PhysicsCBSE Class 12 · Nuclei

Composition, Size and Density of the Nucleus Class 12

Updated 15 October 2026 · Fact-checked

A nucleus contains Z protons and N neutrons (nucleons), with mass number A = Z + N. Its radius is R = R₀A^(1/3), with R₀ ≈ 1.2 × 10⁻¹⁵ m, so its volume is proportional to A. As a result every nucleus has nearly the same density, about 2.3 × 10¹⁷ kg/m³, independent of A.

Radius grows as A^(1/3), so volume grows in step with A and every nucleus has the same density, about 2.3×10¹⁷ kg/m³.

Illustration drawn for QuestionBuddy.

Understand Composition, Size and Density of Nucleus

Rutherford showed that an atom's mass and positive charge sit in a tiny nucleus. This topic asks: what is the nucleus made of, how big is it, and how dense is it?

Composition. A nucleus is made of protons (charge +e, mass about 1.007 u) and neutrons (no charge, mass about 1.009 u), together called nucleons. The neutron was discovered by Chadwick in 1932. For a nucleus written $_{Z}^{A} X$ :

Z = atomic number = number of protons (and of electrons in the neutral atom);
N = number of neutrons;
A = Z + N = mass number = total number of nucleons.

Nuclear masses are measured in the atomic mass unit: 1 u is one-twelfth of the mass of a carbon-12 atom, about $1.66 \times 1 0^{- 27}$ kg. A nucleus of mass number A has a mass of roughly A u.

Isotopes, isobars and isotones.

Isotopes: same Z, different A (same element, different number of neutrons), e.g. $_{1}^{1}$ H, $_{1}^{2}$ H (deuterium), $_{1}^{3}$ H (tritium); $_{17}^{35}$ Cl and $_{17}^{37}$ Cl. They have the same chemical properties.
Isobars: same A, different Z, e.g. $_{1}^{3}$ H and $_{2}^{3}$ He; $_{6}^{14}$ C and $_{7}^{14}$ N.
Isotones: same N, different Z, e.g. $_{80}^{198}$ Hg and $_{79}^{197}$ Au (both N = 118).

Size. Scattering experiments with fast particles show that the nuclear radius is $R = R_{0} A^{1/3}$ , with $R_{0} \approx 1.2 \times 1 0^{- 15}$ m = 1.2 fm (1 fermi = $1 0^{- 15}$ m). So nuclei are a few fermi across, about $1 0^{4}$ to $1 0^{5}$ times smaller than atoms. Since $R^{3} \propto A$ , the volume of a nucleus is proportional to the number of nucleons, as if nucleons were packed like marbles in a bag.

Density. The density of a nucleus is $ρ = \frac{mass}{volume} \approx \frac{A \times m _{u}}{\frac{4}{3} π R _{0}^{3} A} = \frac{3 m _{u}}{4 π R _{0}^{3}}$ . The A cancels, so all nuclei have about the same density, roughly $2.3 \times 1 0^{17}$ kg/m³, about $1 0^{14}$ times the density of water. Ordinary matter is so much less dense because atoms are mostly empty space. Nuclear density is reached in neutron stars, where matter is squeezed until nuclei merge.

Composition, Size and Density of Nucleus: formulas and key facts

Mass number: $A = Z + N$; Notation ^A_Z X.
Nuclear radius: $R = R_{0} A^{1/3}, R_{0} \approx 1.2 \times 1 0^{- 15} m$; 1 fm = 10⁻¹⁵ m.
Nuclear volume: $V = \frac{4}{3} π R_{0}^{3} A \propto A$; Nucleons packed closely.
Nuclear density: $ρ = \frac{3 m _{u}}{4 π R _{0}^{3}} \approx 2.3 \times 1 0^{17} kg/m^{3}$; Independent of A.
Atomic mass unit: $1 u = \frac{1}{12} m (^{12} C) \approx 1.66 \times 1 0^{- 27} kg$; Nuclear mass ≈ A u.
Ratio of radii: $\frac{R _{1}}{R _{2}} = (\frac{A _{1}}{A _{2}})^{1/3}$; Densities equal.

How to solve Composition, Size and Density of Nucleus questions

Use this method for composition, radius and density questions.

