Question

How many orbitals can have the designation$\mathbf{5}\mathit{p}\mathbf{,}\mathbf{3}{\mathit{d}}_{{\mathbf{z}}^{\mathbf{2}}}\mathbf{,}\mathbf{4}\mathit{d}\mathbf{,}\mathit{n}\mathbf{=}\mathbf{5}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{n}\mathbf{=}\mathbf{4}\mathbf{?}$

Short answer

Three orbitals are possible for 5 p

One unique orbital is possible for $3d_{z^2}$.

25 orbitals are possible for n=5.

16 orbitals are possible for n=4.

Step-by-step solution

The quantum number.

There are four quantum numbers that serve basis of quantum aspects related to size, position, orientation, and spin associated with an electron.

Principal quantum number depicted asdetermines size and energy of a particular shell. It may have values from $\mathit{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}$and so on.

Azimuthal quantum number describes shape and determines orbital angular momentum associated with a particular orbital. It is characterized by valuesIequal to 0 for s-orbital,Iequal to 1 for p-orbital,Iequal to 2 for d-orbital andIequal to 3 for f-orbital orbitals and so on.

Magnetic quantum number determines orientation of certain orbital. Its values range from-Ito+I. Each of such orbital gas quantize values of angular momentum that is determined from values ofI.

Spin quantum numbers help to predict intrinsic spin associated with electron. It can take only two values ${\mathit{m}}_{\mathbf{s}}$equal to$+1/2$or ${\mathit{m}}_{\mathbf{s}}$equal to $-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.

Find the number of orbitals that can have the designation5p,3dz2,4d,n=5 and n=4? 

For orbital 5p the value of n=5 and I=1. Since ml ranges from -I to +I so values of can range from -1 to +1 so three orbitals are possible for 5 p that include 5p_x,5p_y and 5p_z.

For orbital $3d_{z^2}$the value of n=3 and I=2. Since m_l ranges from -I to +I so values of m_l can range from -2 to +2. However,$d_{z^2}$ is associated with m_l=0 thus only one unique orbital is possible for $3d_{z^2}$.

For orbital 4d the value of n=4 and I=2. Since ranges from -I to +I so values of m_l can range from -2 to +2 so five orbitals are possible for 4 d.

For n=5 the permitted values of azimuthal quantum number must range from 0 to 4. Therefore I=0,1,2,3,4 are possible. Therefore 5 s, 5 p, 5 d, 5 f and 5 g orbitals are possible. Total 25 orbitals are possible.

For n=4 the permitted values of azimuthal quantum number must range from 0 to 3. Therefore I=0,1,2,3 are possible. Therefore 4 s, 4 p, 4 d, 4 f orbitals are possible. Total 16 orbitals are possible.