Atoms: NEET notes

1. Early models of the atom

NCERT §12.1

• Thomson's discharge experiments of 1897 showed that atoms of every element contain electrons, and that these electrons are identical from one element to the next. A whole atom is electrically neutral, so it must also carry an equal amount of positive charge.
• Thomson's own model (1898): the positive charge is spread evenly through the whole volume of the atom, with the electrons embedded in it like seeds in a watermelon. It is known as the plum pudding model. Later experiments showed the charges are arranged very differently.
• Solids, liquids and dense gases at any temperature give out a continuous spread of wavelengths, with different intensities. This radiation comes from atoms and molecules oscillating under the pull of their neighbours.
• A rarefied gas heated in a flame or excited in a glow tube (a neon sign, a mercury vapour lamp) gives out only certain discrete wavelengths: a spectrum of bright lines. Its atoms are far apart, so the light comes from individual atoms, not from interactions between them.
• Every element has its own characteristic spectrum; hydrogen always gives lines at fixed relative positions. This hinted that the spectrum is tied to the atom's internal structure. In 1885 Balmer found a simple empirical formula for the wavelengths of one group of hydrogen lines.
• Rutherford, a former research student of Thomson, proposed an alpha-particle scattering experiment in 1906. Geiger and Marsden performed it around 1911; Marsden was then a 20-year-old student without his bachelor's degree.
• From its results came Rutherford's nuclear (planetary) model: all the positive charge and most of the mass sit in a tiny central nucleus, with electrons revolving round it like planets round the sun. The model could not explain why atoms emit only discrete wavelengths.

2. Alpha-particle scattering

NCERT §12.2

• Geiger and Marsden aimed a beam of 5.5 MeV alpha-particles from a radioactive ²¹⁴₈₃Bi source at a thin gold foil. Lead bricks collimated the particles into a narrow beam, and the whole apparatus sat in a vacuum chamber.
• The gold foil was 2.1 × 10⁻⁷ m thick. A rotatable detector, a zinc sulphide screen viewed through a microscope, caught the scattered particles: each one made a brief flash of light (a scintillation), so the number scattered could be counted at each angle.
• Most alpha-particles went straight through the foil without any collision. Only about 0.14% were scattered by more than 1°, and about 1 in 8000 were deflected by more than 90°.
• To turn an alpha-particle back, a very large repulsive force is needed. Rutherford argued that this is possible only if most of the atom's mass and all its positive charge are packed tightly at the centre, so an alpha-particle can come very close to that charge without entering it. The data agreed, so Rutherford is credited with discovering the nucleus.
• In Rutherford's model the nucleus holds all the positive charge and most of the mass, and the electrons move in orbits some distance away. The experiments put the nuclear size at about 10⁻¹⁵ m to 10⁻¹⁴ m, while kinetic theory gave the atom a size of about 10⁻¹⁰ m: 10,000 to 100,000 times larger. Most of an atom is empty space.
• That is why most alpha-particles pass straight through. Only one that comes near a nucleus feels the intense field there and is scattered through a large angle. Electrons, being so light, hardly affect the alpha-particles.
• Example 12.1: the electron's orbit (about 10⁻¹⁰ m) is 10⁵ times the nuclear radius (about 10⁻¹⁵ m). A solar system in the same proportion would put the earth at 10⁵ × 7 × 10⁸ m = 7 × 10¹³ m from the sun, more than 100 times its real orbital radius of 1.5 × 10¹¹ m. An atom is far emptier than the solar system.

3. Alpha-particle trajectory

NCERT §12.2, §12.2.1

• The foil is so thin that an alpha-particle is taken to suffer at most one scattering, so it is enough to follow one alpha-particle past one nucleus.
• An alpha-particle is a helium nucleus, with charge 2e and the mass of a helium atom. A gold nucleus has charge Ze with Z = 79, and it is about 50 times heavier than the alpha-particle, so it is treated as stationary.
• The path follows from Newton's second law and Coulomb's law. The repulsive force at separation r is F = (1/4πε₀)(2e)(Ze)/r² (Eq. 12.1), directed along the line joining them; its size and direction change all the time as the particle approaches and recedes.
• The impact parameter b is the perpendicular distance of the alpha-particle's initial velocity from the centre of the nucleus. In a beam, the particles have nearly the same kinetic energy but a spread of impact parameters, so they scatter in different directions with different probabilities.
• Small b means a close pass and a large scattering angle θ. A head-on collision has the smallest b and the particle rebounds (θ ≈ π). A large b leaves it nearly undeviated (θ ≈ 0).
• Only a small fraction of alpha-particles rebound, so few collisions are head-on. That means the mass and positive charge are concentrated in a small volume, and Rutherford scattering gives an upper limit to the size of the nucleus.
• Distance of closest approach (head-on): all the kinetic energy becomes electric potential energy, K = (1/4πε₀)(2e)(Ze)/d, so d = 2Ze²/(4πε₀K).
• Example 12.2: for a 7.7 MeV alpha-particle (1.2 × 10⁻¹² J, the most energetic of natural origin), d = 3.84 × 10⁻¹⁶ Z m, and for gold (Z = 79) d = 3.0 × 10⁻¹⁴ m = 30 fm (1 fm = 10⁻¹⁵ m). The gold nucleus is therefore smaller than 30 fm; its actual radius is about 6 fm, so the alpha-particle turns back without touching it.

