Dual nature, atoms and nuclei for NEET

Modern physics (dual nature of radiation and matter, atoms, and nuclei) is where many NEET candidates pick up their surest physics marks. The chapters are short, the formulas are few, and the questions are mostly direct substitutions or ratios. A little care with units, especially electronvolts, goes a long way.

The main traps are conceptual: what intensity changes in the photoelectric effect and what it does not, and which quantities in the Bohr model grow or shrink with n.

The photoelectric effect

When light of high enough frequency falls on a metal, electrons are emitted. The experiments by Hertz, Hallwachs and Lenard showed four facts that the wave theory could not explain:

• There is a threshold frequency ν₀ for each metal. Below it, no electrons come out, however intense the light.
• The maximum kinetic energy of the electrons depends on the frequency, not the intensity.
• The photocurrent (number of electrons per second) is proportional to the intensity.
• Emission is practically instantaneous.

Einstein's explanation: light comes in photons of energy E = hν. One photon gives all its energy to one electron. Part of it (the work function φ₀) frees the electron; the rest is kinetic energy:

K_max = hν − φ₀ = eV₀, where V₀ is the stopping potential and φ₀ = hν₀.

The graph of V₀ against ν is a straight line with slope h/e, the same for every metal, and an intercept on the ν axis at ν₀.

Photon facts: E = hν = hc/λ, momentum p = h/λ, no rest mass. A handy conversion: E (in eV) ≈ 1240/λ (in nm).

Matter waves

de Broglie proposed that moving particles also have a wavelength:

λ = h/p = h/(mv) = h/√(2mK)

For a charge q accelerated from rest through V volts, K = qV, so λ = h/√(2mqV). For an electron, this simplifies to λ ≈ 1.227/√V nm.

• Heavier particles at the same speed have shorter wavelengths.
• Particles with the same momentum have the same de Broglie wavelength, whatever their mass.
• Everyday objects have wavelengths far too small to detect, which is why we never see a cricket ball diffract.

Atoms: Rutherford and Bohr

In the alpha-particle scattering experiment, most alpha particles passed straight through a thin gold foil, but a very few bounced back through large angles. Rutherford concluded that the positive charge and nearly all the mass of an atom are packed into a tiny nucleus.

The distance of closest approach for an alpha particle of kinetic energy K fired head-on at a nucleus of charge Ze is r₀ = k(2e)(Ze)/K. It gives an upper limit on the nuclear size.

Rutherford's model could not explain why atoms are stable or why they emit line spectra. Bohr's postulates fixed this for hydrogen:

1. Electrons move only in certain orbits in which they do not radiate.
2. In these orbits, angular momentum is quantised: mvr = nh/(2π).
3. An electron jumping from a higher level to a lower one emits a photon with hν = E_i − E_f.

For hydrogen-like atoms (one electron, nuclear charge Z):

For hydrogen: ground state −13.6 eV, first excited state (n = 2) −3.4 eV, n = 3 at −1.51 eV. The ionisation energy is 13.6 eV.

The hydrogen spectrum: 1/λ = RZ²(1/n_f² − 1/n_i²), with R ≈ 1.097 × 10⁷ m⁻¹.

An electron falling from level n to the ground state can produce at most n(n − 1)/2 different spectral lines.

The same energy levels appear in chemistry; see atomic structure for NEET.

Nuclei

A nucleus of mass number A contains Z protons and (A − Z) neutrons.

• Size: R = R₀ × ∛A, with R₀ ≈ 1.2 fm. Since volume ∝ A, the nuclear density is the same for all nuclei, about 2.3 × 10¹⁷ kg m⁻³.
• Nuclear force: very strong, short-range (a few femtometres), attractive, and roughly the same between any pair of nucleons (n–n, p–p, n–p).
• Mass defect: a nucleus weighs less than its separate nucleons. Δm = [Zm_p + (A − Z)m_n] − M.
• Binding energy: BE = Δm c². With masses in atomic mass units, 1 u ≡ 931.5 MeV.

Binding energy per nucleon is low for light nuclei, rises to a peak of about 8.75 MeV near A = 56 (iron), and falls slowly for heavy nuclei (about 7.6 MeV for uranium).

• Fission: a heavy nucleus such as U-235 splits into two middle-sized ones. The products are more tightly bound, so energy is released, roughly 200 MeV per fission.
• Fusion: light nuclei join into a heavier one, again moving towards the peak and releasing energy. It needs very high temperatures to overcome the repulsion between the nuclei, and it powers the Sun.

