Bohr's Model of Atom
Structure of Atom | Postulates, Formulas, and Hydrogen Spectrum
1. Main Postulates of Bohr's Theory
Niels Bohr proposed a model for the Hydrogen atom based on Planck's Quantum Theory.
- Stationary Orbits: Electrons revolve around the nucleus in fixed circular paths called orbits or shells ($K, L, M, N...$). Energy does not radiate while in these orbits.
- Quantization Condition: An electron can revolve only in those orbits where its angular momentum ($L$) is an integral multiple of $h/2\pi$. $$ mvr = \frac{nh}{2\pi} $$
- Energy Transitions: Energy is absorbed or emitted only when an electron jumps from one orbit to another. $$ \Delta E = E_2 - E_1 = h\nu $$
2. Important Formulas (For Hydrogen-like Species)
A. Radius of $n^{th}$ Orbit ($r_n$)
$$ r_n = 0.529 \frac{n^2}{Z} \, \mathring{A} $$
Radius $\propto n^2$ and Radius $\propto 1/Z$.
B. Velocity of Electron ($v_n$)
$$ v_n = 2.18 \times 10^6 \frac{Z}{n} \, m/s $$
Velocity $\propto Z$ and Velocity $\propto 1/n$.
C. Energy of Electron ($E_n$)
$$ E_n = -13.6 \frac{Z^2}{n^2} \, eV/atom $$
Or in Joules: $E_n = -2.18 \times 10^{-18} \frac{Z^2}{n^2} \, J/atom$.
Note: Negative sign indicates the electron is bound to the nucleus.
3. Hydrogen Spectrum
The wave number ($\bar{\nu}$) of emitted radiation when an electron drops from $n_2$ to $n_1$ is given by the Rydberg Formula:
$$ \bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right] $$
Where $R_H \approx 109677 \, cm^{-1}$ (Rydberg Constant).
Spectral Series
| Series | Ground State ($n_1$) | Excited State ($n_2$) | Region |
|---|---|---|---|
| Lyman | 1 | 2, 3, 4... | Ultraviolet (UV) |
| Balmer | 2 | 3, 4, 5... | Visible |
| Paschen | 3 | 4, 5, 6... | Infrared (IR) |
| Brackett | 4 | 5, 6, 7... | Infrared (IR) |
| Pfund | 5 | 6, 7, 8... | Far Infrared |
4. Limitations of Bohr's Model
- It could not explain the spectra of multi-electron atoms (e.g., Helium atom).
- It could not explain the splitting of spectral lines in magnetic fields (Zeeman Effect) or electric fields (Stark Effect).
- It violates the Heisenberg Uncertainty Principle (defines exact position and velocity simultaneously).
Practice Quiz
Test your knowledge on Bohr's Atomic Model.
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