# What fraction of the orbitals in #"1 mol"# of #"Mg"# atoms in a metallic network are occupied at #"0 K"#?

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#a)# Determine the number of metallic bonding orbitals used by #"1 mol"# of #"Mg"# atoms.

#b)# What is the Fermi level? Where does the most likely electronic transition occur? Determine the fraction of the orbitals in #"1 mol"# of #"Mg"# atoms in a metallic network are occupied at #"0 K"# ?

##### 1 Answer

a) This is asking you to apply the **Linear Combination of Atomic Orbitals** (LCAO).

**LINEAR COMBINATION OF ATOMIC ORBITALS (LCAO)**

The idea is that **number of atomic orbitals** (AOs) **yields** **number of molecular orbitals** (MOs).

We can see this in context when we form the *theoretical* dimetal

**Two**#3s# AOs (one from each#"Mg"# ) combine to form one#sigma_(3s)# bonding MO and one#sigma_(3s)^"*"# antibonding MO. That's a**2:2**conversion.**Six**#3p# AOs (three from each#"Mg"# ) combine to form one#pi_(3px)# bonding MO and one#pi_(3px)^"*"# antibonding MO, one#pi_(3py)# bonding MO and one#pi_(3py)^"*"# antibonding MO, and one#sigma_(3pz)# bonding MO and one#sigma_(3pz)^"*"# antibonding MO. Sum that up and you get a**6:6**conversion.- Since we used
**two**#3s# and**six**#3p# AOs, that is#2+6 =# **eight**AOs. We got out**eight**MOs---one each of the following:#sigma_(3s)# ,#sigma_(3s)^"*"# ,#pi_(3px)# ,#pi_(3px)^"*"# ,#pi_(3py)# ,#pi_(3py)^"*"# ,#sigma_(3pz)# , and#sigma_(3pz)^"*"# .

Although the

#["Ne"]3s^2#

... pictorially this nevertheless looks like this:

With the molecular electronic configuration like so:

#(sigma_(1s))^2(sigma_(1s)^"*")^2(sigma_(2s))^2(sigma_(2s)^"*")^2color(blue)((sigma_(3s))^2(sigma_(3s)^"*")^2)# where blue indicates the valence orbitals.

Naturally, this is for one

For one ** atom**, on the other hand (instead of two), divide the number of MOs by two to get

**four**. Then, since we are talking about

#\mathbf("4 mol")# sof#"MOs/Mg atom"#

b) Okay, so some terminology from **Band Theory**. It's not too bad.

**THE HOMO-LUMO GAP**

The **Fermi level** is where the highest-occupied molecular orbital (HOMO) currently lies at

That means the **band gap**. The most likely electronic transition occurs from the HOMO to the lowest-unoccupied molecular orbital (LUMO), across this band gap.

This event constitutes **conduction**, and this energy gap is also called the **HOMO-LUMO gap**. The HOMO-LUMO gap is small for very conductive metals, as seen in the above diagram.

(When electron promotion occurs, it is said that each electron that moves into the empty orbitals above leaves a "hole" in the filled orbitals below the Fermi level, which is what is depicted on the right portion of the above diagram.)

**FRACTION OF MOS FILLED**

When we are at **no electrons have been promoted yet** (due to any thermal energy imparted due to an increase in temperature, for instance), so all the electrons in the MO diagram above are where they should be at

Having

Hence, with

*Therefore, 2/8 = 25% of the MOs are occupied by electron pairs, i.e. filled.*