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 $$\mathrm{Cr}(\mathrm{NCS})_{6}^{3-}+2 \mathrm{en} \longrightarrow \mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}^{+}+4 \mathrm{NCS}^{-}$$ Does \(\mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}^{+}\) exhibit geometric isomerism? Does \(\mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}^{+}\) exhibit optical isomerism?

Step by step solution

01

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 = \(1.75 \times 10^{4} \mathrm{cm}^{-1} \times 1.986 \times 10^{-23} \mathrm{J\, cm}^{-1}\)

02

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: $$\lambda = \frac{6.626 \times 10^{-34}\, \mathrm{Js} \times 2.9979 \times 10^8\, \mathrm{ms}^{-1}}{1.75 \times 10^{4} \mathrm{cm}^{-1} \times 1.986 \times 10^{-23} \mathrm{J\, cm}^{-1}}$$ b. N atom hybridization in the NCS- ligand

03

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+

04

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.

05

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.

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