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ChemWiki: The Dynamic Chemistry E-textbook > Physical Chemistry > Quantum Mechanics > Atomic Theory > Electrons in Atoms > Electronic Orbitals

Atoms are composed of a nucleus containing neutrons and protons with electrons dispersed throughout the atom. Electrons, however, are not simply floating within the atom but are instead fixed within electron orbitals. Electron orbitals are regions within the atom where electrons have the highest probability of being found.

There are multiple electron orbitals within an atom. Each has their own energy level associated to them, and their own properties. Because each orbital is different, we assign them specific quantum numbers: 1s, 2s, 2p 3s, 3p,4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. The numbers, (n=1,2,3, etc.) are called *principal quantum numbers *and can only be positive numbers. The letters (*s,p,d,f*) are representative of the *orbital angular momentum quantum number *(*ℓ*) and the orbital angular momentum quantum number may be 0 or a positive number but can never be greater than n-1. Each letter is paired with a specific *ℓ* value:

S subshell = 0

P subshell = 1

D subshell = 2

F subshell = 3

Orbitals are also described by their *magnetic quantum number *(m_{ℓ}). The magnetic quantum number can range from *– ℓ* to +

Orbitals that have the same or identical energy level are known as **Degenerate. **An example is the 2p orbital: 2p_{x} has the same energy level to 2p_{y}. This concept becomes more important in Molecular Orbitals. The **Pauli exclusion principle** states that no two electrons can have the same exact orbital configuration; in other words, the same exact quantum numbers. However, the electron can exist with spin up (m_{s} = +1/2) or with spin down (m_{s} = -1/2). This means that the s orbital can contain up to two electrons, the p orbital can contain up to six electrons, the d orbital can contain up to 10 electrons, and the f orbital can contain up to 14 electrons.

s subshell | p subshell | d subshell | f subshell |

ℓ = 0 | ℓ = 1 | ℓ = 2 | ℓ = 3 |

m_{ℓ} = 0 | m_{ℓ}= -1, 0, +1 | m_{ℓ}= -2, -1, 0, +1, +2 | m_{ℓ}= -3, -2, -1, 0, +1, +2, +3 |

One s orbital | Three p orbitals | Five d orbitals | Seven f orbitals |

Two s orbital electrons | Six p orbital electrons | 10 d orbital electrons | 14 f orbital electrons |

As discussed in the previous section, the magnetic quantum number (m_{l} ) can range from –l to +l. This tells us how many lobes (orbitals) there are in s, p, d, and f subshells. As you can see from the chart above, the s subshell has one lobe, the p subshell has three lobes, the d subshell has five lobes, and the f subshell has seven lobes. Each of these lobes is labeled differently and is named due to which plane the lobe is resting on. If the lobe lies along the x plane, then it is labeled with an x such as 3p_{x}. If the lobe lies along the xy plane, then it is labeled with an x and a y such as dxy. Electrons will be found where the lobes are found. The plane (or planes) that the orbitals do not fill are called nodes. These are places where there is a 0 probability density of finding electrons. For example, if we have the d orbital of d_{yx}, then there are nodes on planes xz and yz. This can be seen in the diagrams below.

There are two types of nodes that can occur; angular and radial nodes. An **angular node **is a flat plane such as the ones shown in the diagram above. The *ℓ *quantum number determines the number of angular nodes an orbital will have. A r**adial node **is a circular ring that occurs as the principle quantum number increases. Thus, n tells us how many radial nodes an orbital will have and is calculable with the equation: Total # of nodes = n-1.

For example, let us determine the nodes in the 3p_{z} orbital. We are given that n = 3 and *ℓ *= 1 because of the p orbital. We can determine the total number of nodes present in this orbital because: nodes = n-1. In this case, 3-1=2, so there is a total of 2 nodes. The quantum number *ℓ *tells us how many angular nodes there are, so there is 1 angular node, specifically on the xy plane because this is a p_{z} orbital. Since there is one node left, there must be one radial node. To sum up, the 3p_{z} orbital has 2 nodes: 1 angular node and 1 radial node.

