could then lose any amount of energy.

The emitted photons could have almost any energy.

We would expect a:

continuous spectrum

Instead, experiments show:

specific spectral lines

Bohr explained this using quantized energy levels.

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From Energy Levels to Spectral Lines

The process is:

electron begins in higher level

↓

electron moves to lower level

↓

specific energy difference released

↓

photon emitted with E = hf

↓

specific frequency

↓

specific wavelength

↓

spectral line

Therefore:

each spectral line corresponds to a particular electron transition.

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The Hydrogen Spectrum

Different groups of hydrogen spectral lines correspond to transitions ending at different energy levels.

Important series include:

Lyman series

Transitions ending at:

n = 1

These are mainly in the:

ultraviolet

Balmer series

Transitions ending at:

n = 2

Several are in the:

visible spectrum

Paschen series

Transitions ending at:

n = 3

These are mainly in the:

infrared

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The Balmer Series

The visible lines of hydrogen include transitions such as:

n = 3 → n = 2

n = 4 → n = 2

n = 5 → n = 2

n = 6 → n = 2

Each transition releases a different amount of energy.

Therefore each produces a different:

wavelength

This creates the distinctive hydrogen spectrum.

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Absorption Spectra

Atoms can also absorb photons.

If a photon has exactly the right energy to move an electron to a higher allowed state:

photon absorbed → electron moves upward

For example:

n = 2 → n = 3

requires exactly the same energy as is emitted during:

n = 3 → n = 2

Therefore atoms absorb and emit characteristic wavelengths.

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Emission vs Absorption

Emission

Electron:

high → low

Photon:

released

Absorption

Electron:

low → high

Photon:

absorbed

In both cases:

photon energy = difference between allowed energy levels

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Ionization

If enough energy is supplied, an electron can be completely removed from the atom.

This is:

ionization

For hydrogen in its ground state:

E₁ = −13.6 eV

The free-electron level is:

E = 0 eV

Therefore, the minimum energy needed to ionize ground-state hydrogen is:

13.6 eV

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Worked Example: Ionization

How much energy is needed to remove the electron from ground-state hydrogen?

Initial energy:

−13.6 eV

Final energy:

0 eV

Therefore:

ΔE = 0 − (−13.6)

ΔE = 13.6 eV

This is the:

ionization energy

of hydrogen from its ground state.

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Angular Momentum Quantization

Bohr also proposed that electron angular momentum could take only certain values.

In the Bohr model:

L = nħ

where:

• L = electron angular momentum
• n = 1, 2, 3, ...
• ħ = h/(2π)

This restriction helped determine the allowed electron orbits.

It was a major departure from classical mechanics, in which orbital properties could vary continuously.

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Radius of the Hydrogen Atom

Bohr's model also predicts specific allowed orbital radii.

For hydrogen:

rₙ = n²a₀

where:

a₀ ≈ 5.29 × 10⁻¹¹ m

is the:

Bohr radius

For n = 1:

r₁ = 5.29 × 10⁻¹¹ m

For n = 2:

r₂ = 4a₀

For n = 3:

r₃ = 9a₀

Thus higher-energy states correspond, in the Bohr picture, to larger electron orbits.

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Strengths of Bohr's Model

Bohr's model represented a major advance in atomic physics.

One major strength was that it explained:

atomic stability

Electrons could remain in allowed states without continuously radiating energy.

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Explaining Hydrogen's Spectrum

Another major strength was its successful explanation of:

hydrogen's line spectrum

The model correctly connected:

energy levels → electron transitions → photon energies → spectral wavelengths

This was strong evidence that atomic energies are quantized.

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Predicting Hydrogen Energies

Bohr's model correctly predicts the hydrogen energy levels:

Eₙ = −13.6/n² eV

This allowed physicists to calculate:

• transition energies
• photon frequencies
• photon wavelengths
• ionization energy

Its quantitative success for hydrogen was remarkable.

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Explaining the Rydberg Formula

Before Bohr, the wavelengths of hydrogen spectral lines could be described experimentally using mathematical relationships such as the:

Rydberg formula

Bohr's model provided a physical explanation for why this pattern existed.

For hydrogen:

1/λ = R(1/n_f² − 1/n_i²)

where:

• R = Rydberg constant
• n_i = initial higher level
• n_f = final lower level

This connected experimental spectroscopy with atomic structure.

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Hydrogen-Like Ions

Bohr's model also works reasonably well for systems containing only one electron.

Examples include:

• hydrogen, H
• singly ionized helium, He⁺
• doubly ionized lithium, Li²⁺

These are called:

hydrogen-like ions

The model becomes much less successful when several electrons interact with one another.

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Bohr's Model Was Not the Final Model

Despite its successes, Bohr's model had important limitations.

Its picture of electrons travelling around the nucleus in precise circular paths is not the modern description of atomic structure.

Modern quantum mechanics describes electrons using:

wavefunctions and probability distributions

rather than simple planetary-style orbits.

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Limitation: Multi-Electron Atoms

Bohr's model works best for:

hydrogen

and other one-electron systems.

