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Composition and Size of Nucleus

The nucleus is the small, dense, positively charged central part of an atom containing Z protons and N neutrons. The mass number is A = Z + N, and the nuclear radius is approximately R = R0 A1/3, where R0 is about 1.2 fm.

In one line: A counts all nucleons: protons plus neutrons. Radius grows with the cube root, so eight times the mass number gives only twice the radius.

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Understand it

  • Protons determine the atomic number Z and hence the element.
  • Neutrons add to the nuclear mass and influence stability without changing the element.
  • Protons and neutrons together are called nucleons.
  • Since R is proportional to A1/3, nuclear volume is proportional to A, so nuclear density is nearly constant for most nuclei.
  • This is a key difference from ordinary matter, where size and density vary strongly with composition.

Example

For carbon-12, Z = 6 and A = 12, so N = A - Z = 6. Its approximate radius is R = 1.2 × 121/3 fm, which is about 2.7 fm.

Formula and rule

A = Z + N, where A is mass number, Z is number of protons, and N is number of neutrons.

R = R0 A1/3, where R is nuclear radius in metre, R0 is about 1.2 × 10−15 m, and A is mass number.

Use this formula for approximate nuclear size, especially when comparing radii of two nuclei.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Compare the radii of oxygen-16 and lead-128.

  1. 1Since R is proportional to A1/3, Rlead/Roxygen = (128/16)1/3 = 81/3 = 2.
  2. 2The lead-128 nucleus has twice the radius of the oxygen-16 nucleus.

Method you can reuse

  1. 1Identify Z and A from the given nuclide.
  2. 2Find N using N = A - Z.
  3. 3For radius, write R = R0 A1/3.
  4. 4For ratios, cancel R0 and take the cube root of the mass-number ratio.
  5. 5State the answer with correct units or ratio.

Visual tip

A simple labelled nucleus sketch may show protons and neutrons packed in a small central region with electrons outside the nucleus only for atomic context. Important labels: proton, neutron, nucleon, nucleus, atomic scale versus nuclear scale.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If a nucleus has atomic number 17 and mass number 35, how many neutrons does it contain?

Think first, then show answer
It contains 18 neutrons because the number of neutrons is N = A - Z = 35 - 17 = 18.

Common wrong answer

For radius ratio, a common wrong answer is R2/R1 = A2/A1. The correct relation uses the cube root, so R2/R1 = (A2/A1)1/3.

Don't mix this up

Students often treat atomic radius and nuclear radius as similar quantities. Nuclear radius is about 10−15 m in scale, much smaller than atomic radius, which is about 10−10 m.

Exam tip

When a question gives isotope notation, first identify A and Z, then calculate N = A - Z. For radius questions, compare using R proportional to A1/3, not directly proportional to A.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

This concept appears as isotope notation questions, radius ratio numericals, neutron-number calculations, and reasoning questions on why nuclear density is nearly constant.
  1. 1

    Enough for 1 mark

    The nucleus is the small, dense, positively charged central part of an atom containing Z protons and N neutrons. The mass number is A = Z + N, and the nuclear radius is approximately R = R0 A1/3, where R0 is about 1.2 fm.

  2. 2

    Add for 2 marks

    Example: For carbon-12, Z = 6 and A = 12, so N = A - Z = 6. Its approximate radius is R = 1.2 × 121/3 fm, which is about 2.7 fm.

  3. 3

    Add for 3 marks

    Protons determine the atomic number Z and hence the element.

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