Another example is the 5d_{xy} orbital. We know that there are four nodes total (5-1=4) and that there are two angular nodes (d orbital has a quantum number *ℓ*=2) on the xz and zy planes. This means there there must be two radial nodes. You can only calculate the number of radial and angular nodes if the principle quantum number, type of orbital (s,p,d,f), and the plane that the orbital is resting on (x,y,z, xy, etc.) are given.

5d orbital (two radial nodes and two angular nodes)

We can think of an atom like a hotel. The nucleus is the lobby where the protons and neutrons are, and in the floors above, we find the rooms (orbitals) with the electrons. The principle quantum number is the floor number, the subshell type lets us know what type of room it is (s being a closet, p being a single room, d having two adjoining rooms, and f being a suit with three rooms) , the magnetic quantum number lets us know how many beds there are in the room, and two electrons can sleep in one bed (this is because each has a different spin; -1/2 and 1/2). For example, on the first floor we have the s orbital. The s orbital is a closet and has one bed in it so the first floor can hold a total of two electrons. The second floor has the room styles s and p. The s is a closet with one bed as we know and the p room is a single with three beds in it so the second floor can hold a total of 8 electrons.

Each orbital, as previously mentioned, has its own energy level associated to it. The lowest energy level electron orbitals are filled first and if there are more electrons after the lowest energy level is filled, they move to the next orbital. The order of the electron orbital energy levels, starting from least to greatest, is as follows: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p.

Since electrons all have the same charge, they stay as far away as possiible because of repulsion. So, if there are open orbitals in the same energy level, the electrons will fill each orbital singly before filling the orbital with two electrons. For example, the 2p shell has three p orbitals. If there are more electrons after the 1s, and 2s orbitals have been filled, each p orbital will be filled with one electron first before two electrons try to reside in the same p orbital. This is known as **Hund's rule**.

The way electrons move from one orbital to the next is very similar to walking up a flight of stairs. When walking up stairs, you place one foot on the first stair and then another foot on the second stair. At any point in time, you can either stand with both feet on the first stair, or on the second stair but it is impossible to stand in between the two stairs. This is the way electrons move from one electron orbital to the next. Electrons can either jump to a higher energy level by absorbing, or gaining energy, or drop to a lower energy level by emitting, or losing energy. However, electrions will never be found in between two orbitals.

- usually half full and full orbitals are considered stable
- Reason why half full is considered stable: It's preferred to have each electron paired up instead of having an unpaired electron for stability. Example: heavy elements like Copper

- General Chemistry Principles & Modern Applications. 9th ed. New Jersey: Pearson Education, Inc, 2007. Print.
- A new Dictionary of Chemistry. 3rd ed. Great Britian: Longman Green & Co., 1961. Print.
- General Chemistry. USA: Linus Pauling, 1947. Print.

- Which orbital would the electrons fill first? The 2s or 2p orbital?
- How many d orbitals are there in the d subshell?
- How many electrons can the p orbital hold?
- Determine the number of angular and radial nodes of a 4f orbital.
- What is the shape of an orbital with 4 radial nodes and 1 angular node in the xy plane?

- The 2s orbital would be filled before the 2p orbital because orbitals that are lower in energy are filled first. The 2s orbital is lower in energy than the 2p orbital.
- There are 5 d orbitals in the d subshell.
- A p orbital can hold 6 electrons.
- Based off of the given information, n=4 and
*ℓ*=3. Thus, there are 3 angular nodes present. The total number of nodes in this orbital is: 4-1=3, which means there are no radial nodes present. - 1 angular node means
*ℓ*=1 which tells us that we have a p subshell, specifically the p_{z}orbital because the angular node is on the xy plane. The total number of nodes in this orbital is: 4 radial nodes +1 angular node=5 nodes. To find n, solve the equation: nodes=n-1; in this case, 5=n-1, so n=6. This gives us a: 6p_{z}orbital

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