It does not accurately predict the spectra of atoms containing many electrons.

In multi-electron atoms, electrons interact with:

• the nucleus
• other electrons

These interactions are much more complicated than Bohr's simple model allows.

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Limitation: Electron-Electron Interactions

Electrons repel one another electrically.

Bohr's simple model does not adequately describe these:

electron-electron interactions

As more electrons are added to an atom, their behaviour cannot be explained simply by assigning independent circular orbits.

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Limitation: Detailed Spectral Structure

With improved instruments, scientists discovered that some spectral lines actually consist of several closely spaced lines.

This is called:

fine structure

Bohr's original model could not fully explain these details.

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Limitation: Magnetic Fields

When atoms are placed in magnetic fields, some spectral lines split.

This is associated with the:

Zeeman effect

The original Bohr model could not adequately explain all of these observations.

Additional quantum ideas were required.

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Limitation: Electric Fields

Spectral lines can also split when atoms are placed in external electric fields.

This is called the:

Stark effect

Again, the simple Bohr model was insufficient.

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Limitation: Fixed Electron Paths

Perhaps the most important conceptual limitation is the idea of a precisely defined electron orbit.

Modern quantum mechanics does not assign an electron a simple circular path around the nucleus.

Instead, it describes the probability of detecting an electron in different regions of space.

These regions are associated with:

atomic orbitals

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Orbit vs Orbital

These terms are different.

Orbit

In the Bohr model:

a defined path followed by an electron

Orbital

In quantum mechanics:

a quantum state associated with a probability distribution for an electron

An orbital is not simply a circular track around the nucleus.

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The Modern Electron Cloud

The modern atomic model is sometimes represented using an:

electron cloud

This does not mean that electrons are physically spread out like ordinary fog.

It represents the probability distribution associated with where an electron may be detected.

For example, hydrogen's lowest-energy orbital is:

spherically symmetric

rather than a thin circular path.

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From Bohr to Quantum Mechanics

Bohr's model was therefore an important transitional model.

The progression was roughly:

Rutherford nuclear model

↓

Bohr quantized energy levels

↓

wave-particle ideas

↓

modern quantum mechanics

The central idea of quantized atomic energies survived.

The idea of simple fixed circular electron orbits did not.

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What Bohr Got Right

Several key ideas remain important:

• atomic energies are quantized
• atoms have discrete energy states
• photons are emitted or absorbed during transitions
• photon energy corresponds to an energy difference
• hydrogen's energy levels follow a definite pattern

These ideas became part of modern quantum theory.

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What Bohr Got Wrong or Oversimplified

The simple Bohr model incorrectly or incompletely treats electrons as:

• particles following fixed circular paths
• having precisely defined classical-style orbits
• behaving independently in multi-electron atoms

Modern quantum mechanics provides a more accurate description.

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Bohr's Model and Scientific Progress

Bohr's model provides an excellent example of how scientific models develop.

Rutherford's model explained:

nuclear scattering

but could not explain:

atomic stability or line spectra

Bohr added:

quantized electron states

His model successfully explained hydrogen.

Later experiments revealed additional limitations.

Quantum mechanics then provided a more complete theory.

Therefore:

Rutherford → Bohr → quantum mechanics

illustrates how scientific models are refined as new evidence appears.

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Models Can Be Useful Without Being Perfect

Bohr's model is still widely taught because it provides a useful introduction to:

• energy levels
• electron transitions
• emission
• absorption
• ionization
• atomic spectra

A model does not have to describe every aspect of reality perfectly to be useful.

The important question is:

What can the model explain accurately, and where does it stop working?

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Worked Example: Absorption or Emission?

An electron changes from:

n = 2 → n = 5

The electron moves to a higher energy state.

Therefore energy must be:

absorbed

Now consider:

n = 5 → n = 2

The electron moves to a lower energy state.

Therefore energy is:

emitted as a photon

A useful rule is:

up → absorb

down → emit

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Worked Example: Comparing Transitions

Consider two transitions:

A: n = 3 → n = 2

B: n = 4 → n = 2

The n = 4 → n = 2 transition involves a larger energy difference.

Therefore its photon has:

• greater energy
• higher frequency
• shorter wavelength

Remember:

E = hf = hc/λ

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Worked Example: Energy Level

Find the energy of the n = 3 level in hydrogen.

Use:

Eₙ = −13.6/n² eV

E₃ = −13.6/3²

E₃ = −13.6/9

E₃ ≈ −1.51 eV

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Worked Example: Transition Energy

Find the photon energy for:

n = 4 → n = 2

First calculate:

E₄ = −13.6/16

E₄ = −0.85 eV

E₂ = −13.6/4

E₂ = −3.40 eV

Photon energy:

E_photon = E₄ − E₂

= −0.85 − (−3.40)

= 2.55 eV

Because the electron moves downward:

a 2.55 eV photon is emitted.

