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Home , Aufbau principle, Azimuthal quantum number, Energy level, Magnetic quantum number, Quantum number, Spin quantum number

The Mechanical Model of the

Quantum Numbers In order to describe the probable location of , they are assigned four numbers called quantum numbers.

The quantum numbers of an are kind of like the electron’s “address”. No two electrons can be described by the exact same four quantum numbers. This is called The Pauli Exclusion Principle.

• Principle : The principle quantum number describes which orbit the electron is in and therefore how much the electron has. - it is symbolized by the letter n. - positive whole numbers are assigned (not including 0): n=1, n=2, n=3 , etc - the higher the number, the further the orbit from the nucleus - the higher the number, the more energy the electron has (this is sort of like Bohr’s energy levels) - the orbits (energy levels) are also called shells

• Angular (azimuthal) quantum number: The describes the sublevels (subshells) that occur in each of the levels (shells) described above. - it is symbolized by the letter l - positive whole number values including 0 are assigned: l = 0, l = 1, l = 2, etc. - each number represents the shape of a subshell: l = 0, represents an s subshell l = 1, represents a p subshell l = 2, represents a d subshell l = 3, represents an f subshell - the higher the number, the more complex the shape of the subshell.

The picture below shows the shape of the s and p subshells: (notice the electron clouds)

• : All of the subshells described above (except s) have more than one orientation. The magnetic quantum number describes the orientation of the subshell that the electron is found in. - it is symbolized by the letter m - negative and positive numbers, including 0 are assigned - each type of subshell has a particular number of orientations: the s subshell has 1 orientation: m = 0 the p subshell has 3 orientations : m = -1, 0, 1 the d subshell has 5 orientations : m = -2, -1, 0, 1, 2 the f subshell has 7 orientations : m = -3, -2, -1, 0, 1, 2, 3

The picture below shows the shapes and orientations of the s, p, and d subshells. The f subshells are even more complex.

• quantum number – The describes the direction of the spin of an electron – clockwise or counter clockwise. - it is symbolized by the letter s - it is assigned the numbers ½ or -½ - s = ½ for the first electron or s = -½ for the second electron

Energy Levels and Subshells • The number of types of subshells at each is equal to the principle quantum number of that energy level:

energy level (n) # of types of subshells types of subshells 1 1 s 2 2 s, p 3 3 s, p, d 4 4 s, p, d, f

• We can also write the energy level and subshell designation together: 1s 2s, 2p 3s, 3p, 3d 4s, 4p, 4d, 4f etc

Adding Electrons • The Loveseat Analogy : A loveseat is a couch that only holds two people . Imagine that each subshell has a certain number of loveseats as follows (each line represents 1 loveseat)

s subshells have 1 loveseat __ p subshells have 3 loveseats ______d subshells have 5 loveseats ______f subshells have 7 loveseats ______

Now imagine that each loveseat can hold a boy electron (symbolized by an up arrow) and a girl electron (symbolized by a down arrow) – so if all of the couches are filled up, we have

s subshells have 1 loveseat

p subshells have 3 loveseats

d subshells have 5 loveseats

f subshells have 7 loveseats

Notice that: s subshells can hold a total of 2 electrons p subshells can hold a total of 6 electrons d subshells can hold a total of 10 electrons f subshells can hold a total of 14 electrons

Very Important: When we add electrons to our loveseats, we must add all of the boy electrons (up arrows) to all of the loveseats in a subshell , before we add any girl electrons (down arrows). This is called Hund’s Rule .

• Electrons are added to the orbital subshells with the lowest energy first. This is called The .

• Electrons are not added to the next orbital subshell until the previous subshell has been completely filled up.

• The chart to the right shows the order of increasing subshell energy.

• Since electrons are added to the lowest energy subshell first, the order of electron addition to subshells is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d

• Notice that the order of the subshells is not always in numerical order

• An easier way to remember the order that electrons are added to subshells is to memorize the following chart. Follow the arrows to know which subshell comes next. This is called the diagonal rule.

