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Chapter 23: Problem 50

Explain why the \(d_{x y}, d_{x z}\), and \(d_{y z}\) orbitals lie lower in energy than the \(d_{z}^{2}\) and \(d_{x^{2}-y^{2}}\) orbitals in the presence of an octahedral arrangement of ligands about the central metal ion.

Short Answer

Expert verified
In an octahedral arrangement of ligands, the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals lie lower in energy than the \(d_z^2\) and \(d_{x^2-y^2}\) orbitals due to differences in their electron cloud orientations. The former orbitals have their electron clouds in-between the axes, resulting in less repulsion with ligands and lower energy levels, while the latter orbitals' electron clouds align along the axes, causing greater repulsion with the ligands and higher energy levels. Overall, it is the interaction between the d orbitals and the ligands that determines the relative energy levels within the central metal ion.

Step by step solution

01

Recall the Crystal Field Theory:

According to the Crystal Field Theory, the energy levels of the d orbitals in a central metal ion are affected by the presence of ligands. In an isolated metal ion, the five d orbitals have the same energy (degenerate), but when ligands approach the central metal ion, they will cause the energy to split, resulting in the rising of some orbitals' energy and the lowering of others.
02

Visualizing the octahedral arrangement of ligands:

An octahedral arrangement consists of six ligands surrounding the central metal ion, positioned at equal distances along the x, y, and z-axis. There are two ligands directly opposite to each other along each axis, creating an octahedral shape.
03

Understand the interaction between ligands and d orbitals :

The interaction depends on the extent of the overlap between the electron clouds of the metal ion's d orbitals and the electron clouds of the ligands. If the overlap is greater, the repulsion between the ligands and the metal ion will cause an increase in the energy level of the corresponding orbital. Conversely, if the interaction is smaller, the energy level will be lower.
04

Analyze each d orbital's interaction with the ligands :

1. For \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\), these orbitals' electron clouds lie in-between the axes, so they don't point toward the ligands, causing less repulsion and correspondingly lower energy. 2. For \(d_z^2\) and \(d_{x^2-y^2}\), the electron clouds are aligned along the axes. The ligands directly interact with the lobes of these orbitals, causing higher repulsion between ligands and the metal ion and as a result, a higher energy level of these orbitals.
05

Conclude with the explanation:

Based on the analysis of the interactions between d orbitals and ligands in an octahedral arrangement, we can conclude that the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals lie lower in energy than the \(d_z^2\) and \(d_{x^2-y^2}\) orbitals. This is because the electron clouds of the former orbitals are located in-between the axes, causing less repulsion with the ligands and lower energy levels, while the electron clouds of the latter orbitals are aligned along the axes and directly interact with the approaching ligands, leading to higher repulsion and higher energy levels.

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Most popular questions from this chapter

Give the number of (valence) \(d\) electrons associated with the central metal ion in each of the following complexes: (a) \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\), (b) \(\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]\left(\mathrm{NO}_{3}\right)_{2}\) (c) \(\mathrm{Na}\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]\) (d) \(\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Br}_{2}\right] \mathrm{ClO}_{4}\), (e) \([\mathrm{Sr}(\mathrm{EDTA})]^{2-}\)

Which of the following objects is chiral: (a) a left shoe, (b) a slice of bread, \((c)\) a wood screw, (d) a molecular model of \(\mathrm{Zn}(\mathrm{en}) \mathrm{Cl}_{2}\) (e) a typical golf club?

A complex is written as \(\mathrm{NiBr}_{2} \cdot 6 \mathrm{NH}_{3}\). (a) What is the oxidation state of the \(\mathrm{Ni}\) atom in this complex? (b) What is the likely coordination number for the complex? (c) If the complex is treated with excess \(\mathrm{AgNO}_{3}(a q),\) how many moles of \(\mathrm{AgBr}\) will precipitate per mole of complex?

Oxyhemoglobin, with an \(\mathrm{O}_{2}\) bound to iron, is a low-spin \(\mathrm{Fe}(\mathrm{II})\) complex; deoxyhemoglobin, without the \(\mathrm{O}_{2}\) molecule, is a high-spin complex. (a) Assuming that the coordination environment about the metal is octahedral, how many unpaired electrons are centered on the metal ion in each case? (b) What ligand is coordinated to the iron in place of \(\mathrm{O}_{2}\) in deoxyhemoglobin? (c) Explain in a general way why the two forms of hemoglobin have different colors (hemoglobin is red, whereas deoxyhemoglobin has a bluish cast). (d) A 15 -minute exposure to air containing 400 ppm of CO causes about \(10 \%\) of the hemoglobin in the blood to be converted into the carbon monoxide complex, called carboxyhemoglobin. What does this suggest about the relative equilibrium constants for binding of carbon monoxide and \(\mathrm{O}_{2}\) to hemoglobin? (e) \(\mathrm{CO}\) is a strong-field ligand. What color might you expect carboxyhemoglobin to be?

Consider the tetrahedral anions \(\mathrm{VO}_{4}^{3-}\) (orthovanadate ion), \(\mathrm{CrO}_{4}^{2-}(\) chromate ion \(),\) and \(\mathrm{MnO}_{4}^{-}\) (permanganate ion). (a) These anions are isoelectronic. What does this statement mean? (b) Would you expect these anions to exhibit \(d-d\) transitions? Explain. (c) As mentioned in "A Closer Look" on charge-transfer color, the violet color of \(\mathrm{MnO}_{4}^{-}\) is due to a ligand-to-metal charge transfer (LMCT) transition. What is meant by this term? (d) The LMCT transition in \(\mathrm{MnO}_{4}^{-}\) occurs at a wavelength of \(565 \mathrm{nm}\). The \(\mathrm{CrO}_{4}^{2-}\) ion is yellow. Is the wavelength of the LMCT transition for chromate larger or smaller than that for \(\mathrm{MnO}_{4}^{-}\) ? Explain. (e) The \(\mathrm{VO}_{4}{ }^{3-}\) ion is colorless. Do you expect the light absorbed by the LMCT to fall in the UV or the IR region of the electromagnetic spectrum? Explain your reasoning.
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