Question

a. In the absorption spectrum of the complex ion $\mathrm{Cr}(\mathrm{NCS}{)}_6^{3-}$ 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 $\mathrm{Cr}(\mathrm{NCS}{)}_6^{3-}$ is predicted to be ${180}^{\circ }.$ What is the hybridization of the 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? $\mathrm{Cr}(\mathrm{NCS}{)}_6^{3-}$ undergoes substitution by ethylenediamine (en) according to the equation $$\mathrm{Cr}(\mathrm{NCS}{)}_6^{3-}+2\mathrm{en}⟶\mathrm{Cr}(\mathrm{NCS}{)}_2(\mathrm{en}{)}_2^{+}+4{\mathrm{NCS}}^{-}$$ Does $\mathrm{Cr}(\mathrm{NCS}{)}_2(\mathrm{en}{)}_2^{+}$ exhibit geometric isomerism? Does $\mathrm{Cr}(\mathrm{NCS}{)}_2(\mathrm{en}{)}_2^{+}$ exhibit optical isomerism?

Short answer

The wavelength of the absorbed photon is approximately $5.96\times {10}^{-7}$ meters or 596 nm. The nitrogen atom in the NCS- ligand is sp-hybridized when a Lewis acid-base reaction occurs with Cr3+ that gives a ${180}^{\circ }$ Cr-N-C bond angle. The complex ion Cr(NCS)2(en)2+ does not exhibit geometric isomerism due to the presence of bidentate ethylenediamine ligands, but it does exhibit optical isomerism.

Step-by-step solution

Convert energy to Joules

First, we need to convert the energy of the photon from inverse centimeters to Joules. To do this, we use the given conversion factor: Energy in Joules = Energy in ${\mathrm{cm}}^{-1}\times$ Conversion factor Energy in Joules = $1.75\times {10}^4{\mathrm{cm}}^{-1}\times 1.986\times {10}^{-23}{\mathrm{J}\mathrm{cm}}^{-1}$

Calculate wavelength

Using the Planck's equation to relate energy (E) to wavelength (λ): Energy (E) = $h\times c/\lambda$ Where h = Planck’s constant $(6.626\times {10}^{-34}\mathrm{Js})$, c = speed of light $(2.9979\times {10}^8{\mathrm{ms}}^{-1})$, and λ = wavelength in meters Solving the equation for λ: $$\lambda =\frac{h\times c}{E}$$ Substitute the values: $$\lambda =\frac{6.626\times {10}^{-34}\mathrm{Js}\times 2.9979\times {10}^8{\mathrm{ms}}^{-1}}{1.75\times {10}^4{\mathrm{cm}}^{-1}\times 1.986\times {10}^{-23}{\mathrm{J}\mathrm{cm}}^{-1}}$$ b. N atom hybridization in the NCS- ligand

Determine hybridization of nitrogen

Considering a straight Cr-N-C bond angle of 180°, this indicates a linear geometry of the nitrogen atom. In this case, the nitrogen atom is sp-hybridized. c. Isomerism analysis for Cr(NCS)2(en)2+

Analyze geometric isomerism

Geometric isomerism, also known as cis-trans isomerism, occurs in coordination compounds when the ligands are arranged differently around the central metal ion. For the complex ion Cr(NCS)2(en)2+, the presence of bidentate ethylenediamine ligands (en) prevents geometric isomerism since the two en ligands enforce a specific geometry.

Analyze optical isomerism

Optical isomerism occurs when a compound cannot be superimposed on its mirror image. In this case, since the ethylenediamine ligands form a chelating ring with the Cr center, optical isomerism is possible. A mirror image of the molecule can form a non-superimposable structure. Therefore, the complex ion Cr(NCS)2(en)2+ exhibits optical isomerism.

Key concepts

Absorption Spectrum

The absorption spectrum is a fundamental concept in coordination chemistry, providing key insights into how complex ions interact with light. In the context of a complex ion like $\mathrm{Cr}(\mathrm{NCS}{)}_6^{3-}$, this spectrum reveals which wavelengths of light are absorbed by the ion.
This information can be used to determine various properties of the complex, such as the energy levels of electrons within the d-orbitals. The specific band corresponding to the absorption of a photon with energy $1.75\times {10}^4{\mathrm{cm}}^{-1}$ indicates a particular transition in the electron configuration.
Calculating the wavelength using the energy conversion in Joules and applying Planck's Equation helps us understand these transitions better. Through this process, the absorption spectrum becomes an invaluable tool in deciphering the electronic structure and behavior of coordination complexes like $\mathrm{Cr}(\mathrm{NCS}{)}_6^{3-}$.

• Absorption spectrum details energy transitions.
• It helps in understanding electronic behavior.

Ligand Hybridization

Understanding ligand hybridization is essential when examining the structure of complex ions, particularly focusing on bond angles and energy configurations. In the case of $\mathrm{Cr}(\mathrm{NCS}{)}_6^{3-}$, the nitrogen atom in the ${\mathrm{NCS}}^{-}$ ligand exhibits sp hybridization.
This occurs due to the need to maintain a linear geometry with a ${180}^{\circ }$ bond angle in the $\mathrm{Cr}-\mathrm{N}-\mathrm{C}$ arrangement. Such hybridization ensures that orbitals mix appropriately to accommodate both bonding requirements and electron configurations during complex formation.
The nature of sp hybridization involves one s orbital combining with one p orbital, resulting in two degenerate orbitals that provide the necessary linear configuration for the nitrogen atom. This hybridization is crucial for the overall stability and structure of the complex ion.

• Ligand hybridization affects molecular geometry.
• sp hybridization leads to linear shapes.

Geometric Isomerism

In coordination chemistry, geometric isomerism involves the spatial arrangement of ligands around a central metal atom or ion. However, for $\mathrm{Cr}(\mathrm{NCS}{)}_2(\mathrm{en}{)}_2^{+}$, geometric isomerism does not occur. This is primarily due to the nature of the ligands involved.
The ethylenediamine (en) ligands are bidentate, meaning they attach to the chromium ion at two points, creating a stable chelating structure. This prevents the flexibility required for geometric isomerism, as the en ligands enforce a fixed orientation around the chromium center.
In general, geometric isomerism requires ligands to have the possibility of aligning in different spatial configurations, such as cis (adjacent) and trans (opposite). In $\mathrm{Cr}(\mathrm{NCS}{)}_2(\mathrm{en}{)}_2^{+}$, the rigidity of the ethylenediamine coordination hinders such variability.

• Geometric isomerism needs ligand flexibility.
• Bidentate ligands often prevent such isomerism.

Optical Isomerism

Optical isomerism refers to the ability of a compound to exist in two forms that are mirror images of each other and cannot be superimposed, much like left and right hands. This type of isomerism is indeed present in $\mathrm{Cr}(\mathrm{NCS}{)}_2(\mathrm{en}{)}_2^{+}$ due to the spatial arrangement of the bidentate ligands.
When chelating ligands like ethylenediamine coordinate to the central metal, they create a three-dimensional structure that may not be symmetric. This non-superimposable mirror image makes the complexes optically active.
Such molecules possess chiral centers, which in this case arise from the specific 3D arrangement of the ethylenediamine rings around the chromium ion. As a result, these complexes can rotate plane-polarized light, a characteristic property of optical isomers.

• Optical isomerism involves non-superimposable mirror forms.
• Bidentate ligands can create chiral centers.