What atomic number of an element "X" would have to become so that the 4th orbit around X would fit inside the I Bohr orbit of H atom ?

Options

A. 3
B. 4
C. 16
D. 25

Answer

To solve the problem of determining the atomic number \( Z \) of an element \( X \) such that the 4th orbit around \( X \) fits inside the Bohr orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the relationship between the radius of the orbits
The radius of the \( n \)-th orbit for a hydrogen-like atom is given by the formula:
\[ r_n = \frac{n^2 \cdot a_0}{Z} \]
where:
- \( r_n \) is the radius of the \( n \)-th orbit,
- \( n \) is the principal quantum number (orbit number),
- \( a_0 \) is the Bohr radius (approximately \( 0.529 \, \text{Å} \)),
- \( Z \) is the atomic number of the element.

### Step 2: Set the condition for the 4th orbit
We want the radius of the 4th orbit of element \( X \) to be equal to the Bohr radius of hydrogen. Therefore, we set:
\[ r_4 = a_0 \]
Substituting \( n = 4 \) into the radius formula gives:
\[ r_4 = \frac{4^2 \cdot a_0}{Z} \]
Setting this equal to \( a_0 \):
\[ \frac{16 \cdot a_0}{Z} = a_0 \]

### Step 3: Simplify the equation
To eliminate \( a_0 \) from both sides, we can divide both sides by \( a_0 \) (assuming \( a_0 \neq 0 \)):
\[ \frac{16}{Z} = 1 \]

### Step 4: Solve for \( Z \)
Now, we can solve for \( Z \):
\[ Z = 16 \]

### Conclusion
The atomic number \( Z \) of element \( X \) must be 16 for its 4th orbit to fit inside the Bohr orbit of the hydrogen atom.

### Final Answer
The atomic number of element \( X \) is \( 16 \).

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