Nuclear size, radius, and density
Explore how nuclear size is measured, the formula for nuclear radius, its dependence on mass number, and the concept of nuclear density.
Why this shows up in the exam
You need to calculate nuclear radii and understand their properties for NEET questions.
How NEET tests this
Learn the idea
Nuclear radius grows as the cube‑root of the mass number (R = r₀ A^(1/3)); because mass ∝ volume, the density of all nuclei is practically the same.
🧠 Memory hook: Think of a cube of marbles: double the marbles → side length grows by the cube‑root of 2, just like R grows by the cube‑root of A.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- R = r₀ A^(1/3) where r₀ ≈ 1.2 fm (NCERT)
- Mass number A is proportional to nuclear volume → A ∝ R³
- Nuclear density ρ = (mass of nucleus)/(volume) ≈ constant for all nuclei
- In a fission or rupture, linear momentum is conserved: m₁v₁ = m₂v₂
- Since m ∝ A, the ratio of radii is the cube‑root of the mass (or A) ratio
How to approach it
- 1Write the momentum‑conservation equation and obtain the mass (or A) ratio from the given velocity ratio
- 2Replace mass ratio by A‑ratio because m ∝ A
- 3Use R ∝ A^(1/3) to convert the A‑ratio into a radius ratio (take cube‑root)
- 4Express the answer in the required form, keeping track of which fragment is larger
Worked example — watch it click
A nucleus ruptures into two nuclear parts, which have their velocity ratio equal to 2: 1. What will be the ratio of their nuclear size (nuclear radius)?
- A)2¹/³: 1
- ✅1: 2¹/³
- C)3¹/²: 1
- D)1: 3¹/²
The concept behind this problem
The question forces you to link momentum conservation (giving a mass ratio) with the geometric relation R ∝ A^(1/3), showing that size follows the cube‑root of the mass ratio.
Step by step
- 1By momentum conservation, m₁v₁ = m₂v₂, so m₁/m₂ = v₂/v₁ = 1/2.
- 2Since m ∝ A ∝ R³, we have R₁³/R₂³ = 1/2, giving R₁/R₂ = 1/2¹/³ = 2⁻¹/³, or R₁:R₂ = 1:2¹/³.
Watch out
Taking the radius ratio directly as the velocity ratio or using a square‑root instead of a cube‑root.
Common slip-ups that cost marks
- •Mixing up the direction of the ratio – remember m₁/m₂ = v₂/v₁, not v₁/v₂
- •Forgetting the cube‑root when converting A‑ratio to R‑ratio (students often use square‑root)
- •Assuming density changes with size; NCERT states nuclear density is essentially constant
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Practise it
These are real questions from past NEET papers that test this exact idea.
A nucleus ruptures into two nuclear parts, which have their velocity ratio equal to 2: 1. What will be the ratio of their nuclear size (nuclear radius)?
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