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: ∣ΔE∣=∣E<em>f−E</em>i∣=hν=hcλ.|\Delta E| = |E<em>f - E</em>i| = 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: En=−kZ2n2,n=1,2,3,…E_n = -\frac{kZ^2}{n^2}, \quad n = 1, 2, 3, \dots

    • Here, kk 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=2.179×10−18 J.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⇒E1=−kZ212=−kZ2.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: En=−kZ2n2,n=1,2,3,…E_n = -\frac{kZ^2}{n^2}, \quad n = 1, 2, 3, \dots

    • Radius of the circular orbit: r<em>n=a</em>0Zn2,a0=5.292×10−11 m.r<em>n = \frac{a</em>0}{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→∞n \to \infty, En→0E_n \to 0; ionization energy from a given level is the energy required to remove the electron completely.

    • From the ground state, ionization energy: ΔE=0−E1=+kZ2.\Delta E = 0 - E_1 = +kZ^2. For hydrogen (Z = 1), this is simply ΔE=k=2.179×10−18 J.\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:
      En=−kZ2n2,n=1,2,3,…E_n = -\frac{kZ^2}{n^2}, \quad n = 1,2,3,\dots

    • Radius of nth orbit (hydrogen-like):
      r<em>n=a</em>0Zn2,a0=5.292×10−11 m.r<em>n = \frac{a</em>0}{Z} n^2, \quad a_0 = 5.292 \times 10^{-11} \text{ m}.

    • Energy difference for a transition (absorb or emit photon):
      ∣ΔE∣=∣E<em>n</em>f−E<em>n</em>i∣=kZ2∣1n<em>f2−1n</em>i2∣.|\Delta E| = |E<em>{n</em>f} - E<em>{n</em>i}| = kZ^2 \left| \frac{1}{n<em>f^2} - \frac{1}{n</em>i^2} \right|.

    • Photon energy and wavelength:
      ∣ΔE∣=hν=hcλ.|\Delta E| = h\nu = \frac{hc}{\lambda}.

    • Rydberg form for hydrogen-like wavelengths (with R = Rydberg constant):
      1λ=RZ2(1n<em>f2−1n</em>i2),\frac{1}{\lambda} = R Z^2 \left( \frac{1}{n<em>f^2} - \frac{1}{n</em>i^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):
      E<em>1=−kZ212=−kZ2,ΔE</em>ion=−E1=kZ2.E<em>1 = -\frac{kZ^2}{1^2} = -kZ^2, \quad \Delta E</em>{\text{ion}} = -E_1 = kZ^2.

  • Numerical references and example results from the transcript

    • Fundamental constants:

    • k=2.179×10−18 Jk = 2.179 \times 10^{-18} \text{ J}

    • a0=5.292×10−11 ma_0 = 5.292 \times 10^{-11} \text{ m}

    • h=6.626×10−34 J sh = 6.626 \times 10^{-34} \text{ J s}

    • c=2.998×108 m s−1c = 2.998 \times 10^{8} \text{ m s}^{-1}

    • R ≈ 1.097×107 m−11.097 \times 10^{7} \text{ m}^{-1}

    • Example 6.4: energy of electron in Bohr orbit (n = 3) for hydrogen (Z = 1)

    • En=−kZ2n2=−(2.179×10−18)32=−2.421×10−19 J.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: E6=−k62=−6.053×10−20 J.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):
      ΔE=E<em>6−E</em>4=−k62−(−k42)=−k36+k16=k(116−136)=7.566×10−20 J.\Delta E = E<em>6 - E</em>4 = -\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:
      λ=hcΔE=(6.626×10−34)(2.998×108)7.566×10−20≈2.626×10−6 m.\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: ΔE=6.198×10−19 J\Delta E = 6.198 \times 10^{-19} \text{ J}

    • Wavelength: λ=3.205×10−7 m\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:
      En=−kZ2n2E_n = -\frac{kZ^2}{n^2}

    • Radius levels:
      r<em>n=a</em>0Zn2,a0=5.292×10−11 mr<em>n = \frac{a</em>0}{Z} n^2, \quad a_0 = 5.292 \times 10^{-11} \text{ m}

    • Photon energy associated with a transition:
      ∣ΔE∣=hν=hcλ|\Delta E| = h\nu = \frac{hc}{\lambda}

    • Relationship for hydrogen-like wavelengths (Rydberg form):
      1λ=RZ2(1n<em>f2−1n</em>i2)\frac{1}{\lambda} = R Z^2\left( \frac{1}{n<em>f^2} - \frac{1}{n</em>i^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.