Question

a. In the absorption spectrum of the complex ion [Cr(NCS) \(\left._{6}\right]^{3-}\), there is a band corresponding to the absorption of a photon of light with an energy of \(1.75 \times 10^{4} \mathrm{~cm}^{-1}\). Given \(1 \mathrm{~cm}^{-1}=\) \(1.986 \times 10^{-23} \mathrm{~J}\), what is the wavelength of this photon? b. The \(\mathrm{Cr}-\mathrm{N}-\mathrm{C}\) bond angle in \(\left[\mathrm{Cr}(\mathrm{NCS})_{6}\right]^{3-}\) is predicted to be \(180^{\circ}\). What is the hybridization of the \(\mathrm{N}\) atom in the \(\mathrm{NCS}^{-}\) ligand when a Lewis acid-base reaction occurs between \(\mathrm{Cr}^{3+}\) and \(\mathrm{NCS}^{-}\) that would give a \(180^{\circ} \mathrm{Cr}-\mathrm{N}-\mathrm{C}\) bond angle? \(\left[\mathrm{Cr}(\mathrm{NCS})_{6}\right]^{3-}\) undergoes substitution by ethylenediammine (en) according to the equation \(\left[\mathrm{Cr}(\mathrm{NCS})_{6}\right]^{3-}+2 \mathrm{en} \longrightarrow\left[\mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}\right]^{+}+4 \mathrm{NCS}^{-}\) Does \(\left[\mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}\right]^{+}\) exhibit geometric isomerism? Does \(\left[\mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}\right]^{+}\) exhibit optical isomerism?

Short answer

The wavelength of the photon is approximately \(570.59 nm\). The hybridization of the nitrogen atom in the NCS- ligand is sp. The complex [Cr(NCS)2(en)2]+ exhibits both geometric (cis-trans) and optical isomerism.

Step-by-step solution

Part a: Wavelength calculation

First, we need to convert the energy from wavenumbers to joules using the conversion factor given, and then use the energy-wavelength formula to calculate the wavelength. The energy-wavelength formula can be written as: \( E = h c/\lambda \) Where E is the energy, h is the Planck's constant (\(6.626\times10^{-34} Js \)), c is the speed of light in a vacuum (\(2.998\times10^8 m/s \)), and λ is the wavelength. 1. Convert the energy from wavenumbers to joules: \(1.75\times10^4 cm^{-1} \times 1.986\times10^{-23}J\cdot cm = 3.4785\times10^{-19} J\) 2. Rearrange the energy-wavelength formula to solve for λ: \( \lambda = h c/E \) 3. Plug in the values and calculate the wavelength: \( \lambda = (6.626\times10^{-34} Js)\cdot(2.998\times10^8 m/s) / (3.4785\times10^{-19} J) = 570.59 nm \)

Part b: Hybridization of nitrogen atom

The molecule has a linear Cr-N-C bond angle of \(180^{\circ}\). To achieve this bond angle, the hybridization of the nitrogen atom in NCS- ligand must involve the combination of atomic orbitals that lead to a linear geometry. The most appropriate hybridization to achieve a linear geometry is sp hybridization, which consists of mixing one s and one p orbital. Thus, the hybridization of the N atom in the NCS- ligand is sp.

Part c: Geometric and optical isomerism

Geometric isomerism occurs when the ligands surrounding a central metal ion can adopt different spatial arrangements. In the complex [Cr(NCS)2(en)2]+, both ethylenediammine (en) ligands bind to the chromium through two nitrogen atoms, acting as bidentate ligands. As a result, there are only two possible arrangements for the NCS- ligands: either they are adjacent to each other (cis configuration) or opposite to one another (trans configuration). Hence, the complex exhibits geometric isomerism (cis-trans isomerism). Optical isomerism arises when a molecule is chiral, which means it cannot be superimposed on its mirror image. In the case of the complex, the trans configuration is symmetrical and can be superimposed on its mirror image, so it does not exhibit optical isomerism. However, the cis configuration is non-symmetrical and cannot be superimposed on its mirror image, which means that it does exhibit optical isomerism. In conclusion, the complex [Cr(NCS)2(en)2]+ exhibits both geometric (cis-trans) and optical isomerism.

Key concepts

Absorption Spectrum

The absorption spectrum of a complex ion like \[\text{Cr}(\text{NCS})_{6}^{3-}\] reveals the wavelengths of light that the complex can absorb. This absorption occurs due to electronic transitions within the complex, where electrons are promoted to higher energy levels when they absorb photons of specific energies.

Each band in the spectrum corresponds to a different transition, contributing to our understanding of the electronic structure of the complex. By studying the absorption spectrum, we can gain insights into the types of bonds within the complex, their arrangements, and overall symmetry.

Photon Wavelength Calculation

To determine the wavelength of a photon absorbed by a complex ion, one must use the relationship between energy and wavelength. The energy (\(E\) in the exercise is given in units of wavenumbers, which can be converted to joules using the provided conversion factor.

Once the energy is in joules, you can use the energy-wavelength formula \(E = hc/\lambda\), where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. Solving this equation for \(\lambda\) gives the wavelength of the light absorbed by the complex. It's an essential piece of information for understanding the electronic transitions taking place.

Ligand Hybridization

In coordination chemistry, ligand hybridization refers to the atomic orbitals on the ligand atom that overlap with the orbitals on the central metal ion to form a coordination bond. For a linear arrangement such as the \[\text{Cr}-\text{N}-\text{C}\] bond angle in \[\text{Cr}(\text{NCS})_{6}^{3-}\], the nitrogen atom utilizes sp hybridization.

This involves the mixing of one s and one p orbital, creating two sp hybrid orbitals that are oriented 180° apart. The type of hybridization directly influences the geometry of the ligand-metal bonding and is critical for predicting the structure of complex ions.

Geometric Isomerism

Geometric isomerism, also known as cis-trans isomerism, occurs in coordination compounds where ligands can have different spatial orientations around the central metal ion. For a complex such as \[\text{Cr}(\text{NCS})_{2}(\text{en})_{2}^{+}\], where the chromium is coordinated with bidentate ligands (ethylenediammine) and two monodentate ligands (NCS-), geometric isomers can form.

They can be cis, with the same ligands adjacent, or trans, with the same ligands opposite each other. The formation of these isomers affects the chemical and physical properties of the complex, which is pivotal in fields such as medicinal chemistry where the isomer type can determine the effectiveness of a drug.

Optical Isomerism

Optical isomerism arises in coordination compounds that can exist as non-superimposable mirror images, known as enantiomers. This form of isomerism is a type of chirality, essentially like how left and right hands are mirror images but not identical.

In the case of \[\text{Cr}(\text{NCS})_{2}(\text{en})_{2}^{+}\], the cis configuration is non-symmetrical and gives rise to enantiomers, thus displaying optical isomerism. This characteristic is crucial in determining the complex's interaction with polarized light and its biological activity, as one enantiomer may be biologically active while the other is not.