The ratio of the nuclear radius, of an atom with mass number A and` ( 4)/( 2) H e` is `( 14)^(1//3)` . What is the value of A ?

Answer

To solve the problem, we need to find the mass number \( A \) of an atom given the ratio of its nuclear radius to that of helium. The nuclear radius \( R \) is related to the mass number \( A \) by the formula:

\[ R = R_0 A^{1/3} \]

where \( R_0 \) is a constant.

### Step-by-Step Solution:

1. **Write the formula for the nuclear radius of the atom with mass number \( A \)**:
\[ R_A = R_0 A^{1/3} \]

2. **Write the formula for the nuclear radius of helium (\( ^4_2He \))**:
\[ R_{He} = R_0 (4)^{1/3} \]

3. **Set up the ratio of the nuclear radii**:
According to the problem, the ratio of the nuclear radius of the atom with mass number \( A \) to that of helium is given as:
\[ \frac{R_A}{R_{He}} = \left( \frac{14}{1} \right)^{1/3} \]

4. **Substitute the expressions for \( R_A \) and \( R_{He} \)**:
\[ \frac{R_0 A^{1/3}}{R_0 (4)^{1/3}} = \left( \frac{14}{1} \right)^{1/3} \]

5. **Cancel \( R_0 \) from both sides**:
\[ \frac{A^{1/3}}{(4)^{1/3}} = \left( \frac{14}{1} \right)^{1/3} \]

6. **Cross-multiply to eliminate the fraction**:
\[ A^{1/3} = (14)^{1/3} \cdot (4)^{1/3} \]

7. **Combine the terms on the right-hand side**:
\[ A^{1/3} = (14 \cdot 4)^{1/3} \]

8. **Calculate \( 14 \cdot 4 \)**:
\[ 14 \cdot 4 = 56 \]

9. **Now we have**:
\[ A^{1/3} = (56)^{1/3} \]

10. **Cube both sides to solve for \( A \)**:
\[ A = 56 \]

Thus, the value of \( A \) is \( 56 \).