Question

Give the names and symbols of the four quantum numbers required to define the energy of electrons in atoms. What do these quantum numbers relate to, and what numerical values are possible for each? Show how the shape of the periodic table is related to these quantum numbers.

Answer 1

Hint: Quantum numbers don't tell the exact location of electrons but rather it tells where the highest probability of finding electrons is.
Complete solution: Before understanding the concept of Quantum numbers, we need to keep in mind certain points, such as- electron doesn't revolve in shell or orbit rather it revolves in an orbital; an orbital is a 3-dimensional space in which probability of finding the electron is maximum.

Complete step by step answer:
Quantum numbers can be defined as the values that describe the energy or state of energy of an electron, present in an atom. There are in total 4 quantum numbers. These numbers indicate spin, energy, magnetic moment and angular moment of the electron.
Let us try to understand about each quantum numbers in brief-
•Principal quantum number- It is designated as$n$. It identifies the distance of an electron from the nucleus of an atom, and also the size of the orbital. It also describes the energy of an orbital as electrons are in a constant state of motion, have opposite charges, and are attracted to the nucleus. For example, When $n = 1$ , an electron is said to be in the ground state. For $n = 2$ , the orbitals are in an excited state. Smaller value of $n$ means the orbital is closer to the nucleus and thus having less energy. Greater the distance of the electron from the nucleus, higher will be the energy of the orbital.
Energy expression- ${E_{{n^{th}}}} = - 13.6 \times \dfrac{{{Z^2}}}{{{n^2}}}$, where $n$corresponds to shell number and $Z$ is atomic number.
Possible values of $n$ = all positive integers excluding 0.
•Azimuthal quantum number- it is also known as the angular momentum quantum number. It is designated as $l$ . It tells about the shape of the orbital. It also gives information on which suborbital, or atomic shell layer, we can find an electron in. for $l = 0$ orbital have spherical shapes , polar shapes for $l = 1$ and cloverleaf shapes for $l = 2$ . $l = 3$ is for clover-shaped orbital having an extra petal. Orbital tend to have more complex shapes in presence of additional petals. Angular quantum numbers can take any integral values between 0 and $n - 1$ that describe the shape of an orbital. When subshells or suborbitals are present, a letter represents each type: $s$ for $l = 0$ , $p$ for $l = 1$ , $d$ for $l = 2$ and $f$ for $l = 3$.
•Magnetic quantum number- It is designated as $m$ . it gives information about an orbital’s orientation on the basis of its shape and energy. it can have integral values of -2, -1, 0, +1 or +2. . In addition, each subshell has $2l + 1$ orbitals.
•Spin quantum number- According to the Pauli Exclusion Principle , no two electrons can have the same values of $n,l,m$ or $s$.Therefore, a maximum of two electrons can be occupied in the same orbital. When two electrons are present in the same orbital, they need to have spin in opposite directions, as they create a magnetic field. The spin quantum number, designated as $s$gives the information about the direction in which an electron is spinning. Since, there are only two possible directions in which an electron can spin, clockwise or counterclockwise - these directions are represented by $ + \dfrac{1}{2}$ or $ - \dfrac{1}{2}$ .

Note:
The filling of Orbitals is as per $n + l$ rule, where $n$ is the principal quantum number and $l$ corresponds to the subsidiary quantum number. This rule explains the energy content of orbitals, for example,why the $4s$ orbital has a lower energy than that of $3d$ orbital, and in this way, it gives the periodic table its characteristic appearance.