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Quantum Numbers
Principal Quantum Number (n)
Angular Momentum Quantum Number (l)
Magnetic Quantum Number (ml)
Spin Quantum Number (ms)
For each of the four quantum numbers, n, l, ml, and ms you’ll need to know the name and symbol
of the quantum number, the possible values of the number, what the numbers tell you about the
energy of the electron, and what the number tells you about the volume of space where the
electron is most likely to exist. Each electron can be described by a unique combination of values
of the four quantum numbers.
(n) This quantum number is what we think of as the shell number.
Possible values: 1, 2, 3, 4, 5, etc.
Energy of electrons: Higher numbers indicate higher energy levels.
Physical description: Higher numbers suggest that the electron is further away from the nucleus
of the atom.
(l) This quantum number is what we think of as the subshell number.
Possible values: 0, 1, 2, 3
0 = s subshell
1 = p subshell
2 = d subshell
3 = f subshell
Energy: Larger l values correspond to slightly higher energies.
Physical description: l refers to the shape of the space where the electron is likely located:
l = 0 or s is shaped like a sphere (one lobe, no nodes)
s
l = 1 or p is shaped like a dumb-bell (two lobes, one node)
p
l = 2 or d is shaped like a four-leaf clover (four lobes, two nodes)
l = 3 or f is complicated structure that your instructor will probably not ask you to have to
draw (eight lobes, four nodes).
Typically the first two quantum numbers (n, l) are indicated together such as 1s or 4d.
(ml) This quantum number is what we think of as the orbital number.
Possible values: –l ..., –2, –1, 0, +1, +2, ... +l; all the integers from the negative l value to the
positive l value are possible:
l = 0, ml = 0—there is only one value
l = 1, ml = –1, 0, or +1—there are three possible values
l = 2, ml = –2, –1, 0, +1, +2—there are five possible values
l = 3, ml = –3, –2, –1, 0, +1, +2, +3—there are seven possible values.
Each possible value corresponds to an orbital. Note that each orbital can hold a maximum of
two electrons. Therefore, an s subshell can hold 2 electrons, p can hold 6, d can hold 10, and
f can hold 14 electrons when completely filled.
Energy = the orbitals for a given l are said to be equal in energy (degenerate).
Physical description = orientation of the space where the corresponding electrons are most likely
to be. Each orbital corresponds to a different orientation the electrons could have.
(ms) This quantum number refers to the electron spin.
Possible values: +1/2, -1/2
Energy: The energy for spinning either direction is equivalent. However, the energy is lower
when electrons are unpaired and when they are spinning in the same direction.
Physical description: the electron is either spinning one way (+) or the other (-).
Practice Problems
1) Which of the following (n, l, ml, ms) are NOT possible:
c.
c.
c.
(2, 3, 0, +1/2)
(3, 1, –1, –1/2)
(4, 0, 1, +1/2)
Answer
2) How many electrons fit each of the following descriptions:
a. n = 1
b. n = 3
c. n = 2, l = 0
d. n = 4, l = 2
e. n = 4, l = 2, ml = -1
f. n = 4, l = 2, ml = -3
g. n = 3, l = 1, ml = 0, ms = +1/2
h. n = 3, l = 1, ml = 0, ms = -1/2
Answer
Answers
1) a. The possible l values can be up to n-1. Since n = 2, 3 is not a valid l number. Said another
way, there is no f subshell in the second shell.
c. ml can be –l to +l; if l = 0, the only possible value for ml is 0. Said another way, l=0 refers
to the s subshell which contains only one orbital which is named 0.
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2) How many electrons fit each of the following descriptions:
a. (2 electrons)
b. (18 electrons)
c. (2 electrons max in an s subshell)
d. (10 electrons max in a d subshell)
e. (2 electrons max in an orbital)
f. (0 electrons. This is not a valid quantum number sequence).
g. (1 electron)
h. (1 electron)
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