Question

Consider the \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule depicted here, where \(\mathrm{A}\) and \(\mathrm{X}\) are elements. The \(A-A\) 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 \(X-X\) bond length of an \(X_{2}\) molecule? [Section \(\left.7.3\right]\)

Short answer

The bonding atomic radii of atoms A and X in terms of the given bond lengths are: Bonding atomic radius of A (r_A) = \( \frac{d_1}{2} \) and Bonding atomic radius of X (r_X) = \( \frac{d_2}{2} \). The predicted X-X bond length for an X₂ molecule is equal to d₂.

Step-by-step solution

Define bonding atomic radii

Bonding atomic radii are the distance between the nucleus of two bonded atoms, divided by two. In other words, it's the distance from the nucleus to the point where the electron cloud of the bonded atom's outermost shell makes contact with another atom. When two atoms are bonded, the bonding atomic radius is half of the bond length between them. Now let's apply this concept to the given molecule A₂X₄.

Find the bonding atomic radii of A and X in terms of d₁ and d₂

According to the given information, the A-A bond length is d₁, and the A-X bond length is d₂. Using the definition of bonding atomic radii, we can define the bonding atomic radii of atoms A and X as follows: Bonding atomic radius of A (r_A) = d₁ / 2 Bonding atomic radius of X (r_X) = d₂ / 2

Predict the X-X bond length for an X₂ molecule

When two X atoms form a bond in an X₂ molecule, the bonding atomic radii of both X atoms contribute to the bond length. Hence, the X-X bond length (d₃) can be calculated as: d₃ = Bonding atomic radius of X (r_X) + Bonding atomic radius of X (r_X) d₃ = r_X + r_X As we already defined r_X in step 2, we can substitute its value to calculate d₃: d₃ = (d₂ / 2) + (d₂ / 2) d₃ = d₂ Therefore, the X-X bond length for an X₂ molecule is predicted to be equal to d₂.

Key concepts

A2X4 molecule

In chemistry, an \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule refers to a particular molecular structure where two atoms of \(\mathrm{A}\) are connected to four atoms of \(\mathrm{X}\). This molecular arrangement can lead to various geometric shapes depending on the spatial orientation of the bonds. Typically, such molecules are studied to understand how the different types of atoms interact and bond within the structure.
The molecular formula, \(\mathrm{A}_{2} \mathrm{X}_{4}\), indicates that for every two atoms of \(\mathrm{A}\), there are four atoms of \(\mathrm{X}\). Thus, each \(\mathrm{A}\) atom is likely bonded to two \(\mathrm{X}\) atoms. This helps define the internal symmetry and bond distribution within the molecule, which can affect both the physical and chemical properties.
Understanding molecules like \(\mathrm{A}_{2} \mathrm{X}_{4}\) is particularly useful for predicting reactivity and bonding patterns in complex compounds. It demonstrates how variations in molecular composition can result in different chemical behaviors.

• Visualizing the structure helps in predicting bonding interactions.
• Each \(\mathrm{A}\) atom connects potentially in a symmetric or asymmetric pattern.
• Considerations include the angles between bonds and the energies associated with these arrangements.

Bond length

Bond length is a fundamental concept in chemistry that represents the average distance between the nuclei of two bonded atoms. In the context of \(\mathrm{A}_{2} \mathrm{X}_{4}\), there are two types of bond lengths: the \(A-A\) bond length \(d_1\) and the \(A-X\) bond length \(d_2\). These bond lengths can tell us a lot about the strength and the nature of the bonds.
In general, shorter bond lengths suggest stronger bonds due to the nuclei being closer together, which allows the sharing of electrons to be stronger. For instance, the \(A-A\) bond length \(d_1\) in an \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule might indicate a particular type of interaction specific to atoms \(A\).
Similarly, the \(A-X\) bond length \(d_2\) indicates how the atoms of \(A\) and \(X\) are bonded. It suggests the extent of interaction and overlap between the electron clouds of \(A\) and \(X\).

• Shorter lengths typically indicate stronger and more stable bonds.
• Longer bonds usually result from weaker interactions.
• Each type of bond length in a molecule contributes to its overall geometric structure.

Atomic structure

Atomic structure refers to the composition and arrangement of subatomic particles within an atom. Each atom comprises a nucleus made of protons and neutrons, with electrons orbiting this nucleus in various energy levels or shells. The arrangement of these electrons significantly influences how atoms bond and form molecules.
In molecules like \(\mathrm{A}_{2} \mathrm{X}_{4}\), understanding the atomic structure of elements \(A\) and \(X\) is crucial. The number of valence electrons, which are the electrons in the outer shell, determines how an atom can bond with others. For example, if \(A\) has two valence electrons available for bonding, this would affect the type of bonds it can form with \(X\).
This structure leads to specific properties and reactivity patterns. By exploring the atomic structure, we gain insight into why certain atoms form particular bonds and how these bonds dictate the molecule's stability and reactivity.

• Protons determine the elemental identity and are equal to the atomic number.
• Neutrons contribute to the atomic mass but not to the charge.
• Electrons form the outer electron cloud, dictating bonding behaviors.

Chemical bonding

Chemical bonding is the process through which atoms combine to form molecules. It arises due to interactions between the electromagnetic forces of the participating atoms' electrons and nuclei. There are several types of chemical bonds, including covalent, ionic, and metallic bonds. Each type has distinct characteristics and leads to different molecular shapes and properties.
In the \(\mathrm{A}_{2} \mathrm{X}_{4}\) molecule, chemical bonding involves the formation of bonds between \(A\) atoms and \(X\) atoms. These bonds could be covalent if there's a sharing of valence electrons or ionic if electrons are transferred from one atom to another, creating ions.
The type of bond impacts how easily a molecule will react with other substances, its melting and boiling points, and its physical state at room temperature. Being able to predict these properties based on bonding helps chemists understand how molecules will behave in different environments.

• Covalent bonds involve sharing of electron pairs between atoms.
• Ionic bonds occur when electrons are transferred, creating charged ions.
• Complex molecules can contain a mixture of bond types.