4. Electron orbits

NCERT §12.2.2

• Rutherford's atom is a neutral sphere with a very small, massive, positive nucleus at the centre and electrons revolving round it in dynamically stable orbits.
• In hydrogen, the electrostatic attraction supplies the centripetal force: mv²/r = (1/4πε₀)e²/r² (Eq. 12.2). So the orbit radius and the speed are linked by r = e²/(4πε₀mv²) (Eq. 12.3).
• Kinetic energy K = ½mv² = e²/(8πε₀r); potential energy U = −e²/(4πε₀r). The minus sign in U says the force points towards the nucleus (along −r).
• Total energy E = K + U = −e²/(8πε₀r) (Eq. 12.4). So K = −E and U = 2E: the potential energy is twice the total energy, and the kinetic energy is its magnitude.
• E is negative, which means the electron is bound to the nucleus. With a positive total energy the electron would not follow a closed orbit.
• Example 12.3: 13.6 eV separates a hydrogen atom into a proton and an electron, so E = −13.6 eV = −2.2 × 10⁻¹⁸ J. Then r = −e²/(8πε₀E) = 5.3 × 10⁻¹¹ m, and with m = 9.1 × 10⁻³¹ kg the speed is v = e/√(4πε₀mr) = 2.2 × 10⁶ m/s.

5. Atomic spectra

NCERT §12.3

• An atomic gas or vapour excited at low pressure, usually by passing a current through it, emits only certain specific wavelengths. This is an emission line spectrum: bright lines on a dark background.
• Each element's line spectrum is its own, so it works like a fingerprint for identifying a gas.
• Pass white light through the same gas and look at the transmitted light with a spectrometer: dark lines appear in the otherwise continuous spectrum. This is the absorption spectrum of the gas.
• The dark absorption lines fall at exactly the wavelengths of the bright emission lines of that gas: an atom absorbs the same frequencies it emits.
• Contrast: a hot solid gives a continuous spectrum; a rarefied gas of separate atoms gives lines.

6. Why Rutherford's atom fails

NCERT §12.4

• Rutherford's atom copies the sun–planet system, but the forces differ: planets are held by gravity, while the nucleus and electron are charged and interact by Coulomb's law.
• An electron moving in a circle is always accelerating (centripetally). Classical electromagnetic theory says an accelerating charge radiates electromagnetic waves, so the electron should keep losing energy.
• Losing energy, it would spiral inwards and fall into the nucleus (Fig. 12.6). Such an atom cannot be stable, yet real atoms are.
• Classically, the frequency of the emitted wave equals the frequency of revolution. As the electron spirals in, its angular velocity keeps changing, so the emitted frequency changes continuously: the atom would give a continuous spectrum, not the observed line spectrum.
• Example 12.4: with r = 5.3 × 10⁻¹¹ m and v = 2.2 × 10⁶ m/s, the revolution frequency is ν = v/2πr ≈ 6.6 × 10¹⁵ Hz; classically this would be the initial frequency of the emitted light. (NCERT prints the speed as 2.2 × 10⁻⁶ m/s in this example; the value from Example 12.3 is 2.2 × 10⁶ m/s.)
• Both earlier models are unstable: Thomson's electrostatically, Rutherford's because its orbiting electrons radiate.