Worked numericals

Example 1: photoelectric effect

Light of wavelength 400 nm falls on a metal with work function 2.1 eV.

• Photon energy: 1240/400 = 3.1 eV.
• K_max = 3.1 − 2.1 = 1.0 eV, so the stopping potential is 1 V.
• Threshold wavelength: 1240/2.1 ≈ 590 nm. Any light longer than this ejects no electrons.

Example 2: de Broglie wavelength

(a) An electron is accelerated through 100 V. (b) Compare the wavelengths of a proton and an alpha particle accelerated through the same voltage.

• (a) λ = 1.227/√100 = 0.123 nm.
• (b) λ = h/√(2mqV) ∝ 1/√(mq). The alpha particle has 4 times the mass and 2 times the charge, so mq is 8 times larger. λ_p/λ_α = √8 = 2√2.

Example 3: the first Balmer line

Find the wavelength of the photon emitted when a hydrogen electron falls from n = 3 to n = 2.

• ΔE = 13.6 × (1/4 − 1/9) = 13.6 × 5/36 ≈ 1.89 eV.
• λ = 1240/1.89 ≈ 656 nm, the red H-alpha line.

Example 4: binding energy of helium-4

Use m(¹H) = 1.007825 u, m_n = 1.008665 u and m(⁴He) = 4.002603 u.

• Mass of parts: 2 × 1.007825 + 2 × 1.008665 = 2.015650 + 2.017330 = 4.032980 u.
• Δm = 4.032980 − 4.002603 = 0.030377 u.
• BE = 0.030377 × 931.5 ≈ 28.3 MeV, or about 7.07 MeV per nucleon.

Using the hydrogen atom's mass instead of the proton's keeps the electron masses balanced on both sides.

Example 5: nuclear radius

Find the radius of an aluminium-27 nucleus.

• R = 1.2 × ∛27 = 1.2 × 3 = 3.6 fm.

Practice MCQs

1. The energy of the n = 3 level of hydrogen is: (a) −13.6 eV (b) −3.4 eV (c) −1.51 eV (d) −0.85 eV
2. The intensity of light on a photo-surface is doubled at the same frequency. Then: (a) K_max doubles (b) the photocurrent doubles (c) the stopping potential doubles (d) nothing changes
3. The slope of the stopping potential versus frequency graph is: (a) h (b) e/h (c) h/e (d) φ₀/e
4. The ratio of the radius of the n = 2 orbit to the n = 1 orbit in hydrogen is: (a) 2 (b) 4 (c) 1/2 (d) 1/4
5. An electron falls from n = 4 to n = 1. The maximum number of spectral lines possible is: (a) 3 (b) 4 (c) 6 (d) 10
6. The ionisation energy of He⁺ (Z = 2) is: (a) 13.6 eV (b) 27.2 eV (c) 40.8 eV (d) 54.4 eV
7. An electron and a proton have the same de Broglie wavelength. Which has more kinetic energy? (a) the electron (b) the proton (c) both equal (d) cannot be said
8. The ratio of nuclear densities of ²⁷Al and ⁶⁴Cu is: (a) 27 : 64 (b) 3 : 4 (c) 1 : 1 (d) 64 : 27

Answers

1. (c) −13.6/9 ≈ −1.51 eV.
2. (b) More photons per second eject more electrons; each still gets the same energy.
3. (c) eV₀ = hν − φ₀, so V₀ = (h/e)ν − φ₀/e.
4. (b) r ∝ n².
5. (c) n(n − 1)/2 = 4 × 3/2 = 6.
6. (d) 13.6 × Z² = 13.6 × 4 = 54.4 eV.
7. (a) Same λ means the same p. K = p²/2m, so the lighter electron has more KE.
8. (c) Nuclear density does not depend on A.

What to do next

• Put these on one card: K_max = hν − φ₀, 1240/λ, λ = h/p, 1.227/√V, Eₙ = −13.6Z²/n², 1 u = 931.5 MeV.
• Solve ten photoelectric and de Broglie questions entirely in eV and nm, without converting to joules.
• Sketch the binding-energy-per-nucleon curve and mark where fission and fusion release energy.
• Move on to semiconductor electronics for NEET, the last physics unit.

For a final pass over the whole physics syllabus, use the NEET physics revision plan.