Key Postulates and Energy Levels in Bohr’s Hydrogen Atom Theory
Bohr’s theory of the hydrogen atom provided a quantum-based explanation for atomic structure and spectra, defining specific allowed electron orbits and quantised energy levels. This model accurately described key atomic properties and energy transitions in hydrogen and hydrogen-like ions, playing a foundational role in atomic physics and quantum mechanics.
Fundamental Postulates of Bohr's Theory
Bohr’s model introduced key postulates that departed from classical physics to incorporate quantum principles. The first postulate states that an electron in a hydrogen atom revolves around the nucleus in specific circular orbits called stationary states without emitting energy.
The second postulate establishes that each stationary state is associated with a definite energy, remaining constant as long as the electron stays in that orbit. Electrons can only transition between these discrete energy levels by absorbing or emitting fixed quanta of energy.
The third postulate introduces Bohr’s frequency condition. When an electron jumps between two stationary states, the frequency ($\nu$) of the emitted or absorbed radiation is determined by the energy difference ($\Delta E$) between the initial and final states, given as $\nu = \dfrac{\Delta E}{h}$, where $h$ is Planck’s constant.
The fourth postulate is the quantisation of angular momentum. According to Bohr, the angular momentum ($L$) of an electron in any permitted orbit is restricted to discrete values defined by $L = n\dfrac{h}{2\pi}$, where $n$ is a positive integer. This quantisation ensures stable orbits and discrete energy levels.
Quantisation of Angular Momentum and Allowed Orbits
The quantisation condition for angular momentum restricts electrons to specific orbits. For a hydrogen atom, Bohr’s quantisation rule is expressed as $m_e v r = n\dfrac{h}{2\pi}$, with $m_e$ as electron mass, $v$ as velocity, $r$ as radius, and $n$ as the principal quantum number.
These quantised orbits are associated with fixed radii, and their values increase with the square of $n$. The smallest orbit, where $n=1$, is called the ground state or Bohr radius. The stability of each orbit is a direct result of the angular momentum restriction.
Expression for Radius and Energy of Orbits
For hydrogen and hydrogen-like ions, the radius of the $n$th orbit is given as $r_n = n^2 a_0/Z$, where $a_0$ is the Bohr radius and $Z$ is the atomic number. The Bohr radius for hydrogen ($Z=1$) is approximately $0.529 \ \text{Å}$.
The energy of an electron in the $n$th orbit is quantified by the equation $E_n = -13.6 \ \dfrac{Z^2}{n^2} \ \text{eV}$. Here, the negative sign indicates that the electron is bound to the nucleus. For the ground state $(n=1)$ of hydrogen $(Z=1)$, $E_1 = -13.6$ eV.
As $n$ increases, the energy becomes less negative, indicating a decrease in the binding strength between the nucleus and electron. Ionisation energy refers to the energy required to move the electron from $n=1$ $(E=-13.6$ eV$)$ to $n = \infty$ $(E=0$ eV$)$.
Detailed coverage of related concepts can be found in the Understanding Atomic Structure page.
Hydrogen Atom Energy Levels
The Bohr model provides a clear quantification of energy levels for the hydrogen atom. Each level is represented by the principal quantum number $n$, from $n=1$ (ground state) upwards. The energy associated with each level is calculated by $E_n = -13.6 \ \dfrac{1}{n^2}$ eV for hydrogen.
The arrangement of these discrete energy levels explains the observed hydrogen spectra. Electrons can occupy any of these levels but not the intermediate energies. When transitioning between levels, emission or absorption of energy occurs in discrete amounts, supporting the observation of line spectra.
| Value of n | Energy Level (eV) |
|---|---|
| 1 (Ground state) | -13.6 |
| 2 | -3.4 |
| 3 | -1.51 |
| 4 | -0.85 |
| $\infty$ (Ionized) | 0 |
Hydrogen Emission Spectrum and Spectral Series
The hydrogen atom’s spectral lines are organized into series, each corresponding to electron transitions ending at specific energy levels. The Lyman series $(n=1)$ occurs in the ultraviolet region, while the Balmer series $(n=2)$ appears in the visible spectrum.
Paschen, Brackett, and Pfund series represent transitions ending at $n=3, 4,$ and $5$ respectively, and are found in the infrared region. The spectral lines arise due to transitions between allowed energy levels, resulting in photon emission or absorption at characteristic wavelengths.
All such transitions are described by the Rydberg formula: $\dfrac{1}{\lambda} = R_H \left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)$, where $R_H$ is the Rydberg constant ($1.097 \times 10^7 \ \text{m}^{-1}$), $n_2>n_1$ and both are integers.
A comprehensive explanation of spectra-related phenomena is available in the Atoms and Nuclei Overview resource.
Bohr's Frequency Condition for Radiation
According to Bohr’s frequency rule, the frequency ($\nu$) of light emitted or absorbed during an electron transition is given by $\nu = \dfrac{E_2 - E_1}{h}$, where $E_1$ and $E_2$ are the energies of the initial and final states.
This condition establishes a direct relationship between energy transitions within the atom and the electromagnetic spectrum, accurately predicting hydrogen’s line spectra. Each transition corresponds to a photon with energy equal to the difference between two allowed energy states.
Success of Bohr's Model for Hydrogen
Bohr’s model provided precise explanations for the observed spectral lines of hydrogen and hydrogen-like ions, correctly predicting their wavelengths in the ultraviolet, visible, and infrared regions. The model’s quantised energy level structure matched experimental data for hydrogen emission and absorption spectra.
For additional study of quantum aspects in atoms, refer to the Dual Nature of Matter material.
Limitations of Bohr's Model
While Bohr’s theory explained the hydrogen atom’s features, it could not account for finer details like fine structure or the Zeeman and Stark effects arising under magnetic and electric fields. It was also unsuccessful for multi-electron atoms due to electron-electron interactions and other quantum complexities.
The model assumes fixed orbits and does not incorporate the uncertainty principle. It uses classical concepts for electron orbits rather than the wave nature of electrons indicated by modern quantum mechanics.
Bohr’s model does not explain chemical bonding or behaviour in molecules. It applies accurately only to single-electron systems such as hydrogen and ions like He$^+$.
In-depth exploration of atomic structure developments is provided in the Nuclear Fission and Fusion section.
Key Points on Bohr's Theory of Hydrogen Atom
- Electron moves in specific quantised orbits only
- Energy levels are discrete and measurable
- Angular momentum is quantised in units of $h/2\pi$
- Atomic line spectra arise from electron transitions
- Model explains hydrogen, not multi-electron atoms
- Does not account for external field effects
Further Study Topics Related to Bohr's Theory
The Bohr model’s principles lay the groundwork for understanding atomic and nuclear phenomena. To extend understanding, students may refer to supplements such as Magnetic Effects of Current and the Revision Notes on Magnetism.