1From the symbol $_{Z}^{A} X$ , read Z and A; N = A − Z.
2Classify pairs: same Z (isotopes), same A (isobars), same N (isotones).
3For radius, use R = 1.2 × 10⁻¹⁵ A^(1/3) m; take cube roots carefully (27 → 3, 64 → 4, 125 → 5, 216 → 6).
4For ratios, R₁/R₂ = (A₁/A₂)^(1/3); densities are equal.
5For density, ρ = A m_u/((4/3)πR³) ≈ 2.3 × 10¹⁷ kg/m³.
6For 'compressed to nuclear density' questions, use V = M/ρ and r = (3V/4π)^(1/3).

Composition, Size and Density of Nucleus in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

How dense nuclear matter is

Suppose you could fill a 1 cm³ sugar-cube-sized box with matter at nuclear density, about 2.3 × 10¹⁷ kg/m³.

(a) Find the mass of this cube of nuclear matter. (b) Compare it with the mass of the same volume of water. (c) Why is ordinary matter so much less dense?

Try it first, then show the solutionSolution

Mass = ρV = 2.3 × 10¹⁷ × 10⁻⁶ = 2.3 × 10¹¹ kg, over two hundred million tonnes.
Water: 1000 × 10⁻⁶ = 10⁻³ kg (1 g); the ratio is about 2.3 × 10¹⁴.
In ordinary matter, nuclei are separated by mostly empty atoms about 10⁴ to 10⁵ times larger than the nuclei.

Answer: (a) about 2.3 × 10¹¹ kg; (b) about 2.3 × 10¹⁴ times the water; (c) atoms are mostly empty space.

Memory hook: A sugar cube of nuclear matter weighs as much as a mountain.

Neutron stars

Imagine a collapsed star of mass 2.8 × 10³⁰ kg squeezed until its whole mass is at nuclear density, 2.3 × 10¹⁷ kg/m³.

(a) Find its volume. (b) Find its radius. (c) Why does the density not depend on how much matter there is?

Try it first, then show the solutionSolution

V = M/ρ = 2.8 × 10³⁰/2.3 × 10¹⁷ ≈ 1.2 × 10¹³ m³.
r = (3V/4π)^(1/3) ≈ 1.4 × 10⁴ m, about 14 km.
Nucleons pack at a fixed spacing, so volume grows in proportion to mass (as for nuclei, V ∝ A), keeping density constant.

Answer: (a) about 1.2 × 10¹³ m³; (b) about 14 km; (c) volume is proportional to the number of nucleons.

Memory hook: Pack nucleons tight and a star fits inside a city.

Tricks to remember Composition, Size and Density of Nucleus

Short memory aids. Say each one aloud once, then use it on a question the same day.

Isotopes, isobars, isotones: P, A, N
isotoPes: same Protons; isobArs: same A; isotoNes: same Neutrons.
Use it when: Classifying nuclei.
Radius grows as cube root
R = 1.2 × A^(1/3) fm: 8 times the nucleons, only twice the radius.
Use it when: Ratio questions.
Density cancels A
Mass ∝ A and volume ∝ A, so density is the same for all nuclei.
Use it when: Explaining constant density.
2.3 × 10¹⁷
Nuclear density ≈ 2.3 × 10¹⁷ kg/m³, about 10¹⁴ times water.
Use it when: Quick estimates.

Composition, Size and Density of Nucleus: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Radii of two nuclei · 2 marks · Very short answerBoard level

Find the radii of the nuclei ²⁷Al and ⁶⁴Cu, and the ratio of their radii. (R₀ = 1.2 fm)

Try it first, then show the solutionSolution

R(Al) = 1.2 × 27^(1/3) = 1.2 × 3 = 3.6 fm; R(Cu) = 1.2 × 4 = 4.8 fm. (1)
Ratio = 3.6 : 4.8 = 3 : 4. (1)

Answer: 3.6 fm and 4.8 fm; ratio 3 : 4.

Example 2: Composition and classification · 3 marks · Short answerBoard level

(a) Give the number of protons, neutrons and nucleons in ²³⁵₉₂U. (b) Classify each pair: (i) ¹²₆C and ¹⁴₆C, (ii) ¹⁴₆C and ¹⁴₇N, (iii) ³₁H and ⁴₂He.

