Question

The following observations are made about two hypothetical elements \(A\) and \(B\) : The \(A-A\) and \(B-B\) bond lengths in elemental A and B are \(2.36\) and \(1.94 \AA\), respectively. A and B react to form the binary compound \(\mathrm{AB}_{2}\), which has a linear structure (that is \(\angle \mathrm{B}-\mathrm{A}-\mathrm{B}=180^{\circ}\) ). Based on these statements, predict the separation between the two \(\mathrm{B}\) nuclei in a molecule of \(\mathrm{AB}_{2}\).

Short answer

The separation between the two B nuclei in a molecule of AB2 is predicted to be \(4.30 \AA\).

Step-by-step solution

Visualize the AB2 molecule

Since AB2 molecule has a linear structure, we can visualize it as the following: B-A-B The angle between the two B atoms and the A atom is \(180^{\circ}\).

Identify the bond lengths

The bond lengths of A-A and B-B are given in the problem as \(2.36 \AA\) and \(1.94 \AA\), respectively.

Calculate the bond length of A-B

We need to find the bond length A-B in order to calculate the separation between the two B nuclei. We know that the bond length is the average of A-A and B-B bond lengths. So, \(A-B = \frac{A-A + B-B}{2}\) \(A-B = \frac{2.36 \AA + 1.94 \AA}{2}\) \(A-B = 2.15 \AA\)

Calculate the separation between the two B nuclei

To calculate the separation between the two B nuclei, we have to use the linear geometry of the AB2 molecule. If we consider A as the center and the two B atoms on either side of A, the distance between the two B atoms can be obtained by adding the A-B bond length twice, as illustrated below: \(B-B = 2 \times A-B\) \(B-B = 2 \times 2.15 \AA\) \(B-B = 4.30 \AA\) Accordingly, the separation between the two B nuclei in a molecule of AB2 is predicted to be \(4.30 \AA\).

Key concepts

Understanding Bond Lengths

Bond lengths essentially reflect the distance between the nuclei of two bonded atoms within a molecule. These distances can vary depending on the size of the atoms involved, the number of shared electrons (which constitutes the bond order), and the overall electronic environment surrounding the atoms.

For instance, in the exercise, the observed bond lengths of A-A and B-B are given as different values, which suggests that atoms A and B are of different sizes or may have different electronegativities. It is important to note that the bond length can also be influenced by the type of bond formed, be it a single, double, or triple bond, with shorter bond lengths generally found in bonds with higher bond orders due to increased electron sharing between the atoms.

Understanding the concept of bond lengths is crucial because it allows us to predict reactions and the stability of molecules. In practice, the length of the bond has implications for the physical and chemical properties of the compounds. In our example, the average bond length between A and B was key to calculating the separation between the two B atoms in the AB2 molecule.

Deciphering Molecular Geometry

Molecular geometry, also known as molecular structure, is the three-dimensional arrangement of atoms within a molecule. It dictates the shape of a molecule, which can influence the molecule's physical properties and reactivity. Different geometries arise from the repulsion between electron pairs in the valence shell of the central atom, a concept central to VSEPR theory (Valence Shell Electron Pair Repulsion).

In our exercise, the linear geometry of the AB2 molecule is given, indicating that the bond angle between A and both Bs is 180 degrees. This shape is typically associated with molecules that have a central atom with no lone pairs of electrons, bonded to two other atoms. The understanding of molecular geometry is vital for predicting molecular polarity, the way molecules interact with each other, and even their biological function.

Real-World Application of Molecular Geometry

Consider the shape of a water molecule, which is bent. This geometry is partly responsible for water's unique properties, such as its high boiling point and surface tension. If water had a linear structure, its properties—and by extension life on Earth—would be dramatically different.

Binary Compounds Basics

A binary compound is composed of two different elements. These compounds can be classified further based on the types of elements involved—typically into ionic or covalent compounds. In the example, AB2 is a binary molecular compound, which suggests that A and B are nonmetals forming covalent bonds.

Understanding the composition of binary compounds is important because it allows us to predict the properties of the substance, such as conductivity, melting point, and solubility. Binary compounds, depending on their nature, are used in various industries, such as sodium chloride (table salt) in food to silicon carbide in electronics.

Reading Formulas of Binary Compounds

The formula of a binary compound tells us the ratio of the elements present. In AB2, for every one A atom, there are two B atoms, hinting at the valence and combining capacity of the constituent elements. Mastering the composition and naming of binary compounds is fundamental in the study of chemistry, as it forms the foundation for understanding more complex chemical behavior.