Question

The degeneracy of H – atom in a shell is 9. The value of the principal quantum number (n) of the shell is?
A.3
B.9
C.1
D.None of these

Answer 1

Hint: To solve this question, we must first understand the concepts of degeneracy and quantum numbers. Then we must find the relation between the principal quantum number and the degeneracy of the given hydrogen atom, which can be used to find the final answer.

Complete step by step answer:
Before we move forward with the solution of the given question, let us first discuss some important basic concepts.
-In reference to the hydrogen atom, each quantum state of the atom can be represented by 3 quantum numbers, viz. principal quantum number, angular momentum quantum number, and the magnetic quantum number. These quantities are represented using the constants ‘n’, ‘l’, and ‘m’ respectively.
-Now, a relation between the principal quantum number and the degeneracy of the given hydrogen atom can be given as: degeneracy of the hydrogen atom is equal to the square value of the principal quantum number. This relation can be mathematically represented as:
Degeneracy = (principal quantum number) \[^2\]
Degeneracy = \[{n^2}\]
Hence substituting the given values in the above equation, we get,
 \[9 = {n^2}\]
 \[n = \sqrt {9 = 3} \]
Hence the value of the principal quantum number (n) of the shell is 3.

Hence, Option A is the correct option.

Note:
Degeneracy of an atom can be understood as the total number of different states of the same energy level. To explain it in simpler terms, degeneracy can be understood as the possession of the same energy level by different wave functions or atomic orbitals. In another context, it can also be related to the coincidental numerical agreement presented in the Bohr’s atomic model.