Bohr's Model & Hydrogen Spectrum Bohr's Model & Hydrogen Spectrum Atoms radiate light in sharp lines, not in a continuous rainbow. Rutherford’s nuclear model said electrons orbit the nucleus like planets, but a charged particle in circular motion should radiate energy and spiral into the nucleus. That would make atoms unstable and spectra continuous—contradicting experiments. Bohr proposed a bold fix: only certain circular orbits are allowed, and in those allowed orbits the electron does not radiate. Energy is emitted or absorbed only when an electron jumps between two allowed orbits, giving photons with energy exactly equal to the difference of the orbital energies. This simple quantization explains why we see discrete spectral lines for hydrogen and hydrogen-like ions. From this idea flow a set of powerful results: a formula for the radius of the nth orbit, the electron’s speed, total energy, and the Rydberg relation for wavelengths. Together, they reproduce the Lyman, Balmer, Paschen, Brackett, and Pfund series and predict series limits. The model is not the final story (it ignores electron wave nature until de Broglie reinterprets it, and fails for multi-electron atoms, fine structure, and external fields), but it remains a crucial stepping stone. In exams, you will often be asked to compute the radius or energy of an orbit, identify spectral series from given wavelengths, order lines by wavelength or frequency, and estimate ionization energies. This concept also connects back to Rutherford scattering (distance of closest approach) and forward to nuclear binding energy scales, letting you compare atomic and nuclear energies cleanly. remember Think of a guitar string: only certain standing-wave notes ring cleanly. Bohr’s allowed orbits are like those notes—only certain electron “tracks” are stable. Jumps between tracks change the note (photon) you hear (see). Historically, the gold-foil experiment revealed a tiny, massive, positively charged nucleus. But classical electrodynamics then predicted a fatal collapse of atomic orbits. Spectroscopy showed something else: sharp, element-specific lines. Balmer even wrote an empirical formula for visible hydrogen lines before any theory explained why. Bohr’s postulates impose quantization right on the electron’s orbital angular momentum and energy. They are ad hoc in form but extraordinarily predictive for hydrogen. A circular, stationary (non-radiating) electron orbit around the nucleus allowed by Bohr’s quantization rules. Bohr orbit A positive integer (1, 2, 3, …) labeling the allowed Bohr orbit; larger n means larger radius and higher (less negative) energy. Principal quantum number n The lowest energy state with n = 1 for hydrogen-like atoms. Ground state Any allowed state with n ≥ 2; the atom can transition down and emit photons. Excited state Energy needed to take the electron from a bound state (usually n = 1) to n → ∞ (free). For hydrogen, 13.6 eV from ground state. Ionization energy A constant that appears in the hydrogen spectrum formula for wave number; R ≈ 1.097 × 10 7 m⁻¹ for hydrogen. Rydberg constant R The highest-frequency (shortest-wavelength) line in a spectral series as n i → ∞, corresponding to ionization from the final level n f . Series limit Hydrogen-like ion A single-electron species such as He⁺, Li²⁺, Be³⁺. Replace Z = 1 by nuclear charge Z in Bohr formulas. Spectral series (Lyman/Balmer/…) Families of lines produced by transitions that end on a fixed lower level n f (1, 2, 3, 4, 5 for Lyman, Balmer, Paschen, Brackett, Pfund). Bohr’s postulates (operational form) Electrons revolve in certain stable circular orbits without radiating energy. Orbital angular momentum is quantized: m v r = nh/2π, with n = 1, 2, 3, … Radiation is emitted or absorbed only when an electron jumps between two allowed orbits; photon energy hν equals the energy difference. We will work with the electrostatic attraction between the nucleus (+Ze) and electron (−e) and the centripetal requirement for circular motion. The angular momentum quantization then fixes which radii and speeds are allowed. From these we derive the total energy for the nth orbit and show that it scales as −Z²/n², explaining both stability (discrete negative energies) and line spectra (discrete gaps). Orbital angular momentum is restricted to integer multiples of h/2π. Bohr quantization De Broglie reinterpretation: the orbit circumference fits an integer number of matter wavelengths. Standing wave condition Angular momentum is