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: {Determining the principal quantum number}
The energy level for hydrogen-like atoms is defined by: \[ E_n = -\frac{R_H}{n^2} \] Given the energy \( E = -\frac{R_H}{9} \), by comparison with the energy formula: \[ \frac{R_H}{n^2} = \frac{R_H}{9} \Rightarrow n^2 = 9 \Rightarrow n = 3 \]
Step 2: {Calculating the degeneracy}
For the principal quantum number \( n = 3 \), the allowed azimuthal quantum numbers are \( l = 0, 1, 2 \), which define the subshells: \[ (3s, 3p, 3d) \] The number of orbitals within each subshell is: \[ 3s = 1, \quad 3p = 3, \quad 3d = 5 \] The total number of orbitals for \( n = 3 \) is the sum of orbitals in each subshell: \[ 1 + 3 + 5 = 9 \] Therefore, the degeneracy is 9, corresponding to option (D).