Question

Which of the following are predicted by the molecular orbital model to be stable diatomic species? a. \(\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}\) b. \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\)

Short answer

The stable diatomic species predicted by the molecular orbital model are: \(\mathrm{H}_{2}^{+}\), \(\mathrm{H}_{2}\), \(\mathrm{H}_{2}^{-}\), \(\mathrm{He}_{2}^{2+}\), and \(\mathrm{He}_{2}^{+}\).

Step-by-step solution

Determine the number of valence electrons for each species

Count the total number of valence electrons in the given species. Remember that the charge of the species affects the number of electrons: a. \(\mathrm{H_{2}^{+}}\) has \(1 + 1 - 1 = 1\) electron \(\mathrm{H_{2}}\) has \(1 + 1 = 2\) electrons \(\mathrm{H_{2}^{-}}\) has \(1 + 1 + 1 = 3\) electrons \(\mathrm{H_{2}^{2-}}\) has \(1 + 1 + 2 = 4\) electrons b. \(\mathrm{He_{2}^{2+}}\) has \(2 + 2 - 2 = 2\) electrons \(\mathrm{He_{2}^{+}}\) has \(2 + 2 - 1 = 3\) electrons \(\mathrm{He_{2}}\) has \(2+2 = 4\) electrons

Determine the bond order of each species

Apply the bond order formula: \(Bond\: order = \frac{(Number\:of\: electrons\: in\: bonding\: orbitals) - (Number\: of\: electrons\: in\: antibonding\:orbitals)}{2}\). We consider only sigma orbitals for the given species: For H, \(\sigma_{1s}\) is a bonding orbital, and \(\sigma_{1s}^{*}\) is an antibonding orbital. For He, \(\sigma_{1s}\), \(\sigma_{1s}^{*}\), \(\sigma_{2s}\), and \(\sigma_{2s}^{*}\) are the orbitals, where asterisks indicate antibonding orbitals. a. \(\mathrm{H_{2}^{+}}\) has 1 electron in the bonding orbital (\(\sigma_{1s}\)): Bond order \(= \frac{1 - 0}{2} = 0.5\) \(\mathrm{H_{2}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)): Bond order \(= \frac{2 - 0}{2} = 1\) \(\mathrm{H_{2}^{-}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 1 electron in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 1}{2} = 0.5\) \(\mathrm{H_{2}^{2-}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 2 electrons in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 2}{2} = 0\) b. \(\mathrm{He_{2}^{2+}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)): Bond order \(= \frac{2 - 0}{2} = 1\) \(\mathrm{He_{2}^{+}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 1 electron in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 1}{2} = 0.5\) \(\mathrm{He_{2}}\) has 2 electrons in the bonding orbital (\(\sigma_{1s}\)) and 2 electrons in the antibonding orbital (\(\sigma_{1s}^{*}\)): Bond order \(= \frac{2 - 2}{2} = 0\)

Determine the stability of each species

If the bond order is greater than 0, the molecule is considered to be stable. a. Stable diatomic species among \(\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}\) are: \(\mathrm{H}_{2}^{+}\) (bond order \(=0.5\)) \(\mathrm{H}_{2}\) (bond order \(=1\)) \(\mathrm{H}_{2}^{-}\) (bond order \(=0.5\)) b. Stable diatomic species among \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\) are: \(\mathrm{He}_{2}^{2+}\) (bond order \(=1\)) \(\mathrm{He}_{2}^{+}\) (bond order \(=0.5\)) Hence, the stable diatomic species are: \(\mathrm{H}_{2}^{+}\), \(\mathrm{H}_{2}\), \(\mathrm{H}_{2}^{-}\), \(\mathrm{He}_{2}^{2+}\), and \(\mathrm{He}_{2}^{+}\).

Key concepts

Bond Order

In the realm of molecular orbital theory, bond order serves as a quantitative measure. It gives us insight into the stability of a molecule. Essentially, bond order is calculated as half the difference between the number of electrons in bonding orbitals and antibonding orbitals. The formula for calculating bond order is: \[ \text{Bond order} = \frac{(N_b - N_a)}{2} \] Where \(N_b\) is the number of electrons in bonding orbitals, and \(N_a\) is the number of electrons in antibonding orbitals.

A positive bond order suggests that a molecule is stable. If the bond order is zero or negative, the molecule is considered unstable or non-existent under normal conditions. For example:

• When a molecule has a bond order of 1, it indicates a single bond, as seen in hydrogen, \(\text{H}_2\).
• A bond order of 2 suggests a double bond, which enhances molecular stability.

Bond order directly correlates with bond strength and bond length. The higher the bond order, the shorter and stronger the bond. This concept is crucial in predicting and understanding molecular stability, as demonstrated in various diatomic species like \(\text{H}_2^+\) with a bond order of 0.5, making it relatively stable.

Diatomic Species

Diatomic species are molecules composed of only two atoms. They can be the same element, like \(\text{H}_2\) or \(\text{O}_2\), or different elements, such as carbon monoxide, \(\text{CO}\). Diatomic species have unique chemical properties.

Some key points about diatomic species include:

• They have simple structures compared to larger polyatomic molecules, resulting in straightforward molecular orbitals.
• Many diatomic elements exist naturally, as they prefer diatomic states to achieve greater stability. For instance, oxygen is commonly found as \(\text{O}_2\), a necessity for life's survival.
• In molecular orbital theory, diatomic species are an excellent starting point for learning because their electron configurations are less complex. This simplicity makes them ideal for demonstrating concepts like bond order and electron distribution.

Diatomic molecules are foundational to understanding molecular interactions and chemical bonding. They allow us to explore the basics of molecular orbital theory effectively, providing insight into how atoms combine to form stable entities. By calculating bond orders of diatomic species, such as \(\text{H}_2^+, \text{H}_2\), and \(\text{He}_2^{2+}\), one can determine their relative stability.

Valence Electrons

Valence electrons are the outermost electrons of an atom. They play a crucial role in chemical bonding since they are the electrons involved in forming chemical bonds. Understanding valence electrons is pivotal for predicting how elements will interact and bond with each other.

Here are some essential aspects of valence electrons:

• They determine an element's chemical properties and reactivity. Elements in the same group of the periodic table have the same number of valence electrons, leading to similar chemical behavior.
• In molecular compounds, the number of valence electrons affects how atoms bond and arrange themselves. For example, hydrogen, with one valence electron, readily forms diatomic molecules like \(\text{H}_2\) by sharing electrons.
• Charges on ions can change the apparent number of valence electrons. For example, in \(\text{H}_2^{-}\), an extra electron means each hydrogen atom shares more electrons than in neutral \(\text{H}_2\).

Valence electrons can be used to draw Lewis structures or to construct molecular orbitals which, in turn, help us to visualize bonding scenarios and predict molecule's stability. In molecular orbital theory, knowing the starting number of valence electrons in a diatomic species is a crucial first step in determining its bond order and eventual stability.