AQA · A-Level · Physics
Course version: 7408 · Version from Sept 2016
Nuclear radius (A-level only): Calculation of nuclear density.
Nuclear density is found by treating the nucleus as a sphere of uniform matter.
Checked against the AQA 7408 specification (opens in a new tab) on .
- Point 5 of 6
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Nuclear density is found by treating the nucleus as a sphere of uniform matter. Its radius is R = r₀A^(1/3), where r₀ ≈ 1.2 × 10⁻¹⁵ m and A is the nucleon number, so volume V = (4/3)πR³ = (4/3)πr₀³A. Mass m ≈ A × 1.66 × 10⁻²⁷ kg, using the atomic mass unit. Dividing mass by volume gives ρ = m/V = (A × 1.66 × 10⁻²⁷)/((4/3)πr₀³A). The A terms cancel, showing nuclear density is approximately constant at about 2.3 × 10¹⁷ kg m⁻³ for all nuclei. This explains why all nuclei have similar density despite very different sizes and masses.
What earns the marks
- States the nuclear radius relationship R = r₀A^(1/3) with r₀ ≈ 1.2 × 10⁻¹⁵ m and identifies A as the nucleon number.
- Calculates the nuclear volume using V = (4/3)πR³, substituting the expression for R to obtain V = (4/3)πr₀³A.
- Uses nuclear mass m ≈ A × 1.66 × 10⁻²⁷ kg, recognising that the mass of a nucleus is approximately A atomic mass units.
- Divides mass by volume and cancels A to show ρ = (1.66 × 10⁻²⁷)/((4/3)πr₀³), giving a value of about 2.3 × 10¹⁷ kg m⁻³.
- Explains that because A cancels, nuclear density is approximately the same for all nuclei, independent of nucleon number.
Common mistakes and how to put them right
Using the atomic radius (about 10⁻¹⁰ m) instead of the nuclear radius (about 10⁻¹⁵ m).
Put it right: Use R = r₀A^(1/3) with r₀ ≈ 1.2 × 10⁻¹⁵ m.
Forgetting to cube the radius when calculating volume.
Put it right: Apply V = (4/3)πR³ fully, including cubing the numerical value of R.
Treating the mass of the nucleus as A grams rather than A atomic mass units in kilograms.
Put it right: Convert using 1 u ≈ 1.66 × 10⁻²⁷ kg.
Exam tips
- Write the full algebraic expression for density before substituting numbers, so the cancellation of A is visible and can earn method credit.
- Keep powers of ten separate from the numerical calculation to reduce arithmetic errors when cubing r₀.
- State the final answer with the unit kg m⁻³ and an appropriate number of significant figures, typically two or three.
Try the lesson
The start of the Nuclear radius (A-level only): Calculation of nuclear density. lesson, made from checked teaching notes for this specification point.
Have a go first
A nucleus has a mass of 4.0 × 10⁻²⁶ kg and a radius of 3.0 × 10⁻¹⁵ m. Calculate the density of this nucleus.
Listen
Nuclear density is found by treating the nucleus as a sphere of uniform matter. Its radius is R = r₀A^(1/3), where r₀ ≈ 1.2 × 10⁻¹⁵ m and A is the nucleon number, so volume V = (4/3)πR³ = (4/3)πr₀³A. Mass m ≈ A × 1.66 × 10⁻²⁷ kg, using the atomic mass unit. Dividing mass by volume gives ρ = m/V = (A × 1.66 × 10⁻²⁷)/((4/3)πr₀³A). The A terms cancel, showing nuclear density is approximately constant at about 2.3 × 10¹⁷ kg m⁻³ for all nuclei. This explains why all nuclei have similar density despite very different sizes and masses. A strong answer earns marks for these points. States the nuclear radius relationship R = r₀A^(1/3) with r₀ ≈ 1.2 × 10⁻¹⁵ m and identifies A as the nucleon number. Calculates the nuclear volume using V = (4/3)πR³, substituting the expression for R to obtain V = (4/3)πr₀³A. Uses nuclear mass m ≈ A × 1.66 × 10⁻²⁷ kg, recognising that the mass of a nucleus is approximately A atomic mass units. Divides mass by volume and cancels A to show ρ = (1.66 × 10⁻²⁷)/((4/3)πr₀³), giving a value of about 2.3 × 10¹⁷ kg m⁻³. A common slip: Using the atomic radius (about 10⁻¹⁰ m) instead of the nuclear radius (about 10⁻¹⁵ m). Instead, use R = r₀A^(1/3) with r₀ ≈ 1.2 × 10⁻¹⁵ m. Exam tip: Write the full algebraic expression for density before substituting numbers, so the cancellation of A is visible and can earn method credit.
Key words
Say what each word means before you turn the card.
Nuclear radius
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Nuclear volume
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Nuclear mass
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Nuclear density
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See every definition
- Nuclear radius
- Given by R = r₀A^(1/3), where r₀ ≈ 1.2 × 10⁻¹⁵ m and A is the nucleon number.
- Nuclear volume
- Found by treating the nucleus as a sphere of uniform matter, using V = (4/3)πR³ = (4/3)πr₀³A.
- Nuclear mass
- Approximately A × 1.66 × 10⁻²⁷ kg, using the atomic mass unit.
- Nuclear density
- Mass divided by volume, giving about 2.3 × 10¹⁷ kg m⁻³ and approximately constant for all nuclei because the A terms cancel.
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