Hydrogen Spectrum & Bohr's Model
The heart of Atomic Structure. Master the quantization of angular momentum, critical proportionalities, the Rydberg equation, and spectral line counting.
Module Focus
Niels Bohr provided the first atomic model capable of explaining the stability of an atom and the line spectrum of hydrogen. In competitive exams, you will rarely be asked for the full derivations. Instead, you will be heavily tested on the proportionalities (how radius, velocity, and energy vary with $n$ and $Z$), transitions between energy levels, and calculating wavelengths. Remember: Bohr's model is valid only for single-electron species ($H, He^+, Li^{2+}, Be^{3+}$).
1. Bohr's Postulates & Quantization
Bohr combined classical mechanics with Planck's quantum theory. His most critical, paradigm-shifting postulate was the quantization of angular momentum.
The Core Postulates:
- An electron revolves around the nucleus in definite, circular paths called orbits or stationary states.
- As long as an electron remains in a particular orbit, it does not radiate energy.
- Quantization of Angular Momentum: An electron can only revolve in those orbits for which its angular momentum ($mvr$) is an integral multiple of $h/2\pi$.
- Energy is absorbed or emitted only when an electron jumps from one orbit to another ($\Delta E = E_{\text{final}} - E_{\text{initial}} = h\nu$).
Where $n$ = 1, 2, 3... (Principal Quantum Number), $m$ = mass of electron, $v$ = velocity, $r$ = radius.
2. The Big Three: Radius, Velocity, and Energy
Do not derive these in the exam. Memorize the final formulas and, most importantly, the relationships between the variables. $n$ is the orbit number, and $Z$ is the atomic number.
Proportionality: $r_n \propto \frac{n^2}{Z}$
Note: $0.529 \mathring{\text{A}}$ is the Bohr radius ($a_0$)—the radius of the first orbit of Hydrogen ($n=1, Z=1$).
Proportionality: $v_n \propto \frac{Z}{n}$
Notice that velocity decreases as you move to higher orbits.
3. Energy of Electron ($E_n$)
The total energy of an electron is the sum of its Kinetic Energy (KE) and Potential Energy (PE). The negative sign indicates that the electron is bound to the nucleus.
$E_n = -2.18 \times 10^{-18} \times \frac{Z^2}{n^2} \text{ J/atom}$
Let Total Energy (TE) = $E$.
$\mathbf{TE = -KE}$
$\mathbf{PE = 2 \times TE = -2 \times KE}$
If an electron has TE = -13.6 eV, its KE is +13.6 eV, and its PE is -27.2 eV.
3. Hydrogen Line Spectrum & Rydberg Equation
When an electric discharge is passed through hydrogen gas, hydrogen molecules dissociate, and the energetically excited hydrogen atoms emit electromagnetic radiation of discrete frequencies. This creates a line emission spectrum.
The Rydberg Equation
Used to calculate the wave number ($\bar{\nu}$) and wavelength ($\lambda$) of the photon emitted or absorbed during an electron transition.
Rydberg Constant ($R_H$): $109677 \text{ cm}^{-1} \approx 1.1 \times 10^7 \text{ m}^{-1}$
Calculation Shortcut: For rapid calculation, memorize that $\mathbf{\frac{1}{R_H} \approx 912 \mathring{\text{A}}}$.
Spectral Series of Hydrogen ($Z=1$)
| Series Name | $n_{\text{lower}} (n_1)$ | $n_{\text{higher}} (n_2)$ | Spectral 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... | Infrared (IR) |
4. NEET Pro-Tips & Shortcuts
When an electron in a bulk sample of hydrogen atoms drops from an excited state ($n_2$) to a lower state ($n_1$), the maximum number of spectral lines produced is:
If returning to the ground state ($n_1 = 1$), the formula simplifies to: $\mathbf{\frac{n(n-1)}{2}}$
- Longest Wavelength ($\lambda_{max}$): Lowest energy transition. The electron jumps from $n_1$ to the very next level, $n_2 = n_1 + 1$ (First line of the series).
- Shortest Wavelength ($\lambda_{min}$): Highest energy transition. The electron jumps from $n_1$ to $n_2 = \infty$ (Series limit line).
- Ionization Energy (IE): Energy required to remove an electron from the ground state ($n=1$) to infinity. For H-atom, $\text{IE} = +13.6 \text{ eV}$.
- Excitation Energy: Energy required to move an electron from the ground state to any excited state (e.g., First excitation energy is transition from $n=1$ to $n=2$).
- Separation / Binding Energy: Energy required to remove an electron from a specific excited state to infinity (e.g., Binding energy of 2nd state is energy from $n=2$ to $n=\infty$).
NEET Grand Test: Bohr's Model & Spectrum
15 High-Order Thinking Questions testing ratios, spectral line counts, and Rydberg calculations.
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