Putting It All Together – Electron Configurations, Orbital Notations, and Quantum Numbers

• – We say that we have written the electron configuration for an element when we describe its electrons using the the principle quantum numbers (n), the subshell types (s, p, d, or f), and superscripts indicating the number of electrons in each subshell.

i.e. 1s 22s 22p 4

• Orbital Notation – We say that we have written the orbital notation of an element when we describe its electrons using the “loveseat analogy”.

i.e. 1s 2 2s 2

• Quantum Numbers – We say that we are assigning quantum numbers to an electron when we use the four quantum numbers explained above to describe an electron.

i.e. n = 1, l = 1, m = -1, s = -½

Example 1: Write the electron configuration and orbital notation for lithium. Assign quantum numbers for each of lithium’s electrons. Lithium has 3 electrons .

Electron Configuration: Refer to the diagonal rule and add 3 electrons. 1s 22s 1

The 1s subshell is completely filled first. There is one electron left – it goes in the 2s subshell.

Orbital Notation: Refer the diagonal rule and add arrows representing the 3 electrons. You should write the electron configuration under the orbital notation.

1s 2 2s 1

Quantum Numbers : Refer to the explanation of quantum numbers above. Describe the first electron (the first up arrow) n = 1 (because it is 1s) l = 0 (because its an s) m = 0 (because s subshells only have one orientation) s = ½ (because it is an up arrow)

Describe the second electron (the down arrow) n = 1 (because it is 1s) l = 0 (because its an s) m = 0 (because s subshells only have one orientation) s = -½ (because it is a down arrow)

Describe the third electron (the second up arrow) n = 2 (because it is 2s) l = 0 (because its an s) m = 0 (because s subshells only have one orientation) s = ½ (because it is an up arrow)

Example 2: Write the electron configuration and orbital notation for . Assign quantum numbers for each of oxygen’s electrons. Oxygen has 8 electrons .

Electron Configuration : Refer to the diagonal rule and add 8 electrons. 1s 22s 22p 4

The 1s subshell is completely filled first. The 2s subshell is completely filled next. There are four electrons left – they go in the 2p subshell.

Orbital Notation: Refer the diagonal rule to add the electrons. Don’t forget Hund’s rule which says that all of the up arrows within a subshell must be added first.

1s 2 2s 2 2p 4

Quantum Numbers : Describe the first electron (the first up arrow ) n = 1 (because it is 1s) l = 0 (because its an s) m = 0 (because s subshells only have one orientation) s = ½ (because it is an up arrow) Describe the second electron (the first down arrow ) n = 1 (because it is 1s) l = 0 (because its an s) m = 0 (because s subshells only have one orientation) s = -½ (because it is a down arrow)

Describe the third electron (the second up arrow ) n = 2 (because it is 2s) l = 0 (because its an s) m = 0 (because s subshells only have one orientation) s = ½ (because it is an up arrow)

Describe the fourth electron (the second down arrow ) n = 2 (because it is 2s) l = 0 (because its an s) m = 0 (because s subshells only have one orientation) s = -½ (because it is a down arrow)

Describe the fifth electron (the third up arrow ) n = 2 (because it is 2p) l = 1 (because its a p) m = -1 (because p subshells have three orientations – each love seat represents a different orientation and gets a different number ______) -1 0 1 s = ½ (because it is an up arrow)

Describe the sixth electron (the fourth up arrow ) – remember Hund’s rule! n = 2 (because it is 2p) l = 1 (because its an p) m = 0 (because the electron is in the second p seat) s = ½ (because it is an up arrow)

Describe the seventh electron (the fifth up arrow ) – remember Hund’s rule! n = 2 (because it is 2p) l = 1 (because its an p) m = 1 (because the electron is in the third p seat) s = ½ (because it is an up arrow)

Describe the eighth electron (the third down arrow ) n = 2 (because it is 2p) l = 1 (because its an p) m = -1 (because the electron is in the first p seat) s = -½ (because it is a down arrow)

Write the electron configuration and orbital notation for each of the elements below. 1. boron 2. fluorine 3. phosphorous 4. calcium 5. germanium

Write the quantum numbers for the designated electron. (refer to your answers above) 1. the third electron in boron 2. the ninth electron in fluorine 3. the twelfth electron in phosphorous 4. the nineteenth electron in calcium 5. the thirty-second electron in germanium