Problem 85 - Chemistry The Central Science

Question

Given the colors observed for \(\mathrm{VO}_{4}^{3-}\) (orthovanadate ion), \(\mathrm{CrO}_{4}^{2-}\) (chromate ion), and \(\mathrm{MnO}_{4}^{-}\) (permanganate ion (see Exercise \(23.84 ),\) what can you say about how the energy separation between the ligand orbitals and the empty \(d\) orbitals changes as a function of the oxidation state of the transition metal at the center of the tetrahedral anion?

Step-by-Step Solution

Answer

As the oxidation state of the transition metal at the center of the tetrahedral anion increases (from +5 in VO₄³⁻, to +6 in CrO₄²⁻, and +7 in MnO₄⁻), the energy separation between the ligand orbitals and the empty d orbitals increases. This is inferred from the observation that the colors of the complexes change from yellow (orthovanadate ion) to orange/yellow (chromate ion) and finally to violet (permanganate ion), which indicates that the energy of the absorbed light increases with the oxidation state of the transition metal.

Step 1: Identify the transition metal ions and their oxidation states

The complexes given are orthovanadate ion (VO₄³⁻), chromate ion (CrO₄²⁻), and permanganate ion (MnO₄⁻). The transition metals in these complexes are V, Cr, and Mn, and their oxidation states are +5, +6, and +7, respectively.

Step 2: Determine the colors of the complexes

The colors observed for the complexes are: - Orthovanadate ion (VO₄³⁻): Yellow - Chromate ion (CrO₄²⁻): Yellow/orange - Permanganate ion (MnO₄⁻): Violet

Step 3: Relate the colors to energy differences

The color of a complex is related to the energy difference between the ligand orbitals and the d orbitals. This energy difference corresponds to the energy of the absorbed light, causing the complementary color to be observed. The energy of the absorbed light is given by the equation: \[ E = h \cdot f\] Where \(E\) is the energy, \(h\) is Planck's constant, and \(f\) is frequency of the absorbed light. As we go from yellow to violet, we note that the frequency (and hence energy) of the absorbed light increases. On the color wheel, yellow is opposite of violet. Therefore, a yellow compound absorbs light in the violet region, and vice versa.

Step 4: Derive a relationship between energy separation and oxidation state

Since the energy of the absorbed light increases as we move from orthovanadate ion (yellow) to permanganate ion (violet), we can infer that the energy separation between the ligand orbitals and the empty d orbitals also increases. This implies that, as the oxidation state of the transition metal at the center of the tetrahedral anion increases (from +5 in VO₄³⁻, to +6 in CrO₄²⁻, and +7 in MnO₄⁻), the energy separation between the ligand orbitals and the empty d orbitals increases.

Key Concepts

Oxidation States

Transition metal complexes like \(\mathrm{VO}_{4}^{3-}\), \(\mathrm{CrO}_{4}^{2-}\), and \(\mathrm{MnO}_{4}^{-}\) contain metals such as vanadium, chromium, and manganese, each with specific oxidation states. An oxidation state signifies the degree of oxidation of a metal within a complex.

• For orthovanadate ion \(\mathrm{VO}_{4}^{3-}\), vanadium is in a +5 oxidation state.
• In chromate ion \(\mathrm{CrO}_{4}^{2-}\), chromium is in a +6 oxidation state.
• For permanganate ion \(\mathrm{MnO}_{4}^{-}\), manganese is in a +7 oxidation state.

The unique oxidation states influence the electronic structure, impacting the way electrons are distributed among the metal's d orbitals when forming a complex. This results in different chemical and physical properties, such as color.

Ligand Field Theory

Ligand field theory helps explain how electrons are arranged around a transition metal when it forms a complex. When ligands, like \(\mathrm{O}_{4}^{-}\) groups in our examples, approach a metal, they perturb the degenerate d-orbitals (orbitals with the same energy) of the metal.

• This perturbation results in a split of the energy levels of the d orbitals.
• The extent of this splitting influences the stability and color of the complexes.

In tetrahedral complexes, this splitting is typically smaller than in octahedral complexes because there are fewer ligands, resulting in weaker ligand interactions. Ligand field theory provides a framework for predicting and explaining the effects of various ligand types and arrangements on the energy levels of metal d orbitals.

Color of Complexes

The color observed in transition metal complexes arises from the specific wavelengths of light absorbed when electrons transition between different d orbitals. The particular color we see is the complementary color to that absorbed.

• In \(\mathrm{VO}_{4}^{3-}\), the complex appears yellow because it absorbs light in the violet range.
• Chromate ion \(\mathrm{CrO}_{4}^{2-}\) exhibits a yellow/orange color, absorbing in the blue/violet range.
• Within permanganate \(\mathrm{MnO}_{4}^{-}\), the violet appearance comes from the absorption of green/yellow light.

Thus, the transition from yellow to violet among these ions indicates increasing light energy absorbed as the metal's oxidation state rises.

Energy Separation

Energy separation in transition metal complexes refers to the difference in energy between the d orbital levels. This separation is crucial because it dictates how much energy is needed to promote an electron within the metal complex, directly influencing the color of the complex.

• The energy of absorbed light, described by \( E = h \cdot f \), increases with higher energy separation.
• As oxidation states rise from +5 to +7 in our example complexes, the energy separation also increases.

This larger energy separation corresponds to higher frequencies of light being absorbed (and thus more energy), explaining the transition from lower energy absorption (yellow) to higher energy absorption (violet) across \(\mathrm{VO}_{4}^{3-}\), \(\mathrm{CrO}_{4}^{2-}\), and \(\mathrm{MnO}_{4}^{-}\).