Try it first, then show the solutionSolution

(a) Z = 92 protons, N = 235 − 92 = 143 neutrons, A = 235 nucleons. (1)
(b) (i) Same Z, different A: isotopes. (ii) Same A: isobars. (1)
(iii) N = 3 − 1 = 2 and 4 − 2 = 2: isotones. (1)

Answer: (a) 92, 143, 235; (b) isotopes, isobars, isotones.

Example 3: Calculating nuclear density · 3 marks · Short answerHarder board level

Show that the density of nuclear matter is independent of mass number, and calculate it. (m_u = 1.66 × 10⁻²⁷ kg, R₀ = 1.2 × 10⁻¹⁵ m)

Try it first, then show the solutionSolution

Mass of a nucleus ≈ A m_u; volume = (4/3)πR³ = (4/3)πR₀³A. (1)
ρ = A m_u/((4/3)πR₀³A) = 3m_u/(4πR₀³): A cancels, so ρ is the same for all nuclei. (1)
ρ = 3 × 1.66 × 10⁻²⁷/(4π × (1.2 × 10⁻¹⁵)³) ≈ 2.3 × 10¹⁷ kg/m³. (1)

Answer: ρ = 3m_u/(4πR₀³) ≈ 2.3 × 10¹⁷ kg/m³, independent of A.

Example 4: Ratios for two nuclei · 2 marks · Very short answerHarder board level

The mass numbers of two nuclei are in the ratio 1 : 8. Find the ratio of (a) their radii, (b) their volumes and (c) their densities.

Try it first, then show the solutionSolution

(a) R ∝ A^(1/3): 1 : 2. (b) V ∝ A: 1 : 8. (1)
(c) Densities are equal: 1 : 1. (1)

Answer: (a) 1 : 2; (b) 1 : 8; (c) 1 : 1.

Example 5: The Earth at nuclear density · 5 marks · Long answerHOTS / competency

(a) How many times denser is nuclear matter than water? (1)
(b) If the whole mass of the Earth (6 × 10²⁴ kg) were compressed to nuclear density, what would its radius be? (3)
(c) What does this tell you about the structure of ordinary matter? (1)

Try it first, then show the solutionSolution

(a) 2.3 × 10¹⁷/10³ ≈ 2.3 × 10¹⁴ times. (1)
(b) V = M/ρ = 6 × 10²⁴/2.3 × 10¹⁷ ≈ 2.6 × 10⁷ m³. (1)
r = (3V/4π)^(1/3) = (6.2 × 10⁶)^(1/3) m. (1)
r ≈ 184 m, a sphere smaller than a stadium. (1)
(c) Ordinary matter is overwhelmingly empty space; nearly all its mass is in tiny nuclei. (1)

Answer: (a) about 2.3 × 10¹⁴; (b) about 184 m; (c) matter is mostly empty space.

Example 6: An iron nucleus · 4 marks · Case-basedHarder board level

Read the passage and answer.

Iron-56 (²⁶ protons) is one of the most tightly bound nuclei.
(a) How many neutrons does it have? (1)
(b) Estimate its mass in kg. (1)
(c) Find its radius. (1)
(d) Is its density more, less or the same as that of a uranium-238 nucleus? (1)

Try it first, then show the solutionSolution

(a) N = 56 − 26 = 30. (1)
(b) m ≈ 56 × 1.66 × 10⁻²⁷ ≈ 9.3 × 10⁻²⁶ kg. (1)
(c) R = 1.2 × 56^(1/3) ≈ 1.2 × 3.83 ≈ 4.6 fm. (1)
(d) The same: nuclear density does not depend on A. (1)

Answer: (a) 30; (b) about 9.3 × 10⁻²⁶ kg; (c) about 4.6 fm; (d) the same.

Where students lose marks in Composition, Size and Density of Nucleus

The slips examiners see most often, and what to write instead.

Mistake: Using R ∝ A instead of A^(1/3).
Assuming size grows with mass.
Do this: R = R₀A^(1/3); volume (not radius) ∝ A.
Usually costs: The answer mark.
Mistake: Mixing up isobars and isotones.
Similar names.
Do this: Isobars: same A (bar = weight). Isotones: same neutron number (N in 'tones').
Usually costs: 1 mark.
Mistake: Saying heavier nuclei are denser.
Intuition from everyday materials.
Do this: Nuclear density is the same for all nuclei.
Usually costs: 1 mark.
Mistake: Forgetting to convert fm to m.
Unit slip in density calculations.
Do this: 1 fm = 10⁻¹⁵ m; cube it carefully (10⁻⁴⁵ m³).
Usually costs: The answer mark.

