The degeneracy of H-atom in a shell is 9. The value of principal quantum number (n) for the shell is 

Options

A. 3
B. 9
C. 1
D. None of these

Answer

To find the principal quantum number (n) for a hydrogen atom shell given that its degeneracy is 9, we can follow these steps:

### Step 1: Understand the concept of degeneracy
Degeneracy in quantum mechanics refers to the number of different quantum states that can correspond to the same energy level. For a hydrogen atom, the degeneracy of a shell is given by the formula:

\[ \text{Degeneracy} = n^2 \]

where \( n \) is the principal quantum number.

### Step 2: Set up the equation
Given that the degeneracy is 9, we can set up the equation:

\[ n^2 = 9 \]

### Step 3: Solve for n
To find the value of \( n \), we take the square root of both sides of the equation:

\[ n = \sqrt{9} \]

Calculating the square root gives:

\[ n = 3 \]

### Step 4: Conclusion
Thus, the value of the principal quantum number \( n \) for the shell is 3.

### Final Answer:
The principal quantum number \( n \) is 3.

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