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Chapter 13: Problem 11

In a regular octahedral molecule, \(\mathrm{MX}_{6}\) the number of \(\mathrm{X}-\mathrm{M}-\mathrm{X}\) bonds at an angle of \(180^{\circ}\) is (a) three (b) two (c) \(\operatorname{six}\) (d) four

Short Answer

Expert verified
The number of \( \mathrm{X}-\mathrm{M}-\mathrm{X} \) bonds at 180掳 is three.

Step by step solution

01

Understanding Octahedral Geometry

An octahedral molecule, like \( \mathrm{MX}_6 \), has a central atom \( \mathrm{M} \) surrounded by six identical atoms \( \mathrm{X} \). The geometry is octahedral, which means the \( \mathrm{X} \) atoms are placed at the vertices of an octahedron with the central \( \mathrm{M} \) atom at the center.
02

Identifying the Axis with 180掳 Angles

In an octahedral molecule, the atoms are symmetrically arranged. For any \( \mathrm{X}-\mathrm{M}-\mathrm{X} \) alignment along the same axis, the angle between them is \( 180^{\circ} \). Each axis through the center of the octahedron passes through two opposite vertices (atoms).
03

Counting the 180掳 Angles

There are three axes of symmetry in an octahedral molecule that can result in \( 180^{\circ} \) angles. Each axis involves a pair of \( \mathrm{X}-\mathrm{M}-\mathrm{X} \) angles. Hence, there are 3 pairs of 180掳 \( \mathrm{X}-\mathrm{M}-\mathrm{X} \) angles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bond Angles
In the context of octahedral geometry, bond angles play a key role in determining the shape and structure of the molecule. In an octahedral molecule like \( \mathrm{MX}_6 \), the bonds between the central atom \( \mathrm{M} \) and the surrounding \( \mathrm{X} \) atoms are positioned in such a way to minimize repulsion and maximize stability. This arrangement results in bond angles of either \( 90^{\circ} \) or \( 180^{\circ} \).

  • In a regular octahedron, four edges meet at each vertex, creating \( 90^{\circ} \) angles between adjacent edges. This is due to the spatial arrangement where each \( \mathrm{X} \) atom is positioned equidistantly around the central \( \mathrm{M} \) atom.
  • More importantly for this exercise, there are three axes passing through the central atom where \( \mathrm{X}-\mathrm{M}-\mathrm{X} \) form a \( 180^{\circ} \) angle. These axes represent straight lines running through the molecule from one vertex to the opposite vertex directly.
Recognizing these bond angles helps in visualizing the three-dimensional symmetry and structure inherent in octahedral molecules.
Molecular Geometry
Molecular geometry refers to the three-dimensional arrangement of the atoms within a molecule. Octahedral geometry is a type of molecular geometry where the central atom \( \mathrm{M} \) is surrounded symmetrically by six \( \mathrm{X} \) atoms. In an octahedral arrangement, these surrounding atoms form the vertices of an imaginary octahedron.

The structure's symmetry is key, as it dictates both the chemical and physical properties of the molecule. Here are some essential points about octahedral molecular geometry:
  • The molecule is axially symmetric, meaning it can be rotated around certain axes and look the same.
  • The six \( \mathrm{X} \) atoms are split into two sets: four in one plane and two perpendicular to that plane, above and below the central atom.
  • This spatial arrangement allows for the characteristic angles of \( 90^{\circ} \) and \( 180^{\circ} \), contributing to the molecule's overall stability.
Understanding this symmetrical arrangement is crucial when predicting reactivity and interaction with other molecules.
Symmetry in Molecules
Symmetry simplifies the understanding of molecular geometry by offering a visual and conceptual way to predict molecule behavior and properties. In an octahedral geometry, symmetry is particularly significant:

  • An octahedral molecule uses its axes of symmetry to determine many of its physical properties, like polarizability and interaction with electromagnetic fields.
  • There are three axes of symmetry in an octahedral molecule that align opposite pairs of atoms. These contribute to the three pairs of \( 180^{\circ} \) \( \mathrm{X}-\mathrm{M}-\mathrm{X} \) angles.
  • Rotation about these axes shows the molecule's balance and uniformity, making octahedral molecules ideal for complex formation and ligand coordination in chemistry.
Grasping symmetry in such molecules aids in further exploration of complex reactions and interactions in both organic and inorganic chemistry.

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