6.2 The Bohr Model
6.2 The Bohr Model – Study Notes
Historical context and motivation
After Rutherford and colleagues, atoms were pictured as a tiny dense nucleus with electrons orbiting (the planetary model).
Hydrogen is the simplest atom: a single proton nucleus and a single electron.
Classical mechanics predicted that an accelerating electron in an orbit should emit electromagnetic radiation continuously, causing energy loss and an unstable atom (spiraling into the nucleus).
Bohr (1913) resolved the paradox by combining Planck's quantization with Einstein’s idea that light is photons with energy proportional to frequency.
Bohr’s key assumptions:
Stationary state hypothesis: the electron in an atom does not normally emit radiation while in a stationary orbit.
Emission or absorption of a photon occurs only when the electron moves between orbits (different energy levels).
The energy of the photon corresponds to the difference in orbital energies: |\Delta E| = |Ef - Ei| = h\nu = \frac{hc}{\lambda}. The absolute value is used because frequencies and wavelengths are positive.
Quantization of electronic energy levels
Energies of the orbitals are quantized: E_n = -\frac{kZ^2}{n^2}, \quad n = 1, 2, 3, \dots
Here, k is a constant comprising fundamental constants (including electron mass and charge, Planck’s constant). For hydrogen-like atoms, Z is the nuclear charge (+1 for H, +2 for He, +3 for Li, etc.).
For hydrogen, Z = 1 and the constant k = 2.179 \times 10^{-18} \text{ J}.
As n increases, the energy becomes less negative (increases) and approaches zero (ionization limit).
Ground state corresponds to the lowest energy orbit: n=1\Rightarrow E_1 = -\frac{kZ^2}{1^2} = -kZ^2. The atom in this state is the ground electronic state.
Hydrogen-like atoms (one-electron systems)
General energy expression: E_n = -\frac{kZ^2}{n^2}, \quad n = 1, 2, 3, \dots
Radius of the circular orbit: rn = \frac{a0}{Z} n^2, \quad a_0 = 5.292 \times 10^{-11} \text{ m}.
Relationship between energy and radius: higher energy (larger n) implies greater distance from the nucleus, consistent with weaker electrostatic attraction at larger r.
Ionization limit: as n \to \infty, E_n \to 0; ionization energy from a given level is the energy required to remove the electron completely.
From the ground state, ionization energy: \Delta E = 0 - E_1 = +kZ^2. For hydrogen (Z = 1), this is simply \Delta E = k = 2.179 \times 10^{-18} \text{ J}.
Connection to spectroscopy and foundational results
Discrete energy levels lead to line spectra: the wavelengths (or frequencies) of emitted/absorbed photons correspond to transitions between quantized levels.
The model successfully explained three major puzzles of the time that involved Planck’s constant and photons: blackbody radiation, the photoelectric effect, and the hydrogen emission spectrum.
Bohr’s successful derivation of the Rydberg constant and spectrum for hydrogen supported the quantum idea, yielding excellent agreement with the experimental value (R ≈ 1.097 × 10^7 m^-1).
Bohr later generalized to hydrogen-like ions (one-electron systems with higher Z) but could not extend the theory to atoms with more than one electron (where electron–electron interactions become important).
Bohr’s model vs. later quantum mechanics
Strengths:
Introduced quantization of energy levels and the concept of electrons occupying discrete orbits.
Provided a simple explanation for hydrogen’s spectral lines and ionization energies.
Limitations:
Based on classical orbits, which are untenable in the microscopic domain according to later quantum mechanics.
Fails to account for electron-electron interactions in multi-electron atoms (e.g., He, Li, etc.).
Significance: laid the foundation for the quantum mechanical model of the atom and the idea that energy levels are quantized via quantum numbers.
Key equations (summary)
Orbital energy for hydrogen-like atoms:
E_n = -\frac{kZ^2}{n^2}, \quad n = 1,2,3,\dotsRadius of nth orbit (hydrogen-like):
rn = \frac{a0}{Z} n^2, \quad a_0 = 5.292 \times 10^{-11} \text{ m}.Energy difference for a transition (absorb or emit photon):
|\Delta E| = |E{nf} - E{ni}| = kZ^2 \left| \frac{1}{nf^2} - \frac{1}{ni^2} \right|.Photon energy and wavelength:
|\Delta E| = h\nu = \frac{hc}{\lambda}.Rydberg form for hydrogen-like wavelengths (with R = Rydberg constant):
\frac{1}{\lambda} = R Z^2 \left( \frac{1}{nf^2} - \frac{1}{ni^2} \right),
where for hydrogen, Z = 1 and for hydrogen-like ions Z > 1 the factor Z^2 appears.Ground-state energy and ionization energy (from ground state):
E1 = -\frac{kZ^2}{1^2} = -kZ^2, \quad \Delta E{\text{ion}} = -E_1 = kZ^2.
