Question

If \(r_1\) and \(r_2\) are the radii of atomic nuclei of mass numbers 64 and 27 respectively, then the value of \(\left(\frac{r_1}{r_2}\right)\) is :

• A. 1
• B. \(\frac{4}{3}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{27}{64}\)

Hint:

Problems involving nuclear radii usually feature perfect cube mass numbers like 8, 27, 64, or 125. Taking the cube root of these mass numbers gives you the radius ratio immediately.

Answer 1

The Correct Option is B

Concept: Experimental observations demonstrate that nuclear matter has a nearly constant density. As a result, the volume of an atomic nucleus is directly proportional to the total number of nucleons (protons + neutrons) it contains, which is defined as its mass number (\(A\)). Assuming a spherical nuclear geometry, this relationship is given by: \[ \text{Volume} = \frac{4}{3}\pi r^3 \propto A \quad \Rightarrow \quad r \propto A^{1/3} \] This allows us to write the nuclear radius formula as: \[ r = r_0 A^{1/3} \] where \(r_0\) is an empirical constant representing the approximate radius of a single nucleon (\(r_0 \approx 1.2 \times 10^{-15}\text{ m} = 1.2\text{ fm}\)).

Step 1: Setting up the ratio equation based on given mass numbers.
We are given two nuclei with different mass numbers:
• Mass number of the first nucleus: \(A_1 = 64\)
• Mass number of the second nucleus: \(A_2 = 27\) Using the radius proportional formula for both nuclei: \[ r_1 = r_0 (A_1)^{1/3} \] \[ r_2 = r_0 (A_2)^{1/3} \] Dividing the equation for \(r_1\) by the equation for \(r_2\): \[ \frac{r_1}{r_2} = \frac{r_0 (A_1)^{1/3}}{r_0 (A_2)^{1/3}} = \left( \frac{A_1}{A_2} \right)^{1/3} \]

Step 2: Calculating the final numerical value.
Substitute the mass numbers \(A_1 = 64\) and \(A_2 = 27\) into the ratio expression: \[ \frac{r_1}{r_2} = \left( \frac{64}{27} \right)^{1/3} \] Recognizing that both numbers are perfect cubes: \[ 64 = 4^3 \quad \text{and} \quad 27 = 3^3 \] We can rewrite the expression as: \[ \frac{r_1}{r_2} = \left( \frac{4^3}{3^3} \right)^{1/3} = \left[ \left(\frac{4}{3}\right)^3 \right]^{1/3} = \frac{4}{3} \] Thus, the value of the ratio \(\frac{r_1}{r_2}\) is equal to \(\frac{4}{3}\), which matches Option (B).