Bohr's Energy-Level Model

Introduction

The nuclear atom explained scattering but not stable electrons or hydrogen's separated spectral lines. Bohr added a restriction: a hydrogen electron can occupy only certain allowed energy states and does not radiate continuously while in one. Light is emitted or absorbed when the atom changes between states. The model is historically important and quantitatively successful for hydrogen, though its drawn circular orbits are not the modern literal picture.

Core explanation

Bohr's model treats the simplest atom, hydrogen, as a positive nucleus with one electron in allowed states labelled by positive integers n = 1, 2, 3 and so on. In its original picture, each allowed state corresponded to a particular orbit and energy. The lowest state, n = 1, is the ground state. Higher n states have higher energy, meaning they are less tightly bound to the nucleus in the model. “Higher energy” here can mean less negative numerical energy when the zero is set for a separated electron and proton.

An electron in an allowed stationary state does not continuously radiate in Bohr's postulate. This was a deliberate departure from a naive classical orbit, which would radiate as an accelerating charge. When an atom moves from a higher-energy state to a lower one, it emits a photon whose energy equals the level difference. When it absorbs a photon of a matching energy, it can move from a lower state to a higher one. Only certain differences are available, so only certain photon frequencies appear in the atom's spectrum.

For hydrogen, a common energy expression is Eₙ = −2.18 × 10⁻¹⁸ J/n², using the separated electron and proton as zero energy. At n = 1, E₁ ≈ −2.18 × 10⁻¹⁸ J. At n = 2, E₂ ≈ −5.45 × 10⁻¹⁹ J. The n = 2 state is higher because −5.45 × 10⁻¹⁹ is closer to zero than −2.18 × 10⁻¹⁸. A transition from n = 2 to n = 1 releases a photon with energy E₂ − E₁ ≈ 1.64 × 10⁻¹⁸ J. The energy is positive for the emitted photon even though the atom's energy change is negative.

The lines of hydrogen arise from many possible pairs of levels. Transitions down to n = 2 include visible Balmer lines, while transitions down to n = 1 are in the ultraviolet Lyman region. The model predicts a pattern, not merely one colour. An atom may first be excited by an electrical discharge or other energy input and then release light in one or several downward steps. The path between levels is not a tiny visible flight across a drawn diagram.

Bohr's treatment worked especially well for hydrogen and other one-electron ions, such as He⁺. It does not give a complete account of helium with two electrons or larger atoms, where electron–electron interactions matter. Modern quantum mechanics replaces fixed circular tracks with orbitals and probability distributions while retaining the reality of discrete energy states. The Bohr diagram remains a useful instructional model for transitions within its scope.

The model's success should be described precisely. Rutherford supplied a nucleus; Bohr added allowed electron energies and a rule for radiation during transitions. Bohr did not prove that an electron physically circles like a planet at a measured line thickness. A flat diagram with rings is a tool for showing energy order and possible jumps, not a scaled image of an atom.

One more distinction is ionisation. As n becomes very large, Eₙ approaches zero from below in this model. Reaching the separated state requires energy from a bound state. A photon with enough energy can remove the electron; ordinary excitation to a higher but finite n leaves the atom bound. This clarifies why “excited” and “ionised” are not synonyms.

Step-by-step reasoning

1. Label allowed levels by n and identify n = 1 as the ground state. 2. Compare numerical energies carefully; less negative is higher when zero is the separated state. 3. For emission, subtract final atomic energy from initial atomic energy to obtain positive photon energy. 4. State that the fixed-orbit drawing is a model for hydrogen, with modern quantum limits.

Visual explanation

Draw horizontal levels E₁, E₂ and E₃ with increasing energy upward. Put a downward arrow from E₂ to E₁ labelled “emitted photon, E₂ − E₁.” Put an upward arrow from E₁ to E₂ labelled “matching photon absorbed.” Avoid drawing a continuous spiral.

Real-world analogy

A staircase has distinct steps, so one stands at permitted heights rather than every height between them. Bohr's allowed energy levels resemble this discreteness. The analogy does not explain why hydrogen has those energies or make an electron a person moving through space between steps.

Real-world example

Hydrogen gas in a discharge tube emits specific spectral lines after its atoms are excited. Bohr's model connects the measured line energies to downward changes between allowed hydrogen levels, giving a quantitative explanation beyond merely naming the colours.

Why?

Why did Bohr restrict electron energies? Stable atoms and discrete spectral lines contradicted a simple unrestricted classical orbit account. Allowed stationary states avoid continuous radiation within the model, and level differences produce the observed specific photon energies.

Common misconception

“An excited electron must have left the atom.” An excited bound electron occupies a higher allowed energy state. Ionisation requires enough energy to remove it from the atom entirely, represented by the zero-energy limit in the hydrogen model.

Worked example

Use E₁ = −2.18 × 10⁻¹⁸ J and E₂ = −5.45 × 10⁻¹⁹ J. For emission from n = 2 to n = 1, photon energy is E₂ − E₁ = (−5.45 × 10⁻¹⁹) − (−2.18 × 10⁻¹⁸) ≈ 1.64 × 10⁻¹⁸ J. The atom's own energy change is −1.64 × 10⁻¹⁸ J. Equal magnitudes conserve energy.

Quick check

1. Does a hydrogen atom radiate continuously while in one stationary state in Bohr's model? Answer: No. Radiation is associated with a transition between allowed states.

Exam focus

Define ground and excited states, label n values and show the direction of emission or absorption. Handle negative bound-state energies carefully and give the model's one-electron scope rather than presenting circular orbits as modern literal paths.

Advanced insight

The quantum-mechanical hydrogen solution reproduces the same energy formula but assigns different spatial states and angular momentum from Bohr's circular-orbit assumption. This is an instructive example of a model predicting some numbers correctly while its physical picture remains incomplete.

Summary

Bohr's hydrogen model allows stationary states labelled n and photons only when the atom changes between them. Their energy differences explain discrete spectral lines and the ground state's stability within the model. Fixed circular orbits are a historical approximation, not the modern electron description.

Practice questions

1. Which n value is hydrogen's ground state in the Bohr model? Answer: n = 1. 2. Is E₂ = −5.45 × 10⁻¹⁹ J higher or lower than E₁ = −2.18 × 10⁻¹⁸ J? Answer: Higher, because it is less negative and closer to the zero of a separated electron. 3. What happens when hydrogen changes from n = 3 to n = 2? Answer: It emits a photon with energy equal to E₃ − E₂. 4. Why is Bohr's model not a complete description of helium atoms? Answer: Neutral helium has two interacting electrons, beyond the model's simple one-electron scope.