Question

Consider the \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule depicted here, where \(\mathrm{A}\) and \(\mathrm{X}\) are elements. The \(\mathrm{A}-\mathrm{A}\) bond length in this molecule is \(d_{1},\) and the four \(\mathrm{A}-\mathrm{X}\) bond lengths are each \(d_{2}\). (a) In terms of \(d_{1}\) and \(d_{2}\), how could you define the bonding atomic radii of atoms \(A\) and \(X ?(b)\) In terms of \(d_{1}\) and \(d_{2}\), what would you predict for the \(\mathrm{X}-\mathrm{X}\) bond length of an \(\mathrm{X}_{2}\) molecule? \([\) Section 7.3\(]\)

Short answer

The bonding atomic radii of atoms \(A\) and \(X\) in the \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule can be defined as \(r_A = \frac{d_1}{2}\) and \(r_X = d_2 - \frac{d_1}{2}\), respectively. The predicted \(\mathrm{X}-\mathrm{X}\) bond length in an \(\mathrm{X}_2\) molecule is \(d_{X-X} = 2d_2 - d_1\).

Step-by-step solution

(Part a: Determine the bonding atomic radii of atoms A and X in terms of d1 and d2)

First, let's define the bonding atomic radii of atoms \(A\) and \(X\). For any two bonded atoms, the bond length can be considered as the sum of their atomic radii. In the given \(\mathrm{A}_2 \mathrm{X}_4\) molecule, we have a bond length \(d_1\) between the two \(\mathrm{A}\) atoms and a bond length \(d_2\) between each pair of \(\mathrm{A}\) and \(\mathrm{X}\) atoms. Let \(r_A\) be the bonding atomic radius of atom \(A\) and \(r_X\) be the bonding atomic radius of atom \(X\). Using the bond length definition, 1. The bond length between two \(\mathrm{A}\) atoms, \(d_1 = r_A + r_A\), and 2. The bond length between \(\mathrm{A}\) and \(\mathrm{X}\) atoms, \(d_2 = r_A + r_X\).

(Part a: Define atomic radii of atoms A and X in terms of d1 and d2)

Now, we can solve these equations to find expressions for \(r_A\) and \(r_X\) in terms of \(d_1\) and \(d_2\). The first equation can be simplified as: \(r_A= \frac{d_1}{2}\). Substituting the value of \(r_A\) in the second equation: \(d_2 = \frac{d_1}{2} + r_X\). Now, solve for \(r_X\): \(r_X = d_2 - \frac{d_1}{2}\). Thus, we have defined the atomic radii of atoms \(A\) and \(X\) in terms of \(d_1\) and \(d_2\) as follows: \(r_A = \frac{d_1}{2}\), and \(r_X = d_2 - \frac{d_1}{2}\).

(Part b: Predict the X-X Bond Length in an X₂ Molecule)

To predict the \(\mathrm{X}-\mathrm{X}\) bond length in an \(\mathrm{X}_2\) molecule, we can use the atomic radius of atom \(X\) that we derived in the previous part. Recall that the bond length between two atoms is the sum of their atomic radii. So for \(\mathrm{X}_2\) molecule, the \(\mathrm{X}-\mathrm{X}\) bond length will be the sum of the atomic radii of the two \(\mathrm{X}\) atoms. Let \(d_{X-X}\) be the bond length between two \(\mathrm{X}\) atoms. Using the expression for \(r_X\), we can calculate the \(\mathrm{X}-\mathrm{X}\) bond length in an \(\mathrm{X}_2\) molecule as follows: \(d_{X-X} = r_X + r_X = 2 * (d_2 - \frac{d_1}{2})\). Simplifying the expression: \(d_{X-X} = 2d_2 - d_1\). Hence, the \(\mathrm{X}-\mathrm{X}\) bond length in an \(\mathrm{X}_2\) molecule can be predicted as \(d_{X-X} = 2d_2 - d_1\).

Key concepts

Bond Length

Understanding bond length is essential when we begin to discuss the particulars of chemical structures. In the context of molecular bonding, bond length refers to the distance between the centers of two bonded atoms. It's a crucial factor in the stability and reactivity of a molecule. When atoms share electrons to form a bond, their nuclei try to get as close as possible to maximize the attractive forces while repelling each other just enough to maintain a stable arrangement.

As a quick recap of the example given, the bond length between two identical atoms, like the \(A-A\) in \(A_2X_4\), can be defined using their atomic radii. Similarly, when different atoms bond, such as \(A\) and \(X\), their combined atomic radii will equal the distance between them, highlighting an intrinsic relationship between atomic radius and bond length. This concept is not just academic—it paves the way towards predicting how atoms will interact and the shape of the resulting molecules.

Molecular Geometry

Molecular geometry, which also refers to the shapes of molecules, is closely intertwined with chemical bonding and directly affects the physical and chemical properties of substances. The arrangement of atoms in a molecule is determined by the lengths of the bonds as well as the angles between them, which in turn is governed by principles such as the valence shell electron pair repulsion (VSEPR) theory.

In our textbook example involving the molecule \(A_2X_4\), the molecular geometry would be influence by both \(d_1\) and \(d_2\) bond lengths. If we imagine this on a three-dimensional level, we could begin to foresee the spatial relationships between the \(X\) atoms themselves and how the different bond lengths might affect whether they are pushed closer together or further apart. By understanding these geometric aspects, chemists can predict both the geometry and potential chemical behavior of new molecules.

Chemical Bonding

The bedrock of molecular structures lies in the chemical bonds. These bonds are the powerful, yet invisible, forces holding atoms together to form the incredible diversity of chemical compounds we encounter. There are several types of chemical bonds, including ionic, covalent, and metallic bonds, each with distinct characteristics determined by the elements involved and the sharing or transfer of electrons.

Using our \(A_2X_4\) example, the \(A-A\) and \(A-X\) bonds are covalent, meaning the atoms share electron pairs. The ability to define bonding atomic radii \(r_A\) and \(r_X\) is a direct result of our understanding of covalent bonding. The method we used to solve for the atomic radii underpins the atomic scale view of bonding, explaining the lengths of bonds and the overall arrangement of atoms in a molecule.