Question

The ratio of the radii of the nuclei \(^{64}_{29}X\) and \(^{216}_{84}Y\) is:

• A. \(\frac{2}{3}\)
• B. \(\frac{8}{15}\)
• C. \(\frac{8}{3}\)
• D. \(\frac{7}{8}\)

Hint:

Nuclear radius depends only on mass number: \(R \propto A^{1/3}\).

Answer 1

The Correct Option is A

Step 1: Start from the idea that nuclear density is the same for every nucleus.
Since all nuclei are packed nucleons at essentially the same density, nuclear volume is proportional to mass number: $R^3 \propto A$, so $\frac{R^3}{A}$ is the same constant $k$ for every nucleus.

Step 2: Apply this to both nuclei.
\[ R_X^3 = k \times 64, \qquad R_Y^3 = k \times 216 \]
Step 3: Take the ratio, the constant $k$ cancels.
\[ \frac{R_X^3}{R_Y^3} = \frac{64}{216} = \frac{8}{27} \]
Step 4: Recognise $8/27$ as a perfect cube.
\[ \frac{8}{27} = \left(\frac{2}{3}\right)^3 \quad\Rightarrow\quad \frac{R_X}{R_Y} = \frac{2}{3} \]
Final Answer:
\[ \boxed{\frac{2}{3}} \]