Question

The three numbers: (number of protons, number of neutrons, the radius) characterize a nucleus. What is the value of $\frac{r_1}{r_2}$ for two nuclei characterized by $(1, 0, r_1)$ and $(4, 4, r_2)$?

• A. $\frac{1}{2}$
• B. 2
• C. 8
• D. $\frac{1}{8}$

Hint:

The nuclear radius is directly proportional to the cube root of the mass number ($r \propto A^{1/3}$).
Always find the mass number $A = Z + N$ first, and then apply the proportionality.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The radius of a nucleus is directly related to its mass number (total nucleons).
Key Formula or Approach:
\( R = R_{0}A^{1/3} \), where \( A = Z + N \).

Step 2: Detailed Explanation:
1. Nucleus 1: Protons = 1, Neutrons = 0. Mass number \( A_{1} = 1 \).
\( r_{1} = R_{0}(1)^{1/3} \).
2. Nucleus 2: Protons = 4, Neutrons = 4. Mass number \( A_{2} = 8 \).
\( r_{2} = R_{0}(8)^{1/3} = 2R_{0} \).
3. Ratio:
\[ \frac{r_{1}}{r_{2}} = \frac{R_{0} \times 1}{2R_{0}} = \frac{1}{2} \]

Step 3: Final Answer:
The ratio is 1/2.