7. Bohr's postulates

NCERT §12.4

• Niels Bohr spent several months in Rutherford's laboratory in 1912 and accepted the nuclear model. In 1913 he concluded that classical electromagnetism, so successful on a large scale, cannot be applied to processes inside the atom, and he combined classical and early quantum ideas in three postulates.
• First postulate: an electron can revolve in certain stable orbits without radiating, contrary to classical theory. Each such stationary state of the atom has a definite total energy.
• Second postulate: the allowed orbits are those whose angular momentum is a whole multiple of h/2π, L = nh/2π (Eq. 12.5), where h is Planck's constant (NCERT quotes 6.6 × 10⁻³⁴ J s here). Angular momentum is quantised.
• Third postulate: an electron may jump from one stationary orbit to another of lower energy, emitting a photon whose energy equals the difference: hν = Ei − Ef (Eq. 12.6), with Ei > Ef.
• Combining L = nh/2π with the force balance gives the radius of the nth orbit, rn = (n²/m)(h/2π)²(4πε₀/e²) (Eq. 12.7), so rn ∝ n². For n = 1 this is the Bohr radius a₀ = 5.3 × 10⁻¹¹ m.
• Putting rn into E = −e²/(8πε₀r) gives En = −me⁴/(8n²ε₀²h²) (Eq. 12.8), which is −2.18 × 10⁻¹⁸ J/n² (Eq. 12.9), or En = −13.6 eV/n² (Eq. 12.10).
• Since v = nh/(2πmrn) and rn ∝ n², the speed goes as vn ∝ 1/n: 2.2 × 10⁶ m/s in the first orbit, half that in the second.
• The negative En means the electron is bound; energy must be supplied to take it infinitely far from the proton.

8. Energy levels

NCERT §12.4.1

• The atom's energy is lowest (most negative) with the electron in the innermost orbit, n = 1. For n = 2, 3, … the magnitude of E shrinks, so the energy rises in the outer orbits.
• The lowest state is the ground state: n = 1, orbit radius a₀, E₁ = −13.6 eV. The least energy that frees the electron from the ground state, 13.6 eV, is the ionisation energy of hydrogen, and Bohr's prediction agrees very well with experiment.
• At room temperature most hydrogen atoms are in the ground state. Energy from collisions (for example with electrons) can lift the electron to a higher level: the atom is then in an excited state.
• E₂ = −3.40 eV, so excitation to the first excited state needs E₂ − E₁ = 10.2 eV. E₃ = −1.51 eV, so the second excited state (n = 3) needs 12.09 eV.
• From an excited state the electron can fall back to a lower level, emitting a photon. The higher the excitation (the larger n), the less energy is needed to free the electron.
• Fig. 12.7 stacks the levels as horizontal lines, numbered by n from the bottom up in order of rising energy. The top, n = ∞, has E = 0: the electron removed to infinity and at rest. The levels crowd closer together as n increases.
• Above E = 0 the electron is free and can have any energy: a continuum of states.

9. Line spectra of hydrogen

NCERT §12.5

• A transition from a higher level ni to a lower level nf (nf < ni) sends out a photon of frequency νif with hνif = Eni − Enf (Eq. 12.11).
• Both ni and nf are whole numbers, so only certain energy differences, and so only certain discrete frequencies, are possible: this is the line spectrum.
• Emission lines come from electrons dropping from higher to lower states. Absorption happens when an atom takes in a photon whose energy exactly matches the gap to a higher state.
• Light with a continuous range of frequencies passed through a rarefied gas therefore comes out with dark absorption lines at the frequencies the atoms absorbed.
• For hydrogen, hν = 13.6 eV (1/nf² − 1/ni²). Using hc ≈ 1240 eV nm: 2 → 1 gives 10.2 eV (about 122 nm, ultraviolet); 3 → 1 gives 12.09 eV (about 103 nm); 3 → 2 gives 1.89 eV (about 656 nm, red).
• Bohr's explanation of the hydrogen spectrum was a major achievement that drove the development of quantum theory, and he received the Nobel Prize in Physics in 1922.

10. De Broglie's explanation

NCERT §12.6

• Bohr's second postulate, L = nh/2π, is the most puzzling one: why only whole multiples of h/2π? Louis de Broglie explained it in 1923, ten years after Bohr's model.
• De Broglie treated the orbiting electron as a particle wave (λ = h/p). Davisson and Germer confirmed the wave nature of electrons experimentally in 1927.
• On a plucked string only wavelengths with nodes at the ends survive, as standing waves; the others interfere with themselves on reflection and die away. In the same way, only an electron wave that fits the orbit exactly can persist.
• The condition is that the circumference holds a whole number of wavelengths: 2πrn = nλ, n = 1, 2, 3 … (Eq. 12.12). Fig. 12.8 shows four wavelengths fitted round the orbit (n = 4).
• For speeds much less than c, λ = h/mvn, so 2πrn = nh/mvn, which rearranges to mvnrn = nh/2π: exactly Bohr's quantum condition.
• So the quantised orbits and energy levels come from the wave nature of the electron: only resonant standing waves survive.

