Question

What is the orbital degeneracy of the levels in a hydrogen-like atom (atomic number Z in parenthesis) that has an energy of: a) E= -4hcR (Z=2) b) E=-1/4 hcR (Z=4) c) E=-hcR (Z=5) where R is the Rydberg constant

Answer 1

Degeneracy is defined as number of orbitals in shell which are having same energy.

In H like atom energy order of orbital is as follows

1s< (2s=2p) <( 3s= 3p =3d)

now degeneracy of 1 orbit =1

  degeneracy of 2 orbit =4

degeneracy of 3 orbit =9

similarly degeneracy of n^th orbit = n²

Using Rydberg Equation

Multiply this by hc

We know that emitted or absorbed photon energy

Now compare both equation

so

Part (a)

given E = -4hcR

Put Z=2 (given)

so n= 1

Hence degeneracy =1

(b) Similaly solving (b) part

n=8

so

degeneracy =64

(c)

n =5

degeneracy = 25