Nuclei

NucleiNEET Physics · Class 12 · NCERT Chapter 13

Overview Notes NEET Questions Interactive Learning

8 interactive concept widgets for Nuclei. Drag any slider, change any number, and watch the formula and the answer update live. Built so you understand how each NEET problem actually works, not just the final number.

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Nuclear radius and density

Mass defect and binding energy of common nuclei

Binding energy per nucleon vs A

Alpha, beta and gamma decay compared

Exponential decay: N(t), half-life and activity

Half-life, mean life and decay constant

Carbon-14 dating

Energy from nuclear fission and fusion

Nuclear size and density

How big is a nucleus, and why is the density nearly the same for every element.

Nuclear size

Nuclear radius and density

Pick A; the radius follows A^(1/3) and density stays nearly constant.

Nuclear radius scales as the cube root of A: R = R_0 A^(1/3). Nuclear density is nearly the same for every nucleus (~ 2.3 × 10¹⁷ kg/m³).

Mass number A: 56

Nuclear radius R

4.59 fm

R = R_{0} A^{1/3}, R_{0} = 1.2 fm

Nuclear density ρ

2.29e+17 kg/m³

nearly the same for every A

Try this

A = 1 (proton): R ≈ 1.2 fm.
A = 56 (Fe): R ≈ 4.6 fm.
A = 238 (U): R ≈ 7.4 fm.
Volume ∝ A. Mass ∝ A. So density is constant: a teaspoon of nuclear matter would weigh 5 × 10⁸ tons.

Mass defect, binding energy, BE curve

Mass defect translates to binding energy via E = mc². The BE curve peaks at Fe-56.

Mass defect and BE

Mass defect and binding energy of common nuclei

Pick a nucleus; see the mass defect and binding energy.

Mass defect = constituent masses - actual nuclear mass. Multiply by c² (using 1 u = 931.5 MeV) to get binding energy. Divide by A for BE per nucleon.

Deuteron H-2

Helium-4

Lithium-7

Carbon-12

Iron-56

Uranium-235

Uranium-238

Mass defect ΔM

0.0293 u

Binding energy

27.27 MeV

BE per nucleon

6.819 MeV

Δ M = Z m_{p} + (A - Z) m_{n} - M_{nuc}, BE = Δ M \times 931.5 MeV/u

Try this

Deuteron BE = 2.22 MeV; BE/n = 1.11 MeV (very loosely bound).
He-4: BE = 28.3 MeV, BE/n = 7.07 MeV. Very stable, hence alpha emission.
Fe-56: BE/n = 8.79 MeV. Maximum on the curve.
U-235 BE/n drops to 7.6 MeV; less bound than Fe, so fission releases energy.

BE per nucleon

Binding energy per nucleon vs A

The famous curve. Hover dots to read off values.

Hover or click a dot to see that nucleus. Curve peaks near Fe-56 at ~8.8 MeV per nucleon. Light → fusion gains energy (climbing). Heavy → fission gains energy (falling back to peak).

Fe-56: A = 56, BE/A = 8.79 MeV

Try this

Most stable: Fe-56, around 8.79 MeV per nucleon.
Steep climb to A ~ 30 means fusion of light nuclei releases the most energy per nucleon.
Slow drop after Fe means fission of heavy nuclei (e.g. U-235 → Ba + Kr) releases energy.
He-4 is exceptionally stable for its size, which is why alpha decay is favoured.

Radioactivity and decay law

Three decay channels, the exponential decay law, and conversions between λ, t_1/2 and τ.

Decay types

Alpha, beta and gamma decay compared

A reference table for the three decay channels.

Compare the three radioactive decay channels. NEET tests the rules ΔZ, ΔA and the example chain.

