Question 13 of 13beginner💡 UnderstandShort Answer2 marks

From the relation R=R0A1/3R = R_0 A^{1/3}, where R0R_0 is a constant and AA is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of AA).

Correct Answer

The radius of a nucleus is given by the relation R = R₀A^(1/3), where R₀ is a constant and A is the mass number.

Assuming the nucleus is spherical in shape, the volume of the nucleus is V = (4/3)πR³. Substituting the value of R, we get V = (4/3)πR₀³A. This shows that the volume is proportional to A (since R³ is proportional to A).

The mass of the nucleus is proportional to the mass number A. Therefore, nuclear density = Mass/Volume ∝ A/A = constant. Hence, the nuclear matter density is independent of A and is nearly constant for all nuclei, approximately 2.3×102.3 \times 10¹⁷ kg m⁻³.

Exercise: EXERCISES | Q: 13.10 | (Page 17)
For More Understanding

Explanation

This question tests the understanding of how the nuclear radius formula leads to constant density. The key insight from the textbook is that since R ∝ A^(1/3), the volume (proportional to R³) becomes proportional to A. Since mass is also proportional to A, the ratio Mass/Volume becomes independent of A. The textbook explicitly states this conclusion and provides the constant density value of approximately 2.3×102.3 \times 10¹⁷ kg m⁻³.

Solution Steps

  1. Step 1: Write the radius formula: R = R₀A^(1/3)

  2. Step 2: Calculate volume: V = (4/3)πR³ = (4/3)πR₀³A, showing V ∝ A

  3. Step 3: Note that mass M ∝ A (mass number)

  4. Step 4: Calculate density: ρ = M/V ∝ A/A = constant

  5. Step 5: Conclude that nuclear density is independent of A

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