quantized in integral multiples of h/2π, selecting stable orbits and enforcing discrete energies. Coulomb attraction equals centripetal force. Quantization of angular momentum. Substitute v into force balance. Bohr radius a0 gives the hydrogen ground-state radius. Speed scales as Z/n; for H, v 1 2.19 10 6 $ m/s. Bohr radii and speeds in the nth orbit Coulomb force provides centripetal force in a circular orbit Angular momentum quantized: m v r = n Non-relativistic electron; point-like nucleus; single electron r n = n 2 a 0 Z , v n = Z c n The radius grows with n² and shrinks with Z. Hydrogen-like ions have much tighter orbits (smaller r) and higher speeds for the same n because the nucleus pulls harder (larger Z). Note the appearance of the fine-structure constant α, confirming that v is a modest fraction of c in low-n orbits for small Z, justifying the non-relativistic treatment for hydrogen. E n = -13.6 , eV , Z 2 n 2 Same as for radii and speeds Potential energy U(r) = -k Ze 2 / r Total energy in the nth Bohr orbit Kinetic and potential energies. From force balance. Virial-type relation in Coulomb orbits: E = -K = U/2. Insert quantized radius. Numerical value for Z = 1 defines 13.6 eV. Total energy of a bound electron is negative and scales as −Z²/n²; moving to higher n makes energy less negative (closer to zero). Negative total energy means the electron is bound. As n increases, E moves toward 0 from below; at n → ∞, the electron is free with E = 0 (by our convention). Ionization energy from a level n is | E n |. For hydrogen, ionization from ground state needs 13.6 eV; from n = 2 it needs 3.4 eV. Energy (eV) energy-level State label (n) control dependent Discrete horizontal levels at E n = −13.6 Z²/n² for Z = 1. Spacing shrinks with increasing n; dense near E = 0. Hydrogen energy levels: gaps get smaller as n grows; transitions between levels produce spectral lines. -13.6 Ground state -3.4 First excited Ionization limit A downward jump from n i to n f emits a photon with energy hν = E n i − E n f . Equivalently, its wavelength satisfies the Rydberg formula for hydrogen-like ions. Each spectral series fixes n f and lets n i vary over larger integers. Set Z = 1 for hydrogen. n f chooses the series (1: Lyman, 2: Balmer, 3: Paschen, 4: Brackett, 5: Pfund). Rydberg wave number formula Calculates the specific wavelength of the photon emitted when an electron transitions between two energy levels in a hydrogen-like ion. In a given series, as n i increases, the term 1/ n i ² shrinks and 1/λ approaches RZ²(1/ n f ²). Therefore λ decreases toward a finite minimum, the series limit. The lines crowd together near this limit, which is why the high-frequency end of each series looks dense. Hydrogen Spectral Series Series Name Transition ( n 1 ) Spectrum Region Wavelength Formula Longest/Shortest λ Lazy Boys Play Basketball Primarily: Lyman (UV), Balmer (Visible), Paschen/Brackett/Pfund (Infrared). Lyman Series Transition ( n 1=1 , n 2=2,3... ) Ultraviolet (UV) 1 = R ( 1 1 2 - 1 n 2 2 ) Longest = 4 3R ; Shortest = 1 R Balmer Series Transition ( n 1=2 , n 2=3,4... ) Visible 1 = R ( 1 2 2 - 1 n 2 2 ) Longest = 36 5R ; Shortest = 4 R Paschen Series Transition ( n 1=3 , n 2=4,5... ) Infrared (IR) 1 = R ( 1 3 2 - 1 n 2 2 ) Longest = 144 7R ; Shortest = 9 R Brackett Series Transition ( n 1=4 , n 2=5,6... ) Far Infrared 1 = R ( 1 4 2 - 1 n 2 2 ) Longest = 400 9R ; Shortest = 16 R Pfund Series Transition ( n 1=5 , n 2=6,7... ) Far Infrared 1 = R ( 1 5 2 - 1 n 2 2 ) Longest = 900 11R ; Shortest = 25 R hydrogen spectral series Hydrogen-like ions shift all wavelengths by a factor 1/Z² (shorter wavelengths for larger Z). For example, He⁺ has the same pattern of lines but compressed toward higher frequencies because the electron is more tightly bound. Applicability: Bohr’s formulas work quantitatively for single-electron species (H, He⁺, Li²⁺) in the absence of strong external fields and relativistic speeds. They do not capture fine structure, Zeeman/Stark splittings, or multi-electron screening. tip neet-alert Sign trap: Use E photon = E n i - E n f > 0 for emission. In the Rydberg formula, insist on n i > n f . If you accidentally swap them, you get negative 1/λ, which is unphysical. Nuclear binding energy BE = Δm × 931.5 MeV compares mass defect to energy via E = mc²; useful to contrast atomic vs nuclear energy scales. Mass–energy equivalence. Define mass defect as nucleon sum minus actual nuclear mass. Energy equivalent of 1 u in MeV. Convert mass defect to binding