Question

In the quantum mechanical model of the hydrogen atom, if the principal quantum number is 3, then the number of orbitals in that corresponding level will be 
1. 3
2. 7
3. 9
4. 5

Answer 1

Correct Answer - Option 3 : 9

Concept:

Atomic orbitals are precisely distinguished by what are known as quantum numbers.

Each orbital is designated by three quantum numbers labelled as n, l and m_l.

Principal quantum number (n): It is a positive integer with value of n = 1, 2, 3.... The principal quantum number determines the size and to large extent the energy of the orbital.

The principal quantum number also identifies the shell. With the increase in the value of ‘n’, the number of allowed orbital increases and are given by ‘n²’.

All the orbitals of a given value of ‘n’ constitute a single shell of atom and are represented by the following letters n = 1 (K), 2 (L), 3 (M), 4 (N)…

Calculation:

Given n = 3;

⇒ No of orbitals = n² = 3² = 9

Azimuthal quantum number (l): It is also known as orbital angular momentum or subsidiary quantum number.

For a given value of n, l can have n values ranging from 0 to n – 1, that is, for a given value of n, the possible value of l are : l = 0, 1, 2, .......... (n–1);

Each shell consists of one or more subshells or sub-levels. The number of subshells in a principal shell is equal to the value of n.

For example in the first shell (n = 1), there is only one sub-shell which corresponds to l = 0. There are two sub-shells (l = 0, 1) in the second shell (n = 2), three (l = 0, 1, 2) in third shell (n = 3) and so on.

Sub-shells corresponding to different values of l are represented by the following symbols,

l = 0 (s), 1(p), 2(d), 3(f), 4(g),5(h)…

Magnetic orbital quantum number (m_l): It gives information about the spatial orientation of the orbital with respect to standard set of co-ordinate axis.

For any sub-shell (defined by ‘l’ value), 2l+1 values of m_l are possible and these values are given by,

m_l = - l, - (l-1), - (l-2)... 0,1... (l -2), (l-1), l;