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Reading an Energy-Level Diagram

When interpreting an atomic energy-level diagram:

Upward arrow

Means:

absorption

Downward arrow

Means:

emission

Longer energy jump

Means:

larger ΔE

and therefore:

higher photon frequency

and:

shorter photon wavelength

This gives the relationship:

larger ΔE → larger E_photon → larger f → smaller λ

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Common Mistakes

Mistake: Saying Bohr discovered the nucleus.
Rutherford's work established the nuclear model. Bohr introduced quantized atomic energy states.

Mistake: Saying electrons can exist anywhere between Bohr energy levels.
Only certain energies are allowed in the model.

Mistake: Saying electrons continuously lose energy while remaining in an allowed Bohr orbit.
Bohr specifically proposed stable states that do not continuously radiate.

Mistake: Saying an electron emits a photon when moving upward.
An upward transition requires energy absorption.

Mistake: Saying a downward transition absorbs energy.
A downward transition releases energy.

Mistake: Saying every electron transition produces visible light.
Transitions can produce ultraviolet, visible, or infrared radiation.

Mistake: Saying all energy levels are equally spaced.
Hydrogen's levels become closer together as n increases.

Mistake: Treating Bohr's circular electron orbits as the modern atomic model.
Modern quantum mechanics uses orbitals and probability distributions.

Mistake: Saying Bohr's model accurately describes every atom.
It works especially well for hydrogen and other one-electron systems but fails for more complicated atoms.

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Key Terms

Bohr model — atomic model proposing that electrons occupy specific quantized energy states.

Quantization — restriction of a physical quantity to particular discrete values.

Energy level — allowed energy state of an electron in an atom.

Principal quantum number — integer used to identify Bohr energy levels.

Ground state — lowest possible energy state of an atom.

Excited state — atomic state with energy greater than the ground state.

Electron transition — change of an electron from one allowed energy state to another.

Photon — quantum of electromagnetic radiation.

Emission — release of a photon when an atom moves to a lower energy state.

Absorption — uptake of photon energy that can move an atom to a higher energy state.

Line spectrum — pattern of discrete wavelengths emitted or absorbed by atoms.

Ionization — removal of an electron from an atom.

Ionization energy — minimum energy required to remove an electron from a specified state.

Electron volt (eV) — energy unit commonly used in atomic physics; 1 eV = 1.60 × 10⁻¹⁹ J.

Bohr radius — characteristic radius of hydrogen's ground state in the Bohr model, approximately 5.29 × 10⁻¹¹ m.

Orbit — defined electron path used in the Bohr model.

Orbital — quantum state associated with an electron probability distribution in modern quantum mechanics.

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Key Takeaways

• Bohr built on Rutherford's nuclear model of the atom.
• Bohr proposed that electrons can occupy only certain allowed energy states.
• These allowed energies are quantized.
• The lowest allowed state is the ground state.
• Higher-energy states are excited states.
• In the original Bohr model, allowed energies were associated with specific electron orbits.
• Electrons can move between allowed energy levels.
• Upward transitions require energy absorption.
• Downward transitions produce photon emission.
• Photon energy equals the difference between atomic energy levels.
• ΔE = hf
• Because c = fλ, larger energy transitions produce shorter-wavelength photons.
• Bohr's model successfully explained hydrogen's line spectrum.
• Hydrogen's energy levels follow Eₙ = −13.6/n² eV.
• Ground-state hydrogen has an energy of −13.6 eV.
• The ground-state ionization energy of hydrogen is 13.6 eV.
• Bohr's model works particularly well for hydrogen and other one-electron systems.
• It does not accurately describe complex multi-electron atoms.
• It cannot fully explain fine spectral structure and several effects caused by external fields.
• Modern quantum mechanics replaces fixed electron orbits with quantum states and orbitals.
• Bohr's model remains valuable because its concept of quantized atomic energies was fundamentally important to the development of quantum physics.

Check Your Understanding

1. What major problem with Rutherford's atomic model did Bohr attempt to solve?
2. What does it mean to say that atomic energy is quantized?
3. What values can the principal quantum number n take?
4. What is the ground state of an atom?
5. What is an excited state?
6. What happens when an electron moves from a lower energy level to a higher one?
7. What happens when an electron moves from a higher energy level to a lower one?
8. State the relationship between photon energy and frequency.
9. Explain why hydrogen produces a line spectrum rather than a continuous spectrum.
10. Calculate the energy of the n = 2 state of hydrogen using Eₙ = −13.6/n² eV.
11. Calculate the energy of the n = 4 state.
12. Calculate the photon energy for a hydrogen transition from n = 4 to n = 2.
13. Which produces a higher-frequency photon: n = 3 → n = 2 or n = 5 → n = 2? Explain.
14. What is the ionization energy of ground-state hydrogen?
15. Explain one major strength of Bohr's atomic model.
16. Why does Bohr's model work particularly well for hydrogen?
17. Give two observations that the simple Bohr model cannot fully explain.
18. Explain the difference between an orbit and an orbital.
19. Why is the Bohr model still useful even though it is not the modern atomic model?
20. Explain how the progression from Rutherford's model to Bohr's model and then to quantum mechanics demonstrates the development of scientific theories.