Question

What do each of the following quantum numbers describe about an electrons place of residence? Draw a picture to illustrate each.

a) principal quantum number (n)

b) angular momentum quantum number (l)

c) magnetic quantum number (ml)

d) spin quantum number (ms)

Answer 1

The Principal Quantum Number (n)

The principal quantum number, n, 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=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 n 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=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.

k =1,l=2, m=3,n=4......

n=1,2,3,4…

2

The Orbital Angular Momentum Quantum Number (l)

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 l. (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)

3)

The Magnetic Quantum Number (ml)

The magnetic quantum number ml determines the number of orbitals and their orientation within a subshell. Consequently, its value depends on the orbital angular momentum quantum number l. Given a certain l, ml is an interval ranging from –l to +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

	s orbitals	p orbitals	d orbitals	f orbitals
l	0	1	2	3
ml	0	-1, 0, +1	-2, -1, 0, +1, +2	-3, -2, -1, 0, +1, +2, +3
Number of orbitals in designated subshell	1	3	5	7

4)

The Electron Spin Quantum Number (ms)

Unlike n, l, and ml, the electron spin quantum number ms 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 ms 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. (For more information, see Electron Spin.)

ms=±12