A full lesson: a cartoon movie to start, then isotopes, isobars and isotones on the nuclide chart, how alpha particles measure the size of a nucleus, the radius law R = R₀A^(1/3), why every nucleus has the same density, a teaspoon of nuclear matter and neutron stars, NEET shortcuts, problems to solve yourself, and 12 MCQs.
Eight times the boondi makes a laddoo only twice as wide, because volume grows as R³. Nuclei are packed the same way: R = R₀A^(1/3), so every nucleus has the same density, about 2.3 × 10¹⁷ kg/m³.
It is Diwali week. Dadi rolls one small boondi laddoo from one katori of boondi and one big laddoo from eight katoris. Arjun says, "The big one must be eight times wider!" Priya takes a ruler. The small laddoo is 2 cm in radius, the big one only 4 cm: twice, not eight times. Dadi cuts both open. Inside, the tiny boondi pearls are packed exactly the same way.
Try it at home with clay or atta dough: make 1 small ball and 8 identical small balls pressed into one big ball. Measure both diameters. The volume grows as R³, so 8 times the dough gives only 2 times the radius (2³ = 8). A nucleus is a laddoo of nucleons, packed the same way in every nucleus.
"Same packing means same density. Eight times the nucleons, twice the radius. That one sentence is the whole of this lesson's size law."
"If Dadi uses 27 katoris of boondi, how many times wider than the small laddoo will the new laddoo be?"
3 times, because 3³ = 27. Radius goes as the cube root of the amount: R ∝ A^(1/3) for nuclei.
Every nuclide is one square on a chart of neutron number N (across) against proton number Z (up). Tap any square (or drag across the chart) to select a nuclide, then pick a family to light up. Filled squares are stable nuclides; outlined ones are well-known radioactive nuclides.
| Family | Same | Different | Example |
|---|---|---|---|
| Isotopes | Z (protons) | N and A | ¹²C, ¹³C, ¹⁴C · ³⁵Cl, ³⁷Cl · ¹H, ²H, ³H |
| Isobars | A (nucleons) | Z and N | ³H and ³He · ¹⁴C and ¹⁴N · ⁴⁰Ar and ⁴⁰Ca |
| Isotones | N (neutrons) | Z and A | ¹⁹⁸₈₀Hg and ¹⁹⁷₇₉Au (N = 118) · ¹⁴C and ¹⁶O (N = 8) |
"Iso-tope: same place in the periodic table, so same Z. Iso-bar: same weight, so same A. Iso-tone: the n in tone reminds you of neutrons. Isotopes share chemistry because their electron arrangement is the same; isobars and isotones are different elements."
Fire an alpha particle (charge +2e) straight at a nucleus (charge +Ze). It slows down, stops at the distance of closest approach d, and turns back. All its kinetic energy has become electric potential energy. The nucleus must be smaller than d, so d gives an upper limit on its size. Raise the energy and watch when the alpha particle finally touches the nucleus.
At the closest point the alpha particle is momentarily at rest (head-on), so all of K is potential energy.
Gold, K = 5.5 MeV: d ≈ 4.1 × 10⁻¹⁴ m = 41 fm. Bigger K, smaller d.
With fast enough particles, scattering stops following Coulomb's law alone: the particle has reached the nucleus. Experiments with fast alphas and fast electrons gave the size law of Stage 4.
"Closest approach is an energy question, not a force question: kinetic energy equals k times 2e times Ze over d. And remember it only gives an upper limit for the radius."
Scattering experiments show that a nucleus of mass number A has radius R = R₀A^(1/3) with R₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm (1 fermi = 1 femtometre = 10⁻¹⁵ m). Drag the two markers on the graph (or use the sliders) and compare the two nuclei, drawn to the same scale.
So R(⁶⁴Cu) = 1.2 × 4 = 4.8 fm and R(¹²⁵Te) = 1.2 × 5 = 6.0 fm.
R₀ cancels. ²⁷Al : ¹²⁵Te = 3 : 5.
Double the nucleons, double the volume: each nucleon takes the same room. This is the key to Stage 5.
"To make a radius twice as big you need eight times the nucleons. If a question says radius doubles, multiply A by 8; if radius triples, multiply A by 27."
"Uranium-238 has 59.5 times the nucleons of helium-4. About how many times larger is its radius?"
(59.5)^(1/3) ≈ 3.9. R(U) ≈ 7.4 fm and R(He) ≈ 1.9 fm. Even the heaviest natural nucleus has a radius under 10 fm.
The mass of a nucleus is about A × 1 u, and its volume is (4/3)πR₀³A. Both grow in step with A, so their ratio, the density, does not depend on A at all. Slide A from hydrogen to uranium and watch the three meters.
