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 find the atomic number of a nucleus that has 61 neutrons and a radius ratio of 3 compared to a helium nucleus. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the relationship between radius and mass number
The radius \( R \) of a nucleus is related to its mass number \( A \) by the formula:
\[ R \propto A^{1/3} \]
This means that the radius of a nucleus is proportional to the cube root of its mass number.

### Step 2: Set up the ratio of the radii
Given that the ratio of the radius of the nucleus with 61 neutrons (let's call it nucleus 1) to the radius of the helium nucleus (nucleus 2) is 3, we can express this as:
\[ \frac{R_1}{R_2} = 3 \]

### Step 3: Relate the mass numbers using the radius ratio
Using the relationship from Step 1, we can write:
\[ \frac{R_1}{R_2} = \left(\frac{A_1}{A_2}\right)^{1/3} \]
Substituting the radius ratio into this equation gives us:
\[ 3 = \left(\frac{A_1}{A_2}\right)^{1/3} \]

### Step 4: Cube both sides to eliminate the exponent
Cubing both sides of the equation results in:
\[ 3^3 = \frac{A_1}{A_2} \]
\[ 27 = \frac{A_1}{A_2} \]

### Step 5: Find the mass number of the nucleus
Since the mass number of helium \( A_2 \) is 4, we can substitute this value into the equation:
\[ A_1 = 27 \times A_2 = 27 \times 4 = 108 \]

### Step 6: Relate mass number to neutrons and atomic number
The mass number \( A \) is related to the number of neutrons \( N \) and the atomic number \( Z \) by the equation:
\[ A = N + Z \]
We know \( N = 61 \) (the number of neutrons) and \( A = 108 \) (the mass number we just calculated). Thus:
\[ 108 = 61 + Z \]

### Step 7: Solve for the atomic number \( Z \)
Rearranging the equation gives:
\[ Z = 108 - 61 = 47 \]

### Conclusion
The atomic number of the nucleus is \( Z = 47 \).