Question

If the ratio of the nuclear radii of two atoms is $2:3$, then the ratio of their mass numbers is:

• A. $4:9$
• B. $9:4$
• C. $8:27$
• D. $27:8$

Hint:

Radius goes with the cube root ($\sqrt[3]{A}$), so mass numbers will always go with the cube ($R^3$). Cubing the given terms right away ($2^3 = 8$ and $3^3 = 27$) yields the solution in under two seconds.

Answer 1

The Correct Option is C

Concept: The empirical relationship between the structural radius of an atomic nucleus ($R$) and its total mass number ($A$) is given by: \[ R = R_0 A^{1/3} \] where $R_0$ is a fundamental constant ($\approx 1.2\text{ fm}$). This implies that nuclear radius scales with the cube root of the mass number ($R \propto A^{1/3}$), or conversely, $A \propto R^3$.

Step 1: Set up the ratio equation and isolate the mass numbers.
Given the radius ratio: \[ \frac{R_1}{R_2} = \frac{2}{3} \] Using the proportionality relation: \[ \frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3} \implies \frac{2}{3} = \left(\frac{A_1}{A_2}\right)^{1/3} \]

Step 2: Cube both sides to clear the fraction exponent.
\[ \frac{A_1}{A_2} = \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \implies 8:27 \]