Each nucleon is about 1 u (the binding correction is under 1 %).
The cube of A^(1/3) is just A.
1.66 × 10⁻²⁷ / (4.19 × 1.728 × 10⁻⁴⁵) = 2.29 × 10¹⁷ kg/m³, about 2.3 × 10¹⁴ times the density of water.
"Nuclear density is the same for all nuclei. So the ratio of densities of any two nuclei is one is to one: a favourite NEET question with a one-second answer."
Nuclear matter is so dense that a pinhead of it would outweigh a mountain of ordinary things. Choose an example and slide.
About 2.3 lakh tonnes, the mass of roughly 46 000 elephants.
NCERT: a neutron star is like a giant nucleus. 1.4 solar masses packed in a ball about 12 km in radius.
The whole Earth at nuclear density would be a ball about 370 m across, the size of a large stadium complex (a few cricket grounds).
"Ordinary matter is mostly empty space between nuclei, as we saw in Lesson 1. Remove the empty space and you get nuclear density. Nature does exactly that inside a neutron star."
Write Z, N, A for both nuclides in two lines; the equal column names the family.
Perfect cubes: A = 8, 27, 64, 125, 216 give 2.4, 3.6, 4.8, 6.0, 7.2 fm.
Radius ×2 needs A ×8. Volume ratio = mass-number ratio. Density ratio = 1.
A = 8 vs 64: radius 1:2, area 1:4, volume 1:8.
With 1/(4πε₀) = 9 × 10⁹: d (fm) ≈ 1.44 × 2Z / K(MeV), since e²/(4πε₀) = 1.44 MeV fm.
Independent of A. Do not confuse with 10¹⁴ (that is ρ divided by water's density).
"The 1.44 MeV femtometre trick saves time: for gold and 5.5 MeV, d ≈ 1.44 × 158 / 5.5 ≈ 41 femtometres, done in your head."
Cover the solution, write the formula, and solve on paper. Open the hint only if you are stuck for more than a minute.
Find the radius of the ⁶⁴Cu nucleus. (R₀ = 1.2 fm)
64^(1/3) = 4.
Find the ratio of the radii, and of the volumes, of ²⁷Al and ¹²⁵Te nuclei.
R ∝ A^(1/3) and V ∝ A.
A nucleus has a radius twice that of ²⁷Al. Find its mass number.
A ∝ R³.
Calculate the density of nuclear matter. Take the mass of a nucleon as 1.66 × 10⁻²⁷ kg and R₀ = 1.2 × 10⁻¹⁵ m. Compare it with water (10³ kg/m³).
ρ = A × 1 u / ((4/3)πR₀³A); A cancels.
An alpha particle of 7.7 MeV moves head-on towards a gold nucleus (Z = 79). Find the distance of closest approach and compare it with the radius of the gold nucleus (A = 197).
d = (1/4πε₀)(2Ze²)/K. Use K in joules, or the shortcut 1.44 MeV fm.
Take R₀ = 1.2 fm and 1 u = 1.66 × 10⁻²⁷ kg. Write the formula first, then click.
Score: 0 / 12
| If the question says… | Write this first | The trap |
|---|---|---|
| "which pair are isobars / isotones" | Z, N, A of both in a table | Mixing isotones (same N) with isotopes (same Z) |
| "radius of a nucleus of mass number A" | R = 1.2 A^(1/3) fm | Using A or A² instead of the cube root |
| "ratio of radii" | (A₁/A₂)^(1/3) | Writing A₁/A₂ (that is the volume ratio) |
| "ratio of nuclear densities" | 1 : 1 | Answering with a mass or radius ratio |
| "order of nuclear density" | 10¹⁷ kg m⁻³ | Choosing 10¹⁴ (relative to water) or 10³ |
| "radius doubles / triples" | A × 8 / A × 27 | Multiplying A by 2 or 3 |
| "distance of closest approach" | K = k(2e)(Ze)/d | Forgetting the 2 in the alpha charge |
| "closest approach gives" | an upper limit of the nuclear size | Saying it equals the nuclear radius |
Lesson 3 asks why atomic masses are not exact multiples of 1 u: the mass defect, E = mc² and the binding energy per nucleon curve. Lesson 4 explains the short-range nuclear force that packs nucleons at this constant density.
Values used: R₀ = 1.2 fm = 1.2 × 10⁻¹⁵ m, 1 u = 1.66 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C, 1/(4πε₀) = 9 × 10⁹ N m² C⁻², e²/(4πε₀) = 1.44 MeV fm, solar mass 2 × 10³⁰ kg, Earth's mass 5.97 × 10²⁴ kg.
Follow the four steps: understand it, remember it, try it, then prove it with past papers.