Question

The degeneracy of hydrogen atom that has energy equal to \(-\frac{R_H}{9}\) is (where \( R_H \) = Rydberg constant)

• A. 6
• B. 8
• C. 5
• D. 9

Hint:

Degeneracy refers to the number of orbitals with the same energy level. It is given by \( n^2 \) for a hydrogen-like atom.

Answer 1

The Correct Option is D

Step 1: {Understanding degeneracy}
The energy of hydrogen-like atoms is given by: \[ E_n = -\frac{R_H}{n^2} \] Given \( E = -\frac{R_H}{9} \), comparing with the formula: \[ \frac{R_H}{n^2} = \frac{R_H}{9} \Rightarrow n^2 = 9 \Rightarrow n = 3 \] Step 2: {Finding the degeneracy}
For \( n = 3 \), the possible values of \( l \) are \( 0, 1, 2 \), corresponding to subshells: \[ (3s, 3p, 3d) \] Each subshell contains: \[ 3s = 1, \quad 3p = 3, \quad 3d = 5 \] Total orbitals present: \[ 1 + 3 + 5 = 9 \] Thus, the correct answer is (D).

Answer 2

Step 1: Understand energy levels in a hydrogen atom
The energy of an electron in the nth orbit of a hydrogen atom is given by:
\( E_n = -\frac{R_H}{n^2} \)
Where:
- \( R_H \) is the Rydberg constant
- \( n \) is the principal quantum number

Step 2: Use the given energy value
We are told that the energy is:
\( E = -\frac{R_H}{9} \)
Compare with the general formula:
\( -\frac{R_H}{n^2} = -\frac{R_H}{9} \)
So, \( n^2 = 9 \) → \( n = 3 \)

Step 3: Find the degeneracy
Degeneracy of an energy level in hydrogen is given by:
Degeneracy = \( n^2 \)
So, for \( n = 3 \):
Degeneracy = \( 3^2 = 9 \)

Step 4: Final Answer
The degeneracy of the hydrogen atom at energy \( -\frac{R_H}{9} \) is:
9