Question

The following observations are made about two hypothetical elements \(\mathrm{A}\) and \(\mathrm{B}:\) The \(\mathrm{A}-\mathrm{A}\) and \(\mathrm{B}-\mathrm{B}\) bond lengths in the elemental forms of \(\mathrm{A}\) and \(\mathrm{B}\) are 236 and \(194 \mathrm{pm}\), respectively. A and \(B\) react to form the binary compound \(A B_{2}\), which has a linear structure (that is \(\left.\angle \mathrm{B}-\mathrm{A}-\mathrm{B}=180^{\circ}\right) .\) Based on these statements, predict the separation between the two B nuclei in a molecule of \(\mathrm{AB}_{2}\).

Short answer

The separation between the two B nuclei in a molecule of \(\mathrm{AB_{2}}\) is 430 pm.

Step-by-step solution

Understand the structure of AB2 molecule

In the given molecule AB2, element A is at the center and elements B are bonded to it linearly. This means the bond angle between \(\mathrm{B}-\mathrm{A}-\mathrm{B}\) is given as \(180^{\circ}\). Since the molecule has a linear structure, we can use the geometry of the molecule to predict the separation between the two B nuclei.

Use the given bond lengths

We are given the bond lengths in the elemental forms of A and B, which are 236 pm and 194 pm, respectively. We can estimate the bond length of \(\mathrm{A}-\mathrm{B}\) in the AB2 compound using these bond lengths. Since A and B are hypothetical elements, we can assume that their bond length will be the average of the A-A and B-B bond lengths: \[A-B \ bond \ length = \frac{A-A \ bond \ length + B-B \ bond \ length}{2}\]

Calculate the A-B bond length

Now, we will substitute the given bond lengths and calculate the A-B bond length: \[A-B \ bond \ length = \frac{236 \ \mathrm{pm} + 194 \ \mathrm{pm}}{2}\] \[A-B \ bond \ length = \frac{430 \ \mathrm{pm}}{2} = 215 \ \mathrm{pm}\]

Calculate the separation between the two B nuclei

In the AB2 molecule, two B nuclei are bonded with A at a linear angle. Since the bond angle \(\mathrm{B}-\mathrm{A}-\mathrm{B}\) is \(180^{\circ}\), the \(\mathrm{B}-\mathrm{A}\) and \(\mathrm{A}-\mathrm{B}\) bonds are in a straight line. As a result, the distance between the two B nuclei is the sum of the two A-B bond lengths: \[B-B \ separation = 2 \times (A-B \ bond \ length)\]

Calculate the B-B separation

We'll now insert the previously calculated A-B bond length and compute the B-B separation: \[B-B \ separation = 2 \times (215 \ \mathrm{pm}) = 430 \ \mathrm{pm}\] So, the separation between the two B nuclei in a molecule of \(\mathrm{AB_{2}}\) is 430 pm.

Key concepts

Molecular Geometry

Molecular geometry refers to the three-dimensional arrangement of atoms within a molecule. Understanding molecular geometry is essential because it influences the molecule's properties, such as polarity, reactivity, phase of matter, color, magnetism, and biological activity. In the case of the \(AB_2\) molecule, the atomic structure forms a line where the central atom A is flanked by two atoms of element B.
This structure of \(AB_2\) is determined by the bonds and angles that are formed between the participating atoms. The molecular geometry often follows specific patterns, dictated by the number of bonds and lone pairs around the central atom. Identifying and predicting these structures is half strategy and half observation.
The linear shape of the \(AB_2\) molecule means it has a bond angle of \(180^\circ\), as linear molecules are characterized by a straight line, which minimizes electron-pair repulsion according to the VSEPR theory (Valence Shell Electron Pair Repulsion). This angle dictates the overall shape and creates a symmetrical molecule.

Linear Molecular Structure

A linear molecular structure is one where the bond angle between each atom is \(180^\circ\). This results in a straight line of atoms. The linear structure is one of the simplest molecular geometries to identify. Because of this structure, linear molecules have unique properties.
In a linear \(AB_2\) molecule, the central atom A is bonded symmetrically to two B atoms, which are placed directly opposite each other. This symmetry is crucial because it often results in a non-polar molecule despite the individual polarities of bonds.
Linear structures are found commonly in diatomic molecules and some triatomic molecules. The key is the number of outer atoms and bonds showing that when there are two atoms only, helped by the symmetric nature, it distributes the electron density evenly across the molecule, often resulting in a cancellation of dipoles.

Bond Angle

Bond angle plays a key role in molecular geometry by determining the relative positions of connected atoms. It is defined as the angle between three atoms across at least two bonds. In the \(AB_2\) linear molecule, the bond angle between the two B atoms and the central A atom is \(180^\circ\). This angle is pivotal in maintaining the linear geometry.
Because the \(AB_2\) bond angle is \(180^\circ\), the two B atoms are positioned as far apart as possible from each other, given the central A atom. This arrangement results because electron pairs around the central atom repel each other and minimize repulsion by taking positions that put them at maximum distance, supporting the linearity.
Understanding bond angles is crucial when deducing molecular shapes; for instance, when the angle deviates from \(180^\circ\), the molecule could adopt another geometry type, like bent or angular.

Hypothetical Elements

Hypothetical elements are used in theoretical scenarios to simplify the understanding of chemical concepts without the complication of real-world data. In this exercise, elements A and B are presented as hypothetical elements.
Hypothetical scenarios are useful in teaching because they allow learners to focus on understanding principles unfettered by additional complexities of real elements, such as isotopes, real-world applications, and pressures of lab work.
These elements, being hypothetical, provide a clean slate, focusing strictly on the conceptual learning of molecular geometry, bond length, and chemical interactions without unnecessary distractions. When dealing with real elements in a class or lab scenario, one must consider properties like atomic radius, electronegativity, and real molecular interactions.