3.2 The Bohr Model | Chemistry: Atoms First 2e

Learning Objectives

Describe the Bohr model of the hydrogen atom
Use the Rydberg equation to calculate energies of light emitted or absorbed by hydrogen atoms

Core Concepts & Principles

Following Ernest Rutherford’s discovery of the atomic nucleus, scientists viewed the atom like a miniature solar system (the planetary model). However, classical physics presented a fatal flaw: an orbiting electron accelerates and should continuously radiate energy, spiraling into the nucleus and rendering atoms unstable.

In 1913, Niels Bohr resolved this paradox by combining classical mechanics with quantum theory. He proposed that electrons do not radiate energy while in stable circular orbits (stationary states). Instead, energy is only emitted or absorbed when an electron jumps between fixed, quantized energy levels.

Quantized Energy & Electron Jumps

Quantized Orbits: Electron energy levels in atoms are restricted to specific, discrete values determined by integer quantum numbers ( $n = 1, 2, 3, \dots$ ).
Photon Emission/Absorption: Electrons change orbits by absorbing a photon (jumping to a higher $n$ ) or emitting a photon (falling to a lower $n$ ). The photon energy exactly matches the energy difference between the two orbits: $\Delta E = |E_f - E_i| = h\nu = \frac{hc}{\lambda}$ .

Energy Levels and Hydrogen-Like Ions

The energy of an electron in a Bohr orbit depends on its principal quantum number ( $n$ ) and the nuclear charge ( $Z$ ). For hydrogen ( $Z = 1$ ) and hydrogen-like single-electron ions (such as $\text{He}^+$ and $\text{Li}^{2+}$ ), the energy is expressed as:

$E_n = -k \frac{Z^2}{n^2}$

Ground Electronic State: The lowest energy state ( $n = 1$ ), where the atom is most stable.
Excited Electronic State: Any higher energy state ( $n > 1$ ) reached when energy is absorbed.
Ionization Limit: As $n \to \infty$ and $r \to \infty$ , the energy approaches zero ( $E = 0$ ), meaning the electron is completely stripped from the atom (ionization).

Although Bohr’s model successfully explained atomic line spectra and derived the theoretical Rydberg constant, it failed for multi-electron atoms (like helium) because it clung to the classical idea of precise, predictable electron orbits.

Problem-Solving Routines & Methods

Bohr Orbit Energy

E_n = -k \frac{Z^2}{n^2}

Calculates the energy of an electron in a specific orbit level n.

Variables & Constants

E_n

=electron energy (J);

k

2.179 \times 10^{-18}\text{ J}

;

Z

=atomic number;

n

=principal quantum number

Energy Transition & Wavelength

\Delta E = -2.179 \times 10^{-18}\text{ J} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) = \frac{hc}{\lambda}

Calculates energy change and corresponding photon wavelength during an electron jump.

Variables & Constants

\Delta E

=energy change (J);

n_i

=initial orbit;

n_f

=final orbit;

h

=Planck's constant;

c

=speed of light;

\lambda

=wavelength (m)

Calculating Electron Transition Energy and Wavelength

1
Identify the initial level ( $n_i$ ), final level ( $n_f$ ), and nuclear charge ( $Z$ ).
2
Calculate energy change ( $\Delta E$ ) using the Bohr transition equation. A positive $\Delta E$ indicates absorption; negative indicates emission.
3
Use $\lambda = \frac{hc}{|\Delta E|}$ (where $hc = 1.986 \times 10^{-25}\text{ J}\cdot\text{m}$ ) to find the photon wavelength.

Pro-Tip: Always check whether the electron is moving to a higher or lower level to correctly interpret the sign of

\Delta E

Practice & Concept Checks

Concept Check

Why did classical electromagnetic theory predict that Rutherford's planetary model of the atom was unstable?

Concept Check

What happens to the radius and energy of an electron's orbit as the principal quantum number (n) increases?

Key Terms & Vocabulary

RutherfordHistorical Figures

Physicist who established the nuclear model of the atom, depicting a dense positive nucleus surrounded by moving electrons.

BohrHistorical Figures

Physicist who introduced quantization to the atomic model, successfully explaining hydrogen's line spectra.

Bohr’s modelAtomic Models

An early atomic model where electrons travel in specific circular orbits with quantized energies without radiating energy.

Example: En = -k(Z² / n²)

ground electronic stateEnergy States

The lowest energy state of an atom where the electron occupies the n = 1 orbit.

Example: Hydrogen atom with its electron at n = 1

excited electronic stateEnergy States

Any energy state higher than the ground state, attained when an electron absorbs energy and moves to a higher orbit (n > 1).

Example: Hydrogen atom with its electron promoted to n = 3

quantum numbersQuantum Mechanics

Integer values that specify the allowed energy levels and physical characteristics of electrons in atoms.