Question

what are four Quantum numbers and explain it with detail

Answer 1

The Four Electronic Quantum Numbers

Quantum numbers designate specific shells, subshells, orbitals, and spins of electrons. This means that they describe completely the characteristics of an electron in an atom, i.e., they describe each unique solution to the Schrödinger equation, or the wave function, of electrons in an atom. There are a total of four quantum numbers: the principal quantum number (n), the orbital angular momentum quantum number (l), the magnetic quantum number (ml), and the electron spin quantum number (ms). The principal quantum number, nn, describes the energy of an electron and the most probable distance of the electron from the nucleus. In other words, it refers to the size of the orbital and the energy level an electron is placed in. The number of subshells, or ll, describes the shape of the orbital. It can also be used to determine the number of angular nodes. The magnetic quantum number, ml, describes the energy levels in a subshell, and ms refers to the spin on the electron, which can either be up or down.

The Principal Quantum Number (nn)

The principal quantum number, nn, designates the principal electron shell. Because n describes the most probable distance of the electrons from the nucleus, the larger the number n is, the farther the electron is from the nucleus, the larger the size of the orbital, and the larger the atom is. n can be any positive integer starting at 1, as n=1n=1 designates the first principal shell (the innermost shell). The first principal shell is also called the ground state, or lowest energy state. This explains why nn can not be 0 or any negative integer, because there exists no atoms with zero or a negative amount of energy levels/principal shells. When an electron is in an excited state or it gains energy, it may jump to the second principle shell, where n=2n=2. This is called absorption because the electron is "absorbing" photons, or energy. Known as emission, electrons can also "emit" energy as they jump to lower principle shells, where n decreases by whole numbers. As the energy of the electron increases, so does the principal quantum number, e.g., n = 3 indicates the third principal shell, n = 4 indicates the fourth principal shell, and so on.

n=1,2,3,4…(1.1)(1.1)n=1,2,3,4…

EXAMPLE 1.11.1

If n = 7, what is the principal electron shell?

SOLUTION

EXAMPLE 1.21.2

If an electron jumped from energy level n = 5 to energy level n = 3, did absorption or emission of a photon occur?

SOLUTION

Emission, because energy is lost by release of a photon.

The Orbital Angular Momentum Quantum Number (ll)

The orbital angular momentum quantum number l determines the shape of an orbital, and therefore the angular distribution. The number of angular nodes is equal to the value of the angular momentum quantum number ll. (For more information about angular nodes, see Electronic Orbitals.) Each value of l indicates a specific s, p, d, f subshell (each unique in shape.) The value of l is dependent on the principal quantum number n. Unlike n, the value of l can be zero. It can also be a positive integer, but it cannot be larger than one less than the principal quantum number (n-1):

l=0,1,2,3,4…,(n−1)(1.2)(1.2)l=0,1,2,3,4…,(n−1)

EXAMPLE 1.31.3

If n = 7, what are the possible values of l?

Answer: Since l can be zero or a positive integer less than (n-1), it can have a value of 0, 1, 2, 3, 4, 5 or 6.

EXAMPLE 1.41.4

If l = 4, how many angular nodes does the atom have?

Answer: The number of angular nodes is equal to the value of l, so the number of nodes is also 4.

The Magnetic Quantum Number (mlml)

The magnetic quantum number mlml determines the number of orbitals and their orientation within a subshell. Consequently, its value depends on the orbital angular momentum quantum number ll. Given a certain ll, mlml is an interval ranging from –l–l to +l+l, so it can be zero, a negative integer, or a positive integer.

ml=−l,(−l+1),(−l+2),…,−2,−1,0,1,2,…(l–1),(l–2),+l(1.3)(1.3)ml=−l,(−l+1),(−l+2),…,−2,−1,0,1,2,…(l–1),(l–2),+l

EXAMPLE 1.51.5

Example: If n=3, and l=2, then what are the possible values of ml ?

SOLUTION

Answer: Since ml must range from –l to +l, then ml can be: -2, -1, 0, 1, or 2.

The Electron Spin Quantum Number (msms)

Unlike nn, ll, and mlml, the electron spin quantum number msms does not depend on another quantum number. It designates the direction of the electron spin and may have a spin of +1/2, represented by↑, or –1/2, represented by ↓. This means that when msms is positive the electron has an upward spin, which can be referred to as "spin up." When it is negative, the electron has a downward spin, so it is "spin down." The significance of the electron spin quantum number is its determination of an atom's ability to generate a magnetic field or not. (Electron Spin.)

ms=±12(1.4)(1.4)ms=±12

EXAMPLE 1.51.5

List the possible combinations of all four quantum numbers when n=2, l=1, and ml=0ml=0.

Answer 2

principal ouantum number
angular ouantum number
magnetic ouantum number
spin project ouantum number