Type

What is emitted

ΔZ

ΔA

Penetration / ionising

Example

Alpha (α)

⁴₂He nucleus

−2

−4

Few cm of air; stopped by paper; Very high ionising

²³⁸U → ²³⁴Th + α

Beta minus (β⁻)

Electron + antineutrino

A few mm of Al; Medium ionising

¹⁴C → ¹⁴N + β⁻ + ν̄

Beta plus (β⁺)

Positron + neutrino

−1

Same as β⁻; Medium ionising

²²Na → ²²Ne + β⁺ + ν

Gamma (γ)

High-energy photon

Several cm of Pb; Low ionising

⁶⁰Ni* → ⁶⁰Ni + γ

Try this

Alpha: highest ionising, lowest penetration. Mass stopped by paper.
Beta minus is just the conversion: n → p + e + antineutrino.
Gamma is pure photon: no change in Z or A. Often follows alpha or beta as the daughter de-excites.
Penetration order: γ > β > α. Ionising order: α > β > γ.

Radioactive decay

Exponential decay: N(t), half-life and activity

Set the half-life and the time; see how many nuclei remain.

Number of undecayed nuclei drops exponentially. Every half-life, half of what is left disappears.

Half-life t_1/2: 10 s

Time elapsed t: 15 s (1.50 half-lives)

Initial N_0: 1000

Undecayed at t

354 (35.4%)

0.0693 /s

τ (mean life)

14.43 s

Activity = λN = 24.51 per second

N (t) = N_{0} e^{- λ t}, t_{1/2} = (ln 2) / λ, τ = 1/ λ

Try this

After 1 half-life: 50% remain. After 2: 25%. After 3: 12.5%. After n: (1/2)^n.
Mean life τ ≈ 1.44 t_1/2.
Activity also halves every t_1/2: A(t) = A_0 (1/2)^(t/t_1/2).
C-14 has t_1/2 ≈ 5730 years, used for carbon dating.

Half-life and mean life

Half-life, mean life and decay constant

Type any one of λ, t_1/2 or τ; the other two follow.

Three quantities, fully determined by any one. Pick which to type.

t_1/2 value

1.210e-4

t_1/2

5.730e+3

8.267e+3

t_{1/2} = \frac{ln 2}{λ} \approx \frac{0.693}{λ}, τ = \frac{1}{λ} = \frac{t _{1/2}}{ln 2}

Try this

t_1/2 (years): C-14 = 5730, U-238 = 4.5 billion, Ra-226 = 1602.
Mean life ≈ 1.44 × half-life.
After n half-lives, fraction left = (1/2)^n.
Cobalt-60 (used in radiotherapy) t_1/2 = 5.27 years.

Carbon dating

Carbon-14 dating: how old is this sample?

Half-life of C-14 is 5730 years. Set the percentage left to find the age.

Living things absorb C-14 at the same ratio as the atmosphere. After death, no more C-14 is added; what remains decays with t_1/2 = 5730 years. Measure the C-14 left, get the age.

C-14 left: 50.0% of original

Age of sample

5730 years

t = \frac{1}{λ} ln \frac{N _{0}}{N} = \frac{t _{1/2}}{ln 2} ln \frac{N _{0}}{N}

Try this

50% C-14 → exactly one half-life: 5730 years.
25% C-14 → 11460 years (two half-lives).
12.5% C-14 → 17190 years (three half-lives).
Below 1% C-14, the technique becomes unreliable; ages > ~50000 years use other methods (potassium-argon).

Fission and fusion

How heavy nuclei split and light nuclei merge, both releasing energy by climbing the BE curve.

Fission and fusion

Energy from nuclear fission and fusion

Compare the canonical reactions and the energy each releases.

Energy released in nuclear reactions = (BE_after − BE_before) of the system. Both fission of heavy nuclei and fusion of light ones move products closer to Fe-56, releasing the difference.

²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n

Energy released per fission: ~200 MeV. Used in nuclear reactors.

BE/n before

7.60 MeV

BE/n after

8.50 MeV

Energy released Q

~212 MeV

Q = (Δ m) c^{2} = (BE_{after} - BE_{before})

Try this

Fission of 1 g U-235 ≈ energy of burning 3 tonnes of coal.
Fusion needs T ~ 10⁷ K to overcome Coulomb repulsion. Only stars and bombs do it routinely.
D + T → He + n is the fusion reaction studied in tokamaks (ITER).
Both reactions move A towards Fe-56, where BE/n peaks.

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