Question

(a) For a given value of the principal quantum number nfor a hydrogen atom, how many values of the orbital quantum number Iare possible?

(b) For a given value of I, how many values of the orbital magnetic quantum number${\mathbf{m}}_{\mathbf{I}}$are possible?

(c) For a given value of n, how many values of${\mathbf{m}}_{\mathbf{I}}$are possible?

Short answer

(a) There are distinct values of orbital quantum number I are possible For a given value of the principal quantum number for a hydrogen atom.

(b) There are 2I + 1 distinct values of orbital magnetic quantum number $m_I$for a given value of .

(c) There are 2n + 1 distinct values of orbital magnetic quantum number $m_I$for a given value of n .

Step-by-step solution

Principal, orbital, and orbital magnetic quantum number:

Each atom's electron is given one of four quantum numbers, the primary quantum number n , to describe the state of that electron.

A measurement of the angular momentum's magnitude is the orbital quantum number I .

The angular momentum vector's orientation in space is connected to the orbital magnetic quantum number${\mathbf{m}}_{\mathbf{I}}$ .

(a) Finding how many values of the orbital quantum number  are possible for a given value of the principal quantum number  for a hydrogen atom:

For a hydrogen atom, the values of orbital quantum number I for a given value of n are 0,1,2,....,(n-1) .

Hence, there are distinct values of orbital quantum number I are possible For a given value of the principal quantum number n for a hydrogen atom.

(b) Finding how many value the orbital magnetic quantum number mI are possible for a given value of I :

For a hydrogen atom, the values of orbital magnetic quantum number for a given value of I is $-\mathrm{I},-(\mathrm{I}-1),-(\mathrm{I}-2),............,+(\mathrm{I}-1),+\mathrm{I}$.

Hence, there are 2I+1 distinct values of orbital magnetic quantum number $m_I$for a given value of I .

(c) Finding how many value the orbital magnetic quantum number  are possible for a given value of

In step 3, there are 2I+1 distinct values of orbital magnetic quantum number $m_I$for a given value of I .

Also, in step 2, there are n distinct values of orbital quantum number I are possible For a given value of the principal quantum number for a hydrogen atom.

The conclusion is that, there are 2n+1 distinct values of orbital magnetic quantum number $m_I$ for a given value of n .