energy. Binding energy from mass defect and E = mc 2 BE = m , 931.5 , MeV Mass defect measured in atomic mass units (u) 1 u has energy equivalent 931.5 MeV Relates nuclear, not atomic, energetics For a head-on α–nucleus approach, r₀ = (1/4πϵ₀)(2Ze²)/K; used to estimate nuclear size scales in Rutherford scattering. r 0 = 1 4 0 2 Z e 2 K Head-on collision (impact parameter b = 0) Stationary heavy nucleus; only Coulomb repulsion Energy conserved Distance of closest approach in head-on α scattering Far away, only kinetic energy. At the turning point r0, all energy is potential. Solve for r0. Rutherford’s scattering scale (r₀ ~ 10⁻¹⁴ m for MeV α’s) is orders of magnitude smaller than Bohr’s orbital scale (a₀ ~ 10⁻¹⁰ m). This contrast shows why atomic spectra (eV energies) and nuclear processes (MeV energies) live on very different energy and length scales. r1, v1, E1 Z = 1, n = 1 a0 = 5.29×10⁻¹¹ m α ≈ 1/137, c = 3.00×10⁸ m/s E1 = −13.6 eV (to verify) easy Hydrogen ground state parameters Use r n = n² a0 / Z, v n = Z α c / n, and E n = −13.6 Z²/n² eV. Use 1/λ = R (1/ n f ² − 1/ n i ²). Wavelength of H-α line (Balmer series) medium Transition: n i = 3 → n f = 2 (hydrogen) R = 1.097×10⁷ m⁻¹ λ of emitted photon λ min for Lyman series of He⁺ Series limit: n i → ∞, so 1/λ min = R Z² (1/ n f ²). Shortest wavelength in the Lyman series of He⁺ Z = 2 (He⁺), n f = 1 R = 1.097×10⁷ m⁻¹ hard In a series, shorter wavelength corresponds to larger frequency and energy photons. For fixed n f , the shortest wavelength is the series limit ( n i → ∞), and the longest wavelength is the first line ( n i = n f + 1). Always check whether the question asks for shortest or longest wavelength and whether Z = 1 or higher. More negative means more tightly bound. The ground state (n = 1) is the most stable; energy must be supplied to climb up to less negative levels. More negative energy means the electron has less energy than in higher levels, so n = 1 is the least energetic and therefore most unstable. All Balmer lines are visible to the eye. Most prominent Balmer lines are visible, but as you approach the series limit the wavelengths slide into near-UV and may not be distinctly visible. Recall order of series and rough spectral regions. “Lazy Boys Play Brilliant Pranks” → Lyman ( n f =1, UV), Balmer ( n f =2, visible), Paschen ( n f =3, IR), Brackett ( n f =4, IR), Pfund ( n f =5, IR). Higher n means bigger, weaker-binding orbits: r ∝ n², v ∝ 1/n, E ∝ −1/n². Hydrogen-like ions simply scale each by Z (r shrinks by 1/Z, v grows ∝ Z, |E| grows ∝ Z²). remember Use the tool to fix n f and slide n i upward: note how the emitted wavelength decreases and levels bunch near the series limit. Increase Z to see all radii shrink (1/Z), energies deepen (Z²), and wavelengths shift to the ultraviolet. Useful constants (for quick estimates) a0 = 5.29 × 10⁻¹¹ m R = 1.097 × 10⁷ m⁻¹ α ≈ 1/137 hc ≈ 1240 eV·nm 1 eV = 1.602 × 10⁻¹⁹ J tip Unit conversions: For photon energy from wavelength, use E (eV) ≈ 1240/λ(nm). For SI work, convert λ in nm to m and eV to J using 1 eV = 1.602×10⁻19 J. Line positions in H Correct (Rydberg fits perfectly) Excellent quantitative agreement Line intensities Not predicted Requires quantum transition probabilities Fine structure splitting Absent Needs relativistic + spin–orbit corrections Zeeman/Stark effects Absent Requires quantum mechanics with external fields Multi-electron atoms Fails (no electron–electron interactions) Requires full quantum mechanics and screening Aspect Bohr model prediction Reality/limitation De Broglie’s idea makes Bohr’s rule feel natural: the electron behaves as a matter wave with wavelength λ = h/p. A stable orbit must allow a standing wave around the circumference, which enforces 2πr = nλ. Non-integer fits would destructively interfere, radiate, and die out; only standing waves survive as stationary states. De Broglie relation Wave–particle duality provides a physical picture for quantized orbits. This wavelength determines the required standing wave condition for the electron to maintain a stable, quantized orbit. A common spectral task is to identify the series from a given wavelength. Quickly compare λ to known landmarks: Balmer’s visible lines are around 656 nm (H-α), 486 nm (H-β), 434 nm (H-γ), 410 nm (H-δ). Anything much shorter than 364.6 nm is likely Lyman (UV), and much longer than about 800 nm falls into Paschen/Brackett/Pfund (IR), unless Z > 1 compresses the scale. Decide if the atom is hydrogen-like; note Z. Identify whether