Question

The LCAO molecular orbital of the cyclopentadienate system \(\left(\mathrm{C}_{5} \mathrm{H}_{5}\right)\) have schematic energies as follows: Indicate a possible point of difference between \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) and \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) that can be deduced from molecular orbital theory

Short answer

The main difference between the cyclopentadienate ions \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) and \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) deduced from molecular orbital theory is that \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) is a closed-shell ion with higher stability, while \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) is an open-shell ion with lower stability and higher reactivity. This is due to the occupation of their molecular orbitals: \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) has 24 electrons occupying up to the 12th molecular orbital, while \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) has 26 electrons occupying up to the 13th molecular orbital.

Step-by-step solution

Determine the number of electrons in each ion

To begin, we must first determine the number of electrons in each ion. For \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\), each carbon atom has 4 electrons, and each hydrogen atom has 1 electron. Since there are 5 carbon atoms and 5 hydrogen atoms in the ion, there are \((5\times4) + (5\times1) = 20+5 = 25\) electrons in total. However, it is positively charged, which means it lost one electron, leaving it with 24 electrons. For \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\), each carbon atom has 4 electrons, and each hydrogen atom has 1 electron. Since there are 5 carbon atoms and 5 hydrogen atoms in the ion, there are again \((5\times4) + (5\times1) = 25\) electrons. Being negatively charged, this ion gained an electron, resulting in 26 electrons in total.

Apply the LCAO molecular orbital method

The linear combination of atomic orbitals (LCAO) method involves combining atomic orbitals to form new molecular orbitals. We will apply this method for both cyclopentadienate ions: - For \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\): Since it has 24 electrons, it will fill up molecular orbitals up to the 12th level (each orbital can have 2 electrons). - For \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\): It has 26 electrons, and they will fill up molecular orbitals up to the 13th level.

Analyze the differences between the molecular orbitals

Comparing the molecular orbitals of \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) and \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\), we can deduce a difference from their molecular orbital occupation: - \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) has 24 electrons occupying up to the 12th molecular orbital, making it a closed-shell ion with higher stability. - \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) has 26 electrons occupying up to the 13th molecular orbital, which means it has an unpaired electron in its highest occupied molecular orbital (HOMO), making it an open-shell ion with lower stability and higher reactivity. The point of difference deduced from molecular orbital theory for the two cyclopentadienate ions is that \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) is a closed-shell ion with higher stability, while \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) is an open-shell ion with lower stability and higher reactivity.

Key concepts

Molecular Orbital Occupation

Understanding molecular orbital occupation is crucial for analyzing the electronic structure of molecules. In the LCAO molecular orbital theory, atomic orbitals combine to create molecular orbitals, which can then be occupied by electrons.

Electrons will fill the lowest energy molecular orbitals first before moving to higher energy levels, known as the aufbau principle. Each molecular orbital can contain a maximum of two electrons with opposite spins. The way these electrons fill the orbitals is essential for predicting molecular properties such as magnetism and reactivity.

The stability of a molecule is directly related to how these molecular orbitals are occupied. Molecules with fully filled or half-filled orbitals are typically more stable due to electron pairing and exchange energy. Thus, seeing how electrons are distributed among the molecular orbitals gives us insight into the chemical behavior of that molecule.

Cyclopentadienate Ions

The cyclopentadienate ions, specifically \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) and \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\), showcase the result of electron addition or removal from a neutral cyclopentadienyl ring.

These changes in electron count lead to significant differences in their chemical properties. The neutral cyclopentadienyl is aromatic, following Huckel's rule of 4n+2 \(\pi\)-electrons, which renders it highly stable. However, when an electron is removed to form the \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) ion, the molecule loses this aromatic stability, which is reflected in its molecular orbitals. Conversely, adding an electron to form the \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) ion increases the electron count, affecting the orbital occupation and reactivity.

Open-Shell vs Closed-Shell Ions

Distinguishing between open-shell and closed-shell ions is essential for predicting molecular reactivity. An open-shell ion has at least one molecular orbital that is not completely filled with electrons; this often results in unpaired electrons and contributes to a molecule's reactivity and paramagnetic properties.

In contrast, a closed-shell ion has all of its molecular orbitals fully occupied, usually resulting in a more stable arrangement and diamagnetic properties. In the example of \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\), it is a closed-shell ion, suggesting higher stability. On the other hand, the \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) ion, with an additional electron in a previously unoccupied molecular orbital, represents an open-shell ion, indicative of its higher reactivity and potentially less stability.

Electron Configuration

The electron configuration of a molecule or ion tells us how electrons are distributed among the molecular orbitals. Specifically, the electron configuration reflects the principle and angular quantum numbers defining the size and shape of the orbitals occupied by electrons.

For instance, in the cyclopentadienate ions, the difference in electron configuration between \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) and \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) causes a difference in stability and reactivity. The \(\mathrm{C}_{5} \mathrm{H}_{5}^{+}\) with 24 electrons shows a fully occupied molecular orbital configuration, making it a closed-shell system. Whereas, \(\mathrm{C}_{5} \mathrm{H}_{5}^{-}\) with 26 electrons has an extra pair of electrons, potentially leaving one electron unpaired – an open-shell configuration that is less stable but more reactive.