Question

Calculate the sum of ratios of $\sigma $ bonding electrons and $\pi $ bonding electrons in a ${{C}_{2}}$ molecule.

Answer 1

Hint: Calculate the number of electrons present in the molecule and then fill up the bonding and antibonding molecular orbitals according to their energy levels. Then calculate the ratio by counting the number of electrons in the $\sigma $ and $\pi $ orbitals.

Complete step by step solution:
First, let us look at the definitions of the bonding and antibonding molecular orbitals.
- Bonding molecular orbitals are the orbitals where electrons are present between the nuclei of the two atoms involved.
- Anti-bonding molecular orbitals are the orbitals where electrons are present in spaces that are not between the two nuclei of the involved electrons.
To calculate the number of electrons present in the $\sigma $ and $\pi $ bonding orbitals, we will have to look at the electronic configuration of the${{C}_{2}}$ molecule. The number of electrons involved in the molecule will be 12. Since, the number of electrons present on each carbon atom is 6. The electronic configuration is:
${{C}_{2}}:\sigma 1{{s}^{2}},\sigma *1{{s}^{2}},\sigma 2{{s}^{2}},\sigma *2{{s}^{2}},\pi 2{{p}_{x}}^{2}=\pi 2{{p}_{y}}^{2},\pi 2{{p}_{z}}^{0}$
The orbitals marked with a $*$ are the anti-bonding molecular orbitals. So, we can see here that the number of electrons in the bonding molecular orbitals are 8 and the number of electrons in the antibonding molecular orbitals are 4. We are only concerned with the electrons in the bonding molecular orbitals in this question.
The number of electrons present in the $\sigma $- bonding molecular orbitals are 4, and the number of electrons present in the $\pi $- bonding molecular orbitals are also 4. So, the ratio of $\sigma $ bonding electrons and $\pi $ bonding electrons in a ${{C}_{2}}$ molecule is 1:1.

Note: Read the question carefully before attempting to answer as they are asking for the ratio between the sigma and the pi bonding molecular orbitals and not the ratio between the bonding and antibonding molecular orbitals. Also, remember that in carbon the $2{{p}_{x}}$ and $2{{p}_{y}}$ have equal energy and get filled before the $2{{p}_{z}}$ orbital. The exact opposite happens in some other atoms like oxygen and nitrogen.