Bohr's Model & Hydrogen Spectrum Why Bohr’s model matters for NEET Hydrogen shows sharp colored lines when excited gas emits light. Classical physics could not explain these specific lines. Bohr proposed that electrons can only be in certain allowed circular orbits (stationary states) around the nucleus. With just a few postulates, he explained hydrogen’s line spectrum and gave formulas you will calculate with in NEET. We will move from Bohr’s rules to the equations for radius, energy, and wavelength lines — and learn to identify spectral series quickly. Bohr picture of hydrogen: discrete circular orbits labeled by n, with arrows showing emission/absorption between orbits and the visible spectrum bar to relate transitions to color. Bohr’s postulates — the core ideas Electrons move only in fixed circular orbits (stationary states) around the nucleus. Each orbit has a fixed energy. In a stationary orbit, the electron does not radiate energy. Angular momentum is quantized: only specific values are allowed. Light is emitted or absorbed only when an electron jumps between two allowed orbits. The photon energy equals the energy difference between the orbits. Bohr’s four ideas (simple language) n = 1, 2, 3, ... (principal quantum number). This is Bohr’s key quantization rule. Angular momentum quantization Stationary states Only some circular orbits allowed; each with fixed energy No radiation while in an orbit Quantized angular momentum mvr = nh/2π (n ∈ natural numbers) n labels the orbit Emission/absorption by jumps Photon energy equals ΔE between two orbits ΔE = hc/λ Orbit energies More negative for lower n; E increases toward 0 as n → ∞ Ground state is n=1 No. Bohr’s postulates at a glance Postulate What it says NEET cue Photon–energy relation When an electron drops from a higher to a lower orbit, it emits a photon; the wavelength λ is set by the energy gap. Quantized angular momentum: only certain circular paths allowed. The ‘continuous’ orbit idea is crossed out to emphasize Bohr’s quantization (mvr = n h/2π). Derivation outline for r n and E n (hydrogen-like species) We sketch only the key steps (as in NCERT): 1) Balance forces for an electron in a circular orbit: electrostatic attraction provides centripetal force. 2) Apply angular momentum quantization mvr = nh/2π. 3) Solve simultaneously to get allowed radius r n and speed v n . 4) Write total energy E = K + V = (1/2) mv 2 − ke 2 /r (with appropriate constants), substitute r n and v n , and simplify. Result: radii and energies depend only on n and nuclear charge Z (hydrogen-like ion). Allowed radii Ångström (Å). For hydrogen (Z=1), r n = 0.529 n 2 Å. The ground-state radius r 1 = 0.529 Å is the Bohr radius. Å (Hydrogen only; Z = 1). eV per electron. Energy is negative (bound state). More negative means more tightly bound. Allowed energies eV (Hydrogen only; Z = 1). Ground state E 1 = −13.6 eV; as n → ∞, E → 0 eV. Quick proportionalities from Bohr model: r n ∝ n 2 /Z, E n ∝ − Z 2 / n 2 , v n ∝ Z/n. These help compare H, He⁺, Li²⁺ without full calculation. tip Species E n (eV) r n (Å) E 1 (eV) r 1 (Å) Ion Hydrogen-like ions: formulas and ground-state values H (hydrogen) E n = −13.6/ n 2 r n = 0.529 n 2 −13.6 0.529 He⁺ (helium ion) E n = −54.4/ n 2 r n = 0.529 n 2 /2 −54.4 0.265 Li²⁺ (lithium ion) E n = −122.4/ n 2 r n = 0.529 n 2 /3 −122.4 0.176 Bohr restricted electrons to certain allowed circular orbits (quantized angular momentum) to explain spectra. Modern quantum mechanics goes beyond this: electrons are described as probability clouds (orbitals), not fixed paths. Electrons orbit the nucleus exactly like planets around the sun. Hydrogen spectrum and spectral series When an excited hydrogen atom’s electron drops from n2 to a lower n1, it emits a photon. The set of wavelengths for fixed lower level n1 forms a spectral series. Names and regions: - Lyman series (n2 → 1): Ultraviolet (UV) - Balmer series (n2 → 2): Visible - Paschen series (n2 → 3): Infrared (IR) - Brackett (n2 → 4): IR - Pfund (n2 → 5): IR Rydberg formula (hydrogen-like species) R H = 1.097 10 7 m −1 ; integers n 2 > n 1 . For hydrogen, Z = 1. Energy-level diagram for hydrogen with series shown: Lyman (to n=1), Balmer (to n=2), Paschen (to n=3). Note energies: −13.6 eV, −3.4 eV, −1.51 eV, ... approaching 0 eV at n = ∞. Lyman UV ≈ 91.2 nm Lyman-α (2→1) ≈ 121.6 nm Balmer Visible ≈ 364.6 nm H-α (3→2) ≈ 656.3 nm (red) Paschen IR ≈ 820.4 nm Paschen-α (4→3) ≈ 