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-4. The ratio is given as \( \left( \frac{14}{1} \right)^{1/3} \).

### Step-by-Step Solution:

1. **Understand the formula for nuclear radius**:
The nuclear radius \( R \) of an atom is given by the formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant and \( A \) is the mass number of the atom.

2. **Write the expression for the nuclear radius of atom A**:
For the atom with mass number \( A \), the nuclear radius \( R_A \) can be expressed as:
\[ R_A = R_0 A^{1/3} \quad \text{(Equation 1)} \]

3. **Write the expression for the nuclear radius of helium-4**:
For helium-4, which has a mass number of 4, the nuclear radius \( R_{He} \) is:
\[ R_{He} = R_0 (4)^{1/3} \quad \text{(Equation 2)} \]

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

5. **Substituting the expressions for \( R_A \) and \( R_{He} \)**:
Substituting Equations 1 and 2 into the ratio gives:
\[ \frac{R_0 A^{1/3}}{R_0 (4)^{1/3}} = \left( \frac{14}{1} \right)^{1/3} \]
The \( R_0 \) cancels out, leading to:
\[ \frac{A^{1/3}}{(4)^{1/3}} = \left( \frac{14}{1} \right)^{1/3} \]

6. **Cross-multiplying to eliminate the fraction**:
This can be rearranged to:
\[ A^{1/3} = 4^{1/3} \cdot 14^{1/3} \]

7. **Cubing both sides to solve for A**:
To eliminate the cube root, we cube both sides:
\[ A = 4 \cdot 14 \]

8. **Calculating the value of A**:
Now, calculate the product:
\[ A = 56 \]

### Final Answer:
The value of \( A \) is \( 56 \).

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