The radius of germanium `(Ge)` nuclide is measured to be twice the radius of `._(4)^(9)Be`. The number of nucleons in `Ge` are

Options

A. `73`
B. `74`
C. `75`
D. `72`

Answer

To find the number of nucleons in germanium (Ge) given that its radius is twice that of beryllium (Be), we can follow these steps:

### Step 1: Understand the relationship between radius and mass number
The radius \( R \) of a nucleus is directly proportional to the cube root of its mass number \( A \). This can be expressed mathematically as:
\[ R \propto A^{1/3} \]

### Step 2: Set up the equation using the given information
We are given that the radius of germanium is twice that of beryllium:
\[ R_{Ge} = 2 R_{Be} \]

Using the proportionality, we can write:
\[ \frac{R_{Ge}}{R_{Be}} = \left(\frac{A_{Ge}}{A_{Be}}\right)^{1/3} \]

### Step 3: Substitute the known values
Substituting \( R_{Ge} = 2 R_{Be} \) into the equation gives:
\[ \frac{2 R_{Be}}{R_{Be}} = \left(\frac{A_{Ge}}{A_{Be}}\right)^{1/3} \]
This simplifies to:
\[ 2 = \left(\frac{A_{Ge}}{A_{Be}}\right)^{1/3} \]

### Step 4: Cube both sides to eliminate the cube root
Cubing both sides results in:
\[ 2^3 = \frac{A_{Ge}}{A_{Be}} \]
\[ 8 = \frac{A_{Ge}}{A_{Be}} \]

### Step 5: Use the mass number of beryllium
The mass number of beryllium (Be) is 9. Therefore, we can substitute \( A_{Be} = 9 \):
\[ 8 = \frac{A_{Ge}}{9} \]

### Step 6: Solve for the mass number of germanium
Multiplying both sides by 9 gives:
\[ A_{Ge} = 8 \times 9 = 72 \]

### Conclusion
The number of nucleons in germanium (Ge) is \( 72 \).