emission or absorption; set n i and n f accordingly ( n i > n f for emission). Use 1/λ = R Z² (1/ n f ² − 1/ n i ²); for series limits set n i → ∞. For energy in eV, use E = hc/λ with hc ≈ 1240 eV·nm; or use ΔE = 13.6 eV × Z²(1/ n f ² − 1/ n i ²). Check the spectral region and order-of-magnitude for sanity. Solving spectral problems (quick workflow) Series indexing trap: In Lyman, n f = 1 by definition; Balmer has n f = 2, etc. Do not set n f equal to the higher level by mistake. Longest λ in any series is always the first line with n i = n f + 1. neet-alert Reduced mass correction slightly shifts hydrogen’s Rydberg constant because the nucleus is not infinitely heavy. Replacing m by the reduced mass μ = mM/(m + M) nudges predicted wavelengths closer to experiment. For most NEET questions, the textbook value R = 1.097 × 10⁷ m⁻¹ suffices; remember that isotopes (H vs D) have tiny spectral shifts. Energy bookkeeping tips: In bound states, E = −K = U/2. If you know the electron speed in an orbit, you immediately know energies: K = ½mv², E = −K. When comparing different Z or n, scale intelligently—double Z multiplies |E| by 4 and halves r for the same n. Concept bridge to photoelectric effect: Bohr’s model explains atomic line energies, while photoelectric effect uses photon energies to eject electrons from metals. Both rely on E = hν and quantization but operate in different contexts—bound-state jumps versus work function thresholds. When you see a problem mixing atomic and nuclear numbers, check scales. Atomic transitions are a few eV (visible/UV/IR), while nuclear binding energies are MeV (gamma rays). If your computed photon from a Bohr transition comes out in MeV, a unit or Z² mistake is hiding in your work. Hydrogen-like series for Z > 1: replace R by RZ² or simply multiply the hydrogen wave number by Z². For example, the He⁺ Lyman limit is at 1/λ = 4R, so λ is one-quarter of hydrogen’s Lyman limit. All corresponding series lines scale the same way. Spectral intensities are not set by Bohr’s model; they depend on transition probabilities (quantum matrix elements). While not needed for NEET calculations, this explains why some lines (like H-α) appear stronger in emission spectra. Selection rules in modern quantum mechanics restrict which transitions are most likely (e.g., Δl = ±1). Bohr’s original model does not include orbital angular momentum quantum number l, but many observed strong lines are consistent with these rules, further refining spectral predictions. Absorption spectra: If a gas of hydrogen atoms in the ground state is illuminated by white light, only photons matching gaps from n = 1 upward are absorbed, creating dark lines at Lyman wavelengths in a continuous background. Emission lines appear when excited atoms decay. Series identification by wavelength bounds: For hydrogen, any λ shorter than 91.2 nm must be Lyman; between 91.2 nm and 364.6 nm lies Lyman tail and Balmer limit; 364.6–750 nm houses most Balmer lines; beyond ~800 nm, Paschen and higher dominate. Use these guardrails during time-pressured questions. Ionization from excited states: From n, the ionization energy is 13.6 Z²/n² eV. For example, to ionize He⁺ from n = 3 needs 13.6 × 4 / 9 ≈ 6.0 eV. If a photon causes ionization, any extra energy appears as the kinetic energy of the freed electron. Reverse engineering n from a wavelength: Given an observed λ, compute 1/λRZ² and look for a rational difference between 1/ n f ² and 1/ n i ². Start by guessing the series ( n f ) from the region, then solve for n i . If 1/λ is barely larger than RZ²/ n f ², you are near the series limit (large n i ). Comparing orbits: The ratio of radii r m / r n = (m²/n²)( Z n / Z m ) if Z differs, but within the same ion r m / r n = (m/n)². Similarly, the speed ratio v m / v n = (n/m). These clean ratios speed up many multiple-choice eliminations without full calculations. Practice pattern: Many NEET problems hide a quick Rydberg use. Convert the question into 1/λ or ΔE first, decide the series, check for Z² factors, and only then plug numbers. This habit cuts careless errors significantly. Quick recap Bohr radius a0 a 0 Characteristic hydrogen length scale, a0 ≈ 5.29 × 10⁻¹¹ m. Energy level E n For hydrogen-like, E n = −13.6 Z²/n² eV (bound states are negative). E n 1/λ = R Z²(1/ n f ² − 1/ n i ²), n i > n f . Rydberg formula Shortest wavelength (highest frequency) line as n i → ∞ for fixed n f . Series limit Single-electron ion; replace Z appropriately in Bohr relations. Hydrogen-like ion