11. Limits of the Bohr model

NCERT §12.6

• Bohr's model, with its planet-like electron, correctly predicts the gross features of hydrogenic atoms, especially the frequencies they emit or absorb.
• A hydrogenic atom has a nucleus of charge +Ze and a single electron: hydrogen, singly ionised helium (He⁺), doubly ionised lithium (Li²⁺).
• Limit (i): it cannot be extended even to two-electron atoms such as helium. Each electron is attracted by the nucleus and also repelled by every other electron, and the model contains no electron–electron force.
• In the solar system, planet–planet gravity is tiny next to the sun's pull. In an atom, the electron–electron force is of the same order as the electron–nucleus force, because the charges and distances are comparable. That is why the planet picture fails for many-electron atoms.
• Limit (ii): it predicts the frequencies of hydrogen's lines but not their relative intensities; some visible lines are strong and others weak, because some transitions are favoured over others.
• Complex atoms need quantum mechanics, which gives a fuller picture; in it Bohr's orbits become regions where the electron is found with high probability.

Must-know facts

1. Thomson (1898): plum pudding model, positive charge spread through the atom with electrons embedded.
2. Geiger–Marsden (about 1911): 5.5 MeV alpha-particles from ²¹⁴₈₃Bi on gold foil 2.1 × 10⁻⁷ m thick; ZnS screen and microscope.
3. About 0.14% scattered by more than 1°; about 1 in 8000 by more than 90°.
4. Nucleus 10⁻¹⁵ to 10⁻¹⁴ m; atom about 10⁻¹⁰ m, 10,000 to 100,000 times larger.
5. Small impact parameter → large scattering; head-on (b smallest) → rebound, θ ≈ π.
6. Closest approach d = 2Ze²/(4πε₀K); 7.7 MeV alpha on gold: d = 3.0 × 10⁻¹⁴ m = 30 fm; gold radius about 6 fm.
7. Gold: Z = 79, about 50 times heavier than an alpha-particle.
8. Hydrogen orbit: r = e²/(4πε₀mv²); E = −e²/(8πε₀r); K = −E, U = 2E.
9. Ground state: r = 5.3 × 10⁻¹¹ m, v = 2.2 × 10⁶ m/s, E = −13.6 eV.
10. Classical revolution frequency in hydrogen's ground orbit: about 6.6 × 10¹⁵ Hz.
11. Bohr: stationary orbits, L = nh/2π, hν = Ei − Ef.
12. rn ∝ n², vn ∝ 1/n, En = −13.6/n² eV = −2.18 × 10⁻¹⁸/n² J.
13. E₂ = −3.40 eV, E₃ = −1.51 eV; excitation 10.2 eV (1→2) and 12.09 eV (1→3).
14. Ionisation energy of hydrogen: 13.6 eV; n = ∞ has E = 0; continuum above E = 0.
15. Absorption lines fall at the same wavelengths as the gas's emission lines.
16. de Broglie (1923): 2πrn = nλ gives mvr = nh/2π; Davisson–Germer confirmed electron waves in 1927.
17. Bohr model works only for hydrogenic atoms (H, He⁺, Li²⁺) and cannot give line intensities.
18. Bohr received the Nobel Prize in Physics in 1922.

Common traps

Formulas

Key terms

Plum pudding model

Thomson's atom: positive charge spread through the whole atom, electrons embedded in it.

Nuclear model

Rutherford's atom: all the positive charge and most of the mass in a tiny central nucleus.

Scintillation

A brief flash of light when an alpha-particle strikes a zinc sulphide screen.

Impact parameter

Perpendicular distance of an alpha-particle's initial velocity line from the centre of the nucleus.

Scattering angle

The angle θ between the alpha-particle's initial and final directions.

Distance of closest approach

Separation at which a head-on alpha-particle momentarily stops before turning back.

Emission line spectrum

Bright lines at specific wavelengths on a dark background, from an excited rarefied gas.

Absorption spectrum

Dark lines in a continuous spectrum at the wavelengths a gas absorbs.

Stationary state

An allowed orbit in which the electron does not radiate and the atom has a definite energy.

Principal quantum number

The integer n labelling Bohr's orbits and energy levels.

Bohr radius

Radius of hydrogen's innermost orbit, a₀ = 5.3 × 10⁻¹¹ m.

Ground state

The lowest-energy state, n = 1; −13.6 eV for hydrogen.

Excited state

Any state with n > 1, reached by collisions or by absorbing a photon.

Ionisation energy

Least energy to free the electron from the ground state: 13.6 eV for hydrogen.

Hydrogenic atom

A nucleus of charge +Ze with one electron, such as H, He⁺ or Li²⁺.