Atoms — NEET Notes

Alpha-particle scattering — the experiment that opened the atom

At the turn of the twentieth century, J.J. Thomson's plum-pudding model was the accepted picture of the atom: a diffuse cloud of positive charge with electrons embedded in it like raisins in a pudding. The model was electrically reasonable but completely untested. In 1909, on Rutherford's instructions, Hans Geiger and Ernst Marsden — Marsden was only twenty — set up a scattering experiment that would settle the question.

A radioactive bismuth source (²¹⁴Bi) emitted 5.5 MeV alpha-particles, which were collimated by lead bricks into a thin pencil and aimed at a gold foil only 2.1 × 10⁻⁷ m thick. A zinc-sulphide screen mounted on a rotatable microscope detected the scattered particles by flashes of light (scintillations). The whole apparatus sat inside an evacuated chamber so the alphas would not be deflected by air molecules.

If the Thomson model had been right, every alpha-particle should have ploughed through almost undeflected, since the positive charge would have been spread too thinly to deflect a fast 7 MeV projectile. What Geiger and Marsden saw was very different. Most alpha-particles passed straight through, but a small fraction — about 0.14% — scattered by more than 1°, and roughly 1 in 8000 bounced back through angles greater than 90°. A few even returned to within a degree of the source.

"It was as if you fired a fifteen-inch shell at a piece of tissue paper and it came back and hit you."

Ernest Rutherford on the recoiled alpha-particles

The recoil was decisive. To turn a 7.7 MeV alpha-particle around requires a tremendously concentrated repulsive force. That force could exist only if the entire positive charge of the atom — and almost all of its mass — were squeezed into an extraordinarily small region at the centre of the atom. Rutherford named this region the nucleus.

Rutherford's nuclear atom

From the scattering geometry Rutherford could even estimate the size of the nucleus. By equating the kinetic energy of the alpha-particle to the Coulomb potential at its turning point, the distance of closest approach for a head-on collision with a gold nucleus (Z = 79) comes out to about 3.0 × 10⁻¹⁴ m, or 30 fm. The real radius of the gold nucleus is closer to 6 fm — the rest of the gap reflects the fact that the alpha-particle never actually touches the nucleus, it stops short and is repelled back. Either way, the nucleus is roughly 10⁻¹⁵ m to 10⁻¹⁴ m across, while atoms (measured by kinetic theory) are about 10⁻¹⁰ m wide.

Rutherford's nuclear model can be stated in one sentence: the atom is an electrically neutral sphere whose entire positive charge and nearly all the mass sit in a tiny central nucleus, with electrons revolving around it in orbits. Coulomb attraction supplies the centripetal force, just as gravity does in the solar system. For a hydrogen atom this means

F_e = (1 / 4πε₀) · (e² / r²)  =  m v² / r

Coulomb attraction = centripetal force

The total energy of the bound electron (KE + PE) is negative — a hallmark of any bound system — and equals −e²/(8πε₀r). The negative sign tells us that energy must be supplied to free the electron from the nucleus.

Why Rutherford's classical atom cannot stand

Rutherford's model explained why most alphas pass through, why a few bounce back, and why atoms are neutral — but it also contained a fatal contradiction with classical electromagnetism. A revolving electron is continuously accelerated (centripetal acceleration). According to Maxwell's theory, any accelerated charge must radiate electromagnetic waves and so must lose energy. The electron should therefore spiral inwards in a fraction of a second — about 10⁻¹¹ s by the standard estimate — and collapse into the nucleus.

Two glaring contradictions follow. First, atoms in fact are stable; you are made of them. Second, as the electron spirals in, its orbital frequency would change continuously, so the emitted radiation should be a continuous spectrum across all frequencies. But what is observed is the opposite: hot rarefied gases emit at only a small set of discrete wavelengths — sharp bright lines on a dark background — a line spectrum.

Atomic spectra — line spectra and the hydrogen fingerprint

Each element, when its vapour is excited in a discharge tube, emits light at a characteristic set of wavelengths. The result is an emission line spectrum — bright lines on a dark background, fixed for that element, like a chemical fingerprint. Pass white light through a rarefied gas instead and a continuous spectrum will come out with dark gaps at exactly those same wavelengths; this is the corresponding absorption spectrum. The Fraunhofer lines in sunlight, which Kirchhoff used to identify sodium, hydrogen, iron and 60 other terrestrial elements in the Sun's chromosphere, are a famous absorption spectrum.

Hydrogen — the simplest atom, with a single electron and a single proton — gave a spectrum with stunning regularity. In 1885 the Swiss schoolteacher Johann Jakob Balmer found a simple empirical formula that produced the wavelengths of the visible hydrogen lines. Bohr's task, twenty-eight years later, was to derive Balmer's formula from first principles, and in doing so to explain why atoms radiate only at specific frequencies in the first place.

Bohr's three postulates

Niels Bohr had worked briefly in Rutherford's laboratory in 1912 and accepted the nuclear atom. To rescue it from the classical instability, he proposed in 1913 three bold postulates that fused classical mechanics with the new quantum ideas of Planck and Einstein. Of all the postulates, the second — that angular momentum is quantised — is the deepest, and the one NEET tests most often.

"The angular momentum of an electron in a stationary orbit is an integral multiple of h/2π."

