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# Lesson 6.7-6.9: Many-Electron Atoms: Electron Configurations

← Lesson 6.5: Lesson 7.1: →

## Orbitals and Their Energies

Up to now we have described the orbitals available to a single electron in a hydrogen atom. In the case of hydrogen the energy of the electron is given by the energy level or principal quantum number For example, for hydrogen, an electron in a 3s, 3p, or 3d subshell, regardless of orientation, has the same energy

Compare the left and right portions of the diagram below. The diagram on the left is for a hydrogen atom, and the one on the right for an atom with many electrons. For many electron atoms, the energy of an electron in a given shell increases with l value For example, an electron in a 3s orbital has lower energy than one in a 3p, which in turn is lower than one in a 3d orbital. However, the electrons in individual 3p or 3d orbitals have the same energy. Orbitals with the same energy are said to be degenerate.

## Magnetic quantum number: ms

How do the electrons in many-electron atoms occupy orbitals? By careful analysis of spectral lines, it was found that electrons have an intrinsic property called spin – as if each electron was spinning about an axis. The electron can spin in one of two opposite directions This spin is designated with a 4th quantum number, the spin magnetic quantum number, ms. The two possible values are given as +½ and -½ . We can designate the spin with a number, +½ or -½ , or as an up or down arrow.

## Pauli Exclusion Principle

Early in the 20th century, a physicist named Pauli discovered a principle which regulates how the electrons in a many-electron atom are arranged. It is called the Pauli Exclusion Principle, and states that for each electron in a many-electron atom, there must be a unique set of values for n, l, ml, and ms No two electrons in an atom can have the same set of four quantum numbers. For example, in a single atom two electrons could have the following set of quantum numbers.

• n = 1, l = 0, ml = 0, ms = + ½
• n = 1, l = 0, ml = 0, ms = - ½
• But not
• n = 1, l = 0, ml = 0, ms = + ½
• n = 1, l = 0, ml = 0, ms = + ½

Since there are only two possible spin quantum numbers, this means that orbitals can have at most two electrons. Half-filled orbitals are said to be unpaired. Note that up to now we have accepted that orbitals only have two electrons each, but no reason was given to support this fact.

## Hund’s Rule

One last rule is needed for us to make sense of electron configurations in many-electron atoms. It is called Hund’s Rule. It states that for degenerate orbitals, the lowest energy is attained when the number of electrons with the same spin is maximized. This means that, for a set of orbitals in the same sublevel, for example the 2p, or 3d orbitals, the lowest energy is achieved by half-filling each orbital with an electron with one kind of spin, say an up spin, before filling it with a second, down spin electron.

How can we keep track of the relative energies of the subshells? We can use the periodic table to guide us. The order is read from left to right on the table, movinh down as we would read any other text. The order is

1s →

2s → 2p

3s → 3p

4s → 3d → 4p

5s → 4d → 5p

6s → 4f → 5d → 6p

7s → 5f → 6d → 7p

Example 1: What is the electron configuration for titanium? How many unpaired electrons does it have? The red dot shows the location of titanium on the periodic table.There are three ways we can write the electron configuration.The condensed notation is shorter and shows the inner shell core electrons by using the symbol for the previous noble gas. The electrons that are shown after the core electrons are valence electrons.

Normal notation: 1s22s22p63s23p64s23d2

Condensed notation: [Ar] 4s23d2

Orbital notation:

From the orbital notation it is clear that there are 2 unpaired electrons.

If you hover over the elements in the Periodic table → linked here you can see the electron configuration for every element.

There are a whole set of videos that work out the electron configurations of many elements on Khan Academy. Electron Configuration Examples →