1875 nm Brackett IR ≈ 1458 nm Brackett-α (5→4) Pfund IR ≈ 2279 nm Pfund-α (6→5) Series n₁ (lower) Spectral region Series limit λ limit Example prominent line Name Hydrogen spectral series summary neet-alert Identify series fast: n1 tells the series (1=Lyman, 2=Balmer, 3=Paschen, 4=Brackett, 5=Pfund). As n2 increases, lines get closer and approach a series limit (convergence). Hydrogen spectrum bands: horizontal wavelength axis from 90 nm to 2500 nm. Show five colored bands labeled Lyman (UV), Balmer (visible with rainbow colors), Paschen/Brackett/Pfund (IR). Include arrows n2→n1. Vector, clean, textbook style, red arrows. Color-coded series bands with a wavelength scale: UV (Lyman), visible (Balmer), IR (Paschen, Brackett, Pfund). gpt-image-2 2026-05-26T17:04:12.225Z Ionization energy from Bohr’s diagram Ionization energy of a hydrogen atom is the energy needed to move the electron from the ground state (n = 1) to n = ∞ (free). From the energy formula, E 1 = −13.6 eV and E ∞ = 0 eV, so the required energy is 13.6 eV per atom (≈ 1312 kJ mol −1 ). Hydrogen atom ionization energy = 13.6 eV (per atom). From n = 1 to n = ∞. remember gpt-image-2 Ionization shown on an energy ladder: electron rising from n=1 (−13.6 eV) to n=∞ (0 eV). Energy-level ladder for hydrogen with labeled energies: −13.6, −3.4, −1.51, −0.85 eV… Upward arrow from n=1 to n=∞ labeled +13.6 eV. Clean vector, neutral palette, red arrow for ionization. 2026-05-26T17:04:12.418Z Worked example — Balmer H-α line Task: Find the wavelength of the H-α line (transition 3 → 2 in hydrogen). Use Rydberg: 1/λ = R H (1/ 2 2 − 1/ 3 2 ) = R H (1/4 − 1/9) = R H (5/36). With R H = 1.097 × 10 7 m −1 , 1/λ ≈ (1.097 × 10 7 ) × (5/36) ≈ 1.524 × 10 6 m −1 . So λ ≈ 6.56 × 10 −7 m = 656 nm (red, visible). Units trap: Keep R H in m −1 . Convert final λ to nm (1 nm = 10 −9 m) only at the end. tip Real-life links: why spectra are fingerprints Atomic absorption spectroscopy (AAS) uses unique absorption lines of each element to detect trace metals (e.g., Pb, Hg) or essential ions (Ca²⁺, Mg²⁺) in clinical samples, food, and water — just like hydrogen’s lines are unique fingerprints. remember More applications you should know Astrochemistry/astrophysics: identify hydrogen and other elements in stars by their spectral lines. Sodium-vapour street lamps: bright yellow due to Na D-lines (~589 nm doublet). Calibration standards: hydrogen Balmer lines often used as reference in basic spectrometers. Limitations of Bohr’s model Fails for multi-electron atoms (electron–electron repulsion not handled). Cannot explain Zeeman effect (splitting in magnetic field) or Stark effect (splitting in electric field). Cannot explain fine structure (small additional splittings in lines). Does not align with full quantum mechanics of electrons as waves/particles (addressed later in quantum model). Bohr’s model works for all atoms. It works quantitatively only for single-electron systems (hydrogen and hydrogen-like ions such as He⁺, Li²⁺). It fails for multi-electron atoms due to electron–electron repulsions and other effects. Series order by n1 (1→5): Little Boys Play Briskly, Perfectly — Lyman (1), Balmer (2), Paschen (3), Brackett (4), Pfund (5). neet-alert Common traps: (1) Mixing up n1 and n2 in the Rydberg formula — n2 must be greater than n1. (2) Forgetting Z 2 for hydrogen-like ions. (3) Missing the sign: E n is negative; ΔE is positive for absorption, negative for emission. Must-know terms Stationary state An allowed circular orbit of fixed energy where the electron does not radiate. Only specific orbits are permitted, set by mvr = nh/2π; n is an integer (1, 2, 3,...). Quantized orbit Rotational momentum of the electron in orbit; in Bohr’s model it takes only nh/2π values. Angular momentum Rydberg constant ( R H ) 1.097 × 10 7 m −1 for hydrogen; appears in 1/λ calculations. Principal quantum number (n) Orbit label in Bohr’s model; higher n means larger radius and higher (less negative) energy. Ionization energy Energy to remove an electron from the ground state (n=1) to n=∞ (for H: 13.6 eV). Any state with n ≥ 2; less tightly bound (energy closer to 0) than ground state. Excited state Lowest energy level (n = 1) with E 1 = −13.6 eV for hydrogen. Ground state Line spectrum A set of discrete wavelengths (lines) emitted/absorbed by atoms due to quantized energy levels. Hydrogen-like ion Any one-electron species (e.g., He⁺, Li²⁺) obeying Bohr formulas with Z > 1.