The ratio of radii of nuclei `._13 A1^27` and `._52 X^A` is `3 : 5`. The number of neutrons in the nuclei of `X` will be

Options

A. 52
B. 73
C. 125
D. 13

Answer

To solve the problem, we need to find the number of neutrons in the nucleus of element X given the ratio of the radii of the nuclei of aluminum and X.

### Step-by-step Solution:

1. **Understanding 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.

2. **Setting Up the Ratio of Radii**:
Given the ratio of the radii of the nuclei of aluminum (Al) and element X:
\[ \frac{r_{Al}}{r_X} = \frac{3}{5} \]
We can express this in terms of their mass numbers \( A_{Al} \) and \( A_X \):
\[ \frac{r_{Al}}{r_X} = \left(\frac{A_{Al}}{A_X}\right)^{1/3} \]

3. **Substituting Known Values**:
The mass number of aluminum is given as \( A_{Al} = 27 \). Thus, we can write:
\[ \frac{3}{5} = \left(\frac{27}{A_X}\right)^{1/3} \]

4. **Cubing Both Sides**:
To eliminate the cube root, we cube both sides:
\[ \left(\frac{3}{5}\right)^3 = \frac{27}{A_X} \]
This simplifies to:
\[ \frac{27}{125} = \frac{27}{A_X} \]

5. **Solving for \( A_X \)**:
By cross-multiplying, we find:
\[ 27 \cdot A_X = 27 \cdot 125 \]
Dividing both sides by 27 gives:
\[ A_X = 125 \]

6. **Finding the Number of Neutrons**:
The number of neutrons \( N \) in the nucleus can be calculated using the formula:
\[ N = A - Z \]
where \( Z \) is the atomic number (number of protons). For element X, \( Z = 52 \):
\[ N = 125 - 52 = 73 \]

### Final Answer:
The number of neutrons in the nucleus of element X is \( 73 \).

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