Bohr's second postulate — the heart of the quantum atom

It is worth pausing on the audacity of these statements. Postulate 1 violates classical electromagnetism outright by asserting that an accelerated electron simply does not radiate in stationary orbits. Postulate 2 picks out a discrete set of radii from a classical continuum. Postulate 3 explains line spectra: only photons whose energy matches an allowed gap can be emitted or absorbed, so the spectrum can only ever contain a discrete set of wavelengths. The semi-classical hybrid worked because Bohr was willing to ignore the contradictions until experiment caught up.

Deriving the Bohr radius and the energy formula

The Bohr model is one of NEET's favourite derivations because every step is a single line of algebra. Start with Newton's second law applied to the Coulomb force on the orbiting electron (Postulate 1's stable orbit condition), pair it with Postulate 2's quantisation rule, and the radius and energy of every allowed orbit drop out. The derivation is reproduced below in the four-step form NEET asks most.

Energy-level diagram and ionisation energy

From E_n = −13.6 / n² eV the energy of every allowed state can be written down by inspection. The lowest energy state, n = 1, is the ground state at E₁ = −13.6 eV. Higher states form an infinite sequence that converges from below towards zero: E₂ = −3.40 eV, E₃ = −1.51 eV, E₄ = −0.85 eV, and so on. At n = ∞ the electron has zero energy — that is, it has just escaped the nucleus and is free. The crowding of levels near zero is the reason high-energy hydrogen transitions cluster towards a series limit.

To excite hydrogen from the ground state to the first excited state (n = 2) needs E₂ − E₁ = −3.40 − (−13.6) = 10.2 eV. To excite it to n = 3 needs 12.09 eV. To ionise it completely needs 13.6 eV. Any of these numbers can appear on NEET. Note also that for hydrogenic ions like He⁺ or Li²⁺, the formula scales: E_n = −13.6 Z² / n² eV and r_n = (n²/Z) a₀. NEET 2020 asked specifically which atom the Bohr model is not valid for — the answer is any multi-electron species (Ne⁺ has many electrons; He⁺ has just one and is fine).

The five spectral series of hydrogen

When a hydrogen atom in an excited state n₂ drops to a lower state n₁, a photon is emitted whose frequency is fixed by ν = (E_{n₂} − E_{n₁}) / h. Translating energy to wavelength, the result is the celebrated Rydberg formula:

1 / λ  =  R (1 / n₁²  −  1 / n₂²)

Rydberg formula — R = 1.097 × 10⁷ m⁻¹, n₂ > n₁

Fixing n₁ defines a whole series. The five named series exhaust the hydrogen line spectrum.

de Broglie's explanation of Bohr's second postulate

Bohr's quantisation rule, L = nh/2π, was always the least motivated of the three postulates. Why that condition, and not some other? The puzzle stayed open for ten years until Louis de Broglie supplied the answer in 1923, by treating the orbiting electron not as a particle but as a wave.

De Broglie's hypothesis was that every material particle has an associated wavelength λ = h / p = h / mv. An electron moving in a circular orbit therefore has a wavelength, and the orbit is stable only if the wave reinforces itself on each circuit — that is, if the electron's wavefront comes back into phase with itself after travelling once around the circumference. This is the condition for a standing wave on a closed loop. A standing wave on a string requires an integer number of wavelengths between its endpoints; a standing wave on a circular orbit requires an integer number of wavelengths around its perimeter:

2 π r_n  =  n λ  =  n · (h / m v_n)

de Broglie standing-wave condition on a Bohr orbit

Rearrange: m v_n r_n = n h / 2π. That is precisely Bohr's second postulate. The integer n is not pulled out of the air; it is the number of de Broglie wavelengths that fit around the orbit. Orbits with non-integer fits interfere destructively with themselves and cannot survive. The quantisation of angular momentum, the discrete orbits, and the line spectrum of hydrogen all flow from the wave nature of the electron — a hypothesis confirmed in 1927 by Davisson and Germer when they diffracted electrons off a nickel crystal.

Limitations of the Bohr model

Bohr's model was a landmark, and his Nobel Prize in 1922 was deserved. But it is not the last word. Three problems matter for NEET:

First, Bohr's model is valid only for hydrogenic atoms — atoms with a single electron, like H, He⁺, Li²⁺, Be³⁺. For multi-electron atoms the additional electron-electron repulsion cannot be reduced to a simple Coulomb central force, and Bohr's analysis breaks down. NEET 2020 tested this: Bohr is not valid for singly ionised neon (Ne⁺), which has nine electrons.

Second, the Bohr model correctly predicts the frequencies of hydrogen spectral lines but cannot account for their relative intensities. Some lines are bright, some are weak; some transitions are favoured, others almost forbidden. Bohr has nothing to say about this.

Third, Bohr's planet-like orbits violate the Heisenberg uncertainty principle, which forbids the simultaneous specification of an electron's position and momentum to arbitrary precision. The modern quantum-mechanical picture replaces orbits with orbitals — probability distributions where the electron may be found — and labels states with four quantum numbers (n, l, m, s) rather than just n. For a pure Coulomb potential (as in hydrogen) the energy still depends only on n, which is why Bohr's formula E_n = −13.6/n² eV survives unchanged into the full theory.