How can the paramagnetism of \(\mathrm{O}_{2}\) be explained using the molecular orbital model?

The molecular orbital model is a method for describing the electronic structure of molecules by combining atomic orbitals to form molecular orbitals (MOs). These molecular orbitals are then occupied by electrons based on their energies and the Pauli exclusion principle. Paramagnetism is an attribute of molecules that have one or more unpaired electrons, which are attracted by external magnetic fields.

To explain the paramagnetism of \(\mathrm{O}_{2}\), we need to construct its molecular orbital diagram. Atomic orbitals from oxygen atoms will combine to form molecular orbitals. The valence electron configuration for each oxygen atom is \(2s^2 2p^4\). Oxygen atoms combine their atomic orbitals, resulting in the following molecular orbitals, ordered by increasing energy level: 1. Two sigma orbitals formed from the combination of 2s orbitals: \(\sigma_{2s}\) and \(\sigma^*_{2s}\) (bonding and antibonding, respectively). 2. Two sigma orbitals formed from the combination of 2pz orbitals: \(\sigma_{2p_z}\) and \(\sigma^*_{2p_z}\) (bonding and antibonding, respectively). 3. Two sets of two pi orbitals formed from the combination of 2px and 2py orbitals: \(\pi_{2p_x}\), \(\pi_{2p_y}\) (bonding) and \(\pi^*_{2p_x}\), \(\pi^*_{2p_y}\) (antibonding).

Now, we need to fill the molecular orbitals with the valence electrons from the two oxygen atoms. Each oxygen atom contributes a total of 6 valence electrons, so there are 12 electrons to be distributed among the molecular orbitals. Following the Aufbau principle, we fill the lowest energy orbitals first: 1. \(\sigma_{2s}\) - 2 electrons. 2. \(\sigma^*_{2s}\) - 2 electrons. 3. \(\sigma_{2p_z}\) - 2 electrons. 4. \(\pi_{2p_x}\) and \(\pi_{2p_y}\) - 4 electrons (2 in each orbital, Hund's rule). 5. \(\pi^*_{2p_x}\) and \(\pi^*_{2p_y}\) - 2 electrons (1 in each orbital, Hund's rule).

The electron configuration in the molecular orbitals of \(\mathrm{O}_{2}\) now has two unpaired electrons: one in the \(\pi^*_{2p_x}\) orbital and another one in the \(\pi^*_{2p_y}\) orbital. The presence of unpaired electrons makes the molecule paramagnetic because these unpaired electrons have unpaired spins, which can be aligned under the influence of an external magnetic field. In conclusion, the paramagnetism of \(\mathrm{O}_{2}\) can be explained using the molecular orbital model by showing that there are unpaired electrons in its molecular orbitals (\(\pi^*_{2p_x}\) and \(\pi^*_{2p_y}\)). These unpaired electrons result in a paramagnetic behavior when subjected to an external magnetic field.