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Chapter 23: Problem 89

Suppose that a transition-metal ion was in a lattice in which it was in contact with just two nearby anions, located on opposite sides of the metal. Diagram the splitting of the metal \(\bar{d}\) orbitals that would result from such a crystal field. Assuming a strong field, how many unpaired electrons would you expect for a metal ion with six \(d\) electrons? (Hint: Consider the linear axis to be the \(z\) -axis)

Short Answer

Expert verified
In a crystal field with two anions along the \(z\)-axis, the \(\bar{d}\) orbitals split into \(e_g\) (\(d_{z^2}\) and \(d_{x^2-y^2}\)) and \(t_{2g}\) (\(d_{xy}\), \(d_{xz}\), and \(d_{yz}\)) sets. For a metal ion with six \(d\) electrons and a strong field, all electrons will occupy the lower-energy \(t_{2g}\) orbitals, following Hund's rule by filling each orbital once before pairing up. Therefore, there will be no unpaired electrons in this case.

Step by step solution

01

Analyze the crystal field and identify the axis that divides the \(\bar{d}\) orbitals

Since the two anions are located on opposite sides of the metal along the \(z\)-axis, the crystal field will exert an effect mainly along this axis. \(\bar{d}\) orbitals can split into two groups depending on their orientation along this axis: those with a lobe along the \(z\)-axis (known as \(e_g\) orbitals) and those which lie in the xy plane (known as \(t_{2g}\) orbitals). In this case, the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals have lobes along the \(z\)-axis and will form the more energetic \(e_g\) set of orbitals. The remaining \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals will form the less energetic \(t_{2g}\) set.
02

Determine the splitting of the \(\bar{d}\) orbitals and diagram the split orbitals

As the crystal field acts along the \(z\)-axis, the \(d\) orbitals split into \(e_g\) and \(t_{2g}\) sets. The \(e_g\) orbitals include the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals, and the \(t_{2g}\) orbitals include the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals. The \(e_g\) orbitals will have a higher energy level than the \(t_{2g}\) orbitals due to the crystal field from the anions along the \(z\)-axis.
03

Calculate the number of unpaired electrons for a metal ion with six \(d\) electrons in a strong field

Since we are assuming a strong field, the energy gap between the \(t_{2g}\) and \(e_g\) orbitals is large. This means that electrons will fill the lower-energy \(t_{2g}\) orbitals first before occupying the higher-energy \(e_g\) orbitals (following Hund's rule). There are three \(t_{2g}\) orbitals, and they can accommodate up to six \(d\) electrons. Therefore, all six \(d\) electrons will occupy the \(t_{2g}\) orbitals, following Hund's rule by filling each orbital once before pairing up. So, each of the \(t_{2g}\) orbitals will have two electrons with opposite spins. Thus, the number of unpaired electrons for a metal ion with six \(d\) electrons in a strong field is 0.

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Most popular questions from this chapter

A classmate says, "A weak-field ligand usually means the complex is high spin." Is your classmate correct? Explain.

Indicate the coordination number of the metal and the oxidation number of the metal as well as the number and type of each donor atom of the ligands for each of the following complexes: (a) \(\mathrm{K}_{3}\left[\mathrm{Co}(\mathrm{CN})_{6}\right]\) (b) \(\mathrm{Na}_{2}\left[\mathrm{CdBr}_{4}\right]\) (c) \(\left[\mathrm{Pt}(\mathrm{en})_{3}\right]\left(\mathrm{ClO}_{4}\right)_{4}\) (d) \(\left[\mathrm{Co}(\mathrm{en})_{2}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)\right]^{+}\) (e) \(\mathrm{NH}_{4}\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{2}(\mathrm{NCS})_{4}\right]\) (f) \(\left[\mathrm{Cu}(\text { bipy })_{2} \mathrm{I}\right] \mathrm{I}\)

Indicate the coordination number of the metal and the oxidation number of the metal as well as the number and type of each donor atom of the ligands for each of the following complexes: (a) \(\mathrm{Na}_{2}\left[\mathrm{CdCl}_{4}\right]\) (b) \(\mathrm{K}_{2}\left[\mathrm{MoOCl}_{4}\right]\) (c) \(\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{Cl}_{2}\right] \mathrm{Cl}\) (d) \(\left[\mathrm{Ni}(\mathrm{CN})_{5}\right]^{3-}\) (e) \(\mathrm{K}_{3}\left[\mathrm{~V}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{3}\right]\) (f) \(\left[\mathrm{Zn}(\mathrm{en})_{2}\right] \mathrm{Br}_{2}\)

A manganese complex formed from a solution containing potassium bromide and oxalate ion is purified and analyzed. It contains \(10.0 \% \mathrm{Mn}, 28.6 \%\) potassium, \(8.8 \%\) carbon, and \(29.2 \%\) bromine by mass. The remainder of the compound is oxygen. An aqueous solution of the complex has about the same electrical conductivity as an equimolar solution of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right] .\) Write the formula of the compound, using brackets to denote the manganese and its coordination sphere.

(a) Can we see light that is \(300 \mathrm{nm}\) in wavelength? \(500 \mathrm{nm}\) in wavelength? (b) What is meant by the term complementary color? (c) What is the significance of complementary colors in understanding the colors of metal complexes? (d) If a complex absorbs light at \(610 \mathrm{nm},\) what is the energy of this absorption in \(\mathrm{kJ} / \mathrm{mol}\) ?
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