Numerical references and example results from the transcript
Fundamental constants:
k = 2.179 \times 10^{-18} \text{ J}
a_0 = 5.292 \times 10^{-11} \text{ m}
h = 6.626 \times 10^{-34} \text{ J s}
c = 2.998 \times 10^{8} \text{ m s}^{-1}
R ≈ 1.097 \times 10^{7} \text{ m}^{-1}
Example 6.4: energy of electron in Bohr orbit (n = 3) for hydrogen (Z = 1)
E_n = -\frac{kZ^2}{n^2} = -\frac{(2.179 \times 10^{-18})}{3^2} = -2.421 \times 10^{-19} \text{ J}.
Check Your Learning for n = 6: E_6 = -\frac{k}{6^2} = -6.053 \times 10^{-20} \text{ J}.
Example 6.5: transition from n = 4 to n = 6 in a hydrogen atom (Z = 1)
Energy change (absorption):
\Delta E = E6 - E4 = -\frac{k}{6^2} - \left(-\frac{k}{4^2}\right) = -\frac{k}{36} + \frac{k}{16} = k\left(\frac{1}{16} - \frac{1}{36}\right) = 7.566 \times 10^{-20} \text{ J}.Wavelength of absorbed photon:
\lambda = \frac{hc}{\Delta E} = \frac{(6.626 \times 10^{-34})(2.998 \times 10^{8})}{7.566 \times 10^{-20}} \approx 2.626 \times 10^{-6} \text{ m}. (Infrared region).Check Your Learning (He⁺ ion, Z = 2) result (as given):
Energy: \Delta E = 6.198 \times 10^{-19} \text{ J}
Wavelength: \lambda = 3.205 \times 10^{-7} \text{ m} (UV region). Note: the transcript shows a typographical issue with the exponent; the correct wavelength is in the UV, around 321 nm.
Worked figures and concepts referenced
Figure 6.14 (energy levels vs. quantum numbers) illustrates that higher n yields less negative energy (closer to 0) and larger radii.
Figure 6.15 illustrates electron transitions: photons absorbed (left) or emitted (right) as electrons move between orbits.
Important qualitative takeaways
Discrete energy levels and quantization are fundamental outcomes, with quantum numbers uniquely characterizing orbital states.
An electron’s energy increases with distance from the nucleus (higher n).
Spectral lines arise from transitions between quantized energy levels.
Bohr’s model provided a bridge between classical ideas and quantum concepts, but its limitations highlighted the need for a full quantum mechanical treatment.
Summary of why the Bohr model mattered
Introduced the quantization of electronic energy levels in atoms.
Demonstrated a successful calculation of the Rydberg constant and hydrogen spectral lines.
Set the stage for the development of quantum mechanics and the modern atomic model.
Connections to foundational principles and real-world relevance
Planck’s constant (quantization) and photon concept underpin spectroscopy, lasers, LEDs, and many modern technologies.
The concept that energy levels are quantized explains why matter has stable, discrete spectral features rather than a continuum.
Mathematical highlights to remember
Energy levels for hydrogen-like atoms:
E_n = -\frac{kZ^2}{n^2}Radius levels:
rn = \frac{a0}{Z} n^2, \quad a_0 = 5.292 \times 10^{-11} \text{ m}Photon energy associated with a transition:
|\Delta E| = h\nu = \frac{hc}{\lambda}Relationship for hydrogen-like wavelengths (Rydberg form):
\frac{1}{\lambda} = R Z^2\left( \frac{1}{nf^2} - \frac{1}{ni^2} \right)
Key terms to define for exam readiness
Ground state, excited state, stationary state, ionization limit, hydrogen-like atoms, quantum numbers, Rydberg constant, Bohr radius, Planck’s constant, photon, wavelength, frequency, energy level.