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Quantum Numbers
>> Parts of this equation/concept include:
There are four things you need to know about each quantum number:
(1) its name and symbol, (2) the acceptable/possible values of the
number, (3) what the number says about the energy of the electron,
and (4) what the number says about what the electron is doing. The
fourth item will often refer to the "probability density" or volume
of space where the electron is likely to exist.
| A. Principal Quantum Number (n) |
possible values = 1, 2, 3, 4, ...
energy = larger values of n are higher in energy
physical meaning = larger values represent a larger (further from
the nucleus) probability density
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| B. Angular Momentum Quantum Number (l) |
possible values = 0, 1, 2, 3, ... n – 1
The values have alternate names you also need to know:
- l = 0 is s
- l = 1 is p
- l = 2 is d
- l = 3 is f
Hint: The numbers come from the names given to the lines
in the hydrogen spectra. The letters stand for sharp, principal,
diffuse, and fundamental. However, mnemonics are often
useful for remembering such things. I like, "Scott picks dead flowers."
energy = larger values of l represent slightly larger values
of energy
physical meaning = l corresponds to the shape of the probability
density:
- l = 0 or s is spherically shaped (one lobe, no
nodes)
- l = 1 or p is dumbbell shaped (two lobes, one
node)
- l = 2 or d is clover shaped (four lobes, two nodes)
- l = 3 or f is really complicated (eight lobes,
four nodes)
The first two quantum numbers can be expressed together; for example,
1s or 2p.
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| C. Magnetic Quantum Number (ml) |
possible values = –l ..., –2, –1, 0, +1,
+2, ... +l
energy = all ml values have the same energy (are
degenerate)
physical meaning = orientation of probability density
The important part of this is knowing how many orientations are
possible:
- For l = 0, ml = 0—there is only
one way the sphere is oriented or one s orbital
- For l = 1, ml = –1, 0, or +1—there
are three p orbitals
- For l = 2, ml = –2, –1, 0,
+1, +2—there are five d orbitals
- For l = 3, ml = –3, –2, –1,
0, +1, +2, +3—there are seven f orbitals
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| D. Spin Quantum Number (ms) |
possible values = +1/2 or –1/2
energy = both values are the same energy; however, it is lower
if electrons are not paired and spinning in the same direction
physical meaning = spin (sometimes it's obvious)
Note that only the principal and angular momentum quantum numbers
have energy effects. The differences for n are large; the
differences for l are smaller. However, many small steps
may equal a large step. A useful rule is that higher n +
l values are higher in energy. If two values of n
+ l are equal, then the one with the lower n is lower
in energy.
Lower energies are more stable. The lowest possible energy is the
"ground state."
You should be able to use the quantum numbers to describe and compare
electrons.
>> Example 1
Rank the following electrons with quantum numbers (n,
l, ml, ms) from lowest
energy to highest energy.
- (2, 1, 1, +1/2)
- (1, 0, 0, –1/2)
- (4, 1, –1, +1/2)
- (4, 2, –1, +1/2)
- (3, 2, –1, +1/2)
- (4, 0, 0, +1/2)
- (2, 1, –1, +1/2)
- (3, 1, 0, +1/2)
Solution:
Lowest energy has the lowest n + l value. Electrons
C and E have the same value (5), so the one with the lowest n
(E) is lower in energy. Electrons H and F also have the same n
+ l value, and H is lower in energy. Electrons A and G
have the same values of n and l so each electron
has the same energy.
low energy B < A = G < H < F < E < C
< D high energy
>> Example 2
Refer to the electrons from Example 1 to answer the following questions:
- (2, 1, 1, +1/2)
- (1, 0, 0, –1/2)
- (4, 1, –1, +1/2)
- (4, 2, –1, +1/2)
- (3, 2, –1, +1/2)
- (4, 0, 0, +1/2)
- (2, 1, –1, +1/2)
- (3, 1, 0, +1/2)
- Which electron is spinning in a direction different from that of the others?
- Which electron is in a spherically shaped orbital?
- Which electron is in a p orbital?
- Which electron is in a d orbital?
- Which electron is furthest from the nucleus?
- Which two electrons are in the same orbital?
- Which two electrons differ only by the orientation of their orbitals?
- Which two electrons cannot exist in the same atom?
- Which electrons are degenerate?
- Which electrons are in an f orbital?
Solution:
- The ms quantum number refers to spin. Only
electron B has a value of ms that is different
from that of the others.
- The l quantum number refers to shape. An s orbital
(l = 0) is spherically shaped; therefore any electrons
with l = 0 will have a spherically shaped orbital. Electrons
B and F.
- When l = 1, it is also called a p orbital. l
= 1 for electrons A, C, G, and H.
- A d orbital is l = 2. That is electrons D and
E.
- The distance from the nucleus is determined by the n
quantum number. The larger the number, the further from the
nucleus. The highest n among our choices is 4, and three
electrons have that value. These three electrons-C, D, and F-are
equally distant from the nucleus.
- For electrons to be in the same orbital, the first three quantum
numbers must be the same. The first two give the type of orbital;
the third gives its orientation. Two electrons spinning in opposite
directions can exist in the same orbital. In this example there
are no two electrons in the same orbital.
- The orientation of the orbital is the third quantum number,
ml. Electrons A and G differ only by this
quantum number.
- The Pauli exclusion principle says that each electron in an
atom must have a unique set of quantum numbers. Since each of
these electrons has a unique set of quantum numbers, they all
may exist in the same atom.
- Atoms with the same energy (same n and l values)
are degenerate. In this example, A and G are degenerate.
- An f orbital has l = 3. There are no electrons
in this example with an f orbital.
>> Example 3
What is wrong with the quantum numbers (n, l, ml,
ms) of the following electrons?
- (2, 2, 0, +1/2)
- (3, 1, –1, –1/2)
- (3, 1, –2, 1)
- (4, 0, 1, +1/2)
- (+1/2, 1, 1, 1)
Solution:
- The possible values for l only extend to n –
1. Since n = 2 in this electron, l can have a
value of 1 or 0; 2 is not acceptable.
- There is nothing wrong with this set of quantum numbers.
- ml may only have values from –l
to +l. In this case, –1, 0, +1; –2 is not a
valid choice. In addition, the only choices for ms
are +1/2 and –1/2, not 1.
- Since l = 0, the only valid choice for ml
is 0.
- n must be a counting number. Fractions are not permitted
(nor is zero). With that wrong, it is impossible for the other
values to have any valid choices, with the possible exception
of ms (with choices of +1/2 and -1/2), but
even that is wrong.
>> Example 4
Rank the following from highest energy to lowest energy.
3s, 5p, 4d, 1s, 5d, 3p
Solution:
Recall that s is 0, p is 1, and d is 2.
The energies can be ranked by n + l, so
3s = 3, 5p = 6, 4d = 6, 1s = 1, 5d
= 7, 3p = 4
high energy 5d > 5p > 4d
> 3p > 3s > 1s low energy
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