If the ratio of the radius of a nucleus with 61 neutrons to that of helium nucleus is 3, then the atomic number of this nucleus is

Options

A. 27
B. 47
C. 51
D. 61

Answer

To solve the problem, we need to determine the atomic number of a nucleus that has 61 neutrons and a radius ratio of 3 compared to a helium nucleus. Let's break this down step by step.

### Step 1: Understand the relationship between radius and atomic mass
The radius \( r \) of a nucleus is given by the formula:
\[ r = r_0 A^{1/3} \]
where \( r_0 \) is a constant and \( A \) is the atomic mass number (total number of protons and neutrons).

### Step 2: Set up the ratio of the radii
We are given that the ratio of the radius of the nucleus with 61 neutrons (let's call it nucleus 1) to that of the helium nucleus (nucleus 2) is 3:
\[ \frac{r_1}{r_2} = 3 \]
Substituting the formula for radius, we have:
\[ \frac{r_0 A_1^{1/3}}{r_0 A_2^{1/3}} = 3 \]
The \( r_0 \) cancels out, leading to:
\[ \frac{A_1^{1/3}}{A_2^{1/3}} = 3 \]

### Step 3: Cube both sides to eliminate the exponent
Cubing both sides gives us:
\[ \frac{A_1}{A_2} = 27 \]
This means:
\[ A_1 = 27 A_2 \]

### Step 4: Determine the atomic mass of helium
The atomic mass of helium (which has 2 protons and 2 neutrons) is:
\[ A_2 = 4 \]
Substituting this into the equation:
\[ A_1 = 27 \times 4 = 108 \]

### Step 5: Relate atomic mass to neutrons and protons
We know that:
\[ A = Z + N \]
where \( Z \) is the atomic number (number of protons) and \( N \) is the number of neutrons. For nucleus 1:
\[ 108 = Z + 61 \]

### Step 6: Solve for the atomic number \( Z \)
Rearranging the equation gives:
\[ Z = 108 - 61 = 47 \]

### Conclusion
The atomic number of the nucleus with 61 neutrons is:
\[ \boxed{47} \]