Exam tips

Composition, size and density of the nucleus were set in 2023 (55/1/1 Q12 and 55/5/1 Q8), 2024 (55/1/1 Q28), 2025 (55/7/1 Q12) and 2026 (55/5/1 Q8 and Q9).
Show the cancellation of A in the density derivation; that line is what earns the mark.
Remember perfect cubes (8, 27, 64, 125, 216) to simplify radius ratios.
Give an example with every definition of isotope, isobar and isotone.
Radioactivity is not in the 2026-27 syllabus; do not prepare decay laws.

Practice questions on Composition, Size and Density of Nucleus

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The radius of a nucleus is proportional to
(a) A (b) A² (c) A^(1/3) (d) A^(2/3)
Hint
R = R₀A^(1/3).
Answer
(c) A^(1/3).
Q2 · 2 marks · Very short answerBoard level
Define isotopes, isobars and isotones, with one example each.
Hint
Same Z, same A, same N.
Answer
Isotopes: same Z, different A (¹²C, ¹⁴C). Isobars: same A, different Z (¹⁴C, ¹⁴N). Isotones: same N, different Z (³H, ⁴He, both N = 2).
Q3 · 1 mark · Very short answerHarder board level
Find the ratio of the radii of ¹²⁵Te and ²⁷Al nuclei.
Hint
(125/27)^(1/3).
Answer
(125/27)^(1/3) = 5/3, so 5 : 3.
Q4 · 1 mark · Very short answerHarder board level
Find the approximate mass in kg of a nucleus of mass number 64.
Hint
≈ A u.
Answer
≈ 64 × 1.66 × 10⁻²⁷ ≈ 1.06 × 10⁻²⁵ kg.
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The density of nuclear matter is the same for all nuclei.
Reason (R): The volume of a nucleus is proportional to its mass number, as is its mass.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
R³ ∝ A.
Answer
(a).
Q6 · 2 marks · Very short answerHOTS / competency
A nucleus has a radius of 6 fm. How many nucleons does it contain?
Hint
A = (R/R₀)³.
Answer
A = (6/1.2)³ = 5³ = 125.
Q7 · 2 marks · Very short answerHOTS / competency
Two nuclei have radii in the ratio 1 : 3. Find the ratio of their masses and of the number of their neutrons if both have N ≈ Z.
Hint
A ∝ R³.
Answer
A ∝ R³, so the mass ratio is 1 : 27; with N ≈ A/2 in each, the neutron numbers are also in the ratio 1 : 27.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Formula for nuclear radius?Show answer
R = R₀A^(1/3), R₀ ≈ 1.2 fm.
2.Approximate nuclear density?Show answer
2.3 × 10¹⁷ kg/m³.
3.What are isobars?Show answer
Nuclei with the same mass number A but different Z.
4.Value of 1 u in kg?Show answer
About 1.66 × 10⁻²⁷ kg.
5.Number of neutrons in ²³⁵₉₂U?Show answer
143.
6.Who discovered the neutron?Show answer
Chadwick (1932).

Composition, Size and Density of Nucleus: frequently asked questions

Why is nuclear density independent of mass number?

Both the mass (≈ A m_u) and the volume ((4/3)πR₀³A) of a nucleus are proportional to A, so their ratio does not depend on A.

How big is a nucleus?

A few femtometres (10⁻¹⁵ m), from about 1.2 fm for hydrogen to about 7.4 fm for uranium, roughly 10⁴ to 10⁵ times smaller than an atom.

What is the difference between isotopes and isobars?

Isotopes have the same atomic number (same element) but different mass numbers; isobars have the same mass number but different atomic numbers (different elements).

What is an atomic mass unit?

One-twelfth of the mass of a carbon-12 atom, about 1.66 × 10⁻²⁷ kg. Nuclear and atomic masses are expressed in u.

Is radioactivity in the CBSE 2026-27 syllabus?

No. Radioactivity, decay laws and half-life are listed as not to be assessed. The nuclei chapter covers composition, size, nuclear force, mass-energy relation, binding energy, fission and fusion.