Question

Answer the following questions for the Zn atom:
How many electrons have l−m=1?

Answer 1

Hint: The atomic number of zinc (\[Zn\]) is 30 and it is a d-block element. This allows us to write the complete electronic configuration of the zinc atom. The question can then be answered using the concept of quantum numbers, more specifically, the range of quantum numbers and the orbitals indicated by a certain set of quantum numbers.

Complete Step by Step Solution:
The solution of Schrodinger’s wave equation for the hydrogen atom yields three quantum numbers n, l, and m. Let’s look at them in detail now.

The principal quantum number, determines, to a large extent, the energy of the electron and the average distance of the electron from the nucleus. A higher principal quantum number means that the electron is farther away from the nucleus. This quantum number is denoted as\[n\]. The principal quantum number takes integer values from 1 onwards (1, 2, 3, 4 etc). These indicate the orbit/energy level in which the electron resides.
A part of the energy of the electron comes from its orbital motion around the nucleus. This orbital motion is described by the angular momentum quantum number (also called the azimuthal quantum number). This quantum number is denoted by\[l\]. The values \[l\] range from \[0{\rm{ to n - 1}}\]. Electrons in an atom are not only grouped into energy levels (given by n) but also into energy sub-levels (called subshells) which are described by l. The values of l also determine the shape of the orbital in which the electron resides.

Value of I	Subshell
o	s
1	p
2	d
3	f

The orbital motion of an electron resembles the flow of electric current around a closed loop. This creates a magnetic field which interacts with external magnetic fields. The electrons in each energy sublevel orient themselves in certain regions around the nucleus because of this interaction. These regions are called orbitals. The number of orbitals in each sub-level within a specific energy level is given by the magnetic quantum number denoted by \[m\]. A particular value \[l\] \[m\]can range from \[ - l\]to \[ + l\] including 0. Thus, each value \[l\] has \[2l + 1\]values \[m\] associated with it.

The complete electronic configuration of the zinc atom is\[1{s^2}2{s^2}2{p^6}3{s^2}3{p^6}3{d^{10}}4{s^2}\]. To get \[l - m = 1\], we have the following cases:
Case 1: \[n = 2,{\rm{ l = 1, m = 0}}\]which refers to the 2p orbital. The 2p orbital has 6 electrons.
Case 2: \[n = 3,{\rm{ l = 1, m = 0}}\]which refers to the 3p orbital. The 3p orbital has 6 electrons.
Case 3: \[n = 3,{\rm{ l = 2, m = 1}}\]which refers to the 3d orbital. The 3d orbital has 10 electrons.
Thus, the total number of electrons satisfying \[l - m = 1\]is \[6 + 6 + 10 = 22\].

Thus, there are 22 electrons for which \[l - m = 1\].

Note: Please keep in mind that n can take values 1, 2, 3, 4 etc. For each value of n, l ranges from 0 to n-1. For each value of l, m ranges from -l to +l through 0 thus it has 2l+1 total possible values.