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 Question:
We are given two nuclei characterized by the coordinates $(Z, N, r)$, representing the number of protons, number of neutrons, and the radius, respectively.
We need to find the ratio of their radii, $\frac{r_1}{r_2}$.

Step 2: Key Formula or Approach:
The mass number $A$ of a nucleus is the sum of the number of protons $Z$ and the number of neutrons $N$:
\[ A = Z + N \] The empirical formula for the radius $r$ of a nucleus as a function of its mass number $A$ is:
\[ r = R_0 A^{1/3} \] where $R_0$ is a constant.

Step 3: Detailed Explanation:

• For the first nucleus, characterized by $(1, 0, r_1)$:
Number of protons $Z_1 = 1$, and number of neutrons $N_1 = 0$.
The mass number is:
\[ A_1 = Z_1 + N_1 = 1 + 0 = 1 \] The radius is:
\[ r_1 = R_0 (1)^{1/3} \]
• For the second nucleus, characterized by $(4, 4, r_2)$:
Number of protons $Z_2 = 4$, and number of neutrons $N_2 = 4$.
The mass number is:
\[ A_2 = Z_2 + N_2 = 4 + 4 = 8 \] The radius is:
\[ r_2 = R_0 (8)^{1/3} = 2 R_0 \]
• Now, we find the ratio of the two radii:
\[ \frac{r_1}{r_2} = \frac{R_0 (1)^{1/3}}{R_0 (8)^{1/3}} = \left( \frac{1}{8} \right)^{1/3} = \frac{1}{2} \]

Step 4: Final Answer:
The ratio $\frac{r_1}{r_2}$ is equal to $\frac{1}{2}$.