Calculating Racah Parameter B'

The question asks us to determine the Racah parameter B' for the [CrF₆]³⁻ complex using the provided spectral transition data. The complex [CrF₆]³⁻ involves Cr³⁺ ion, which has a d³ electronic configuration. In an octahedral ligand field, the ground state for a d³ ion is ⁴A_2g. The given spectral transitions originate from this ground state to higher energy quartet states: ⁴T_2g, ⁴T_1g (F), and ⁴T_1g (P).

Converting Wavelengths to Wavenumbers

Spectral transitions are often expressed in terms of wavenumbers (cm⁻¹), which are directly proportional to energy. We need to convert the given wavelengths (\(\lambda\)) in nanometers (nm) to wavenumbers (\(\bar{\nu}\)) in cm⁻¹ using the relationship:

\(\bar{\nu} \text{ (in cm}^{-1}\text{)} = \frac{1}{\lambda \text{ (in cm)}}\)

Since the wavelength is given in nm, we use the conversion 1 nm = 10⁻⁷ cm.

\(\bar{\nu} \text{ (in cm}^{-1}\text{)} = \frac{10^7}{\lambda \text{ (in nm)}}\)

• Transition 1 (671 nm): \(^4\text{A}_{2\text{g}} \rightarrow ^4\text{T}_{2\text{g}}\)
• \(\bar{\nu}_1 = \frac{10^7 \text{ cm}^{-1}}{671} \approx 14903.13 \text{ cm}^{-1}\)

• Transition 2 (441 nm): \(^4\text{A}_{2\text{g}} \rightarrow ^4\text{T}_{1\text{g}}\) (F)
• \(\bar{\nu}_2 = \frac{10^7 \text{ cm}^{-1}}{441} \approx 22675.74 \text{ cm}^{-1}\)

• Transition 3 (291 nm): \(^4\text{A}_{2\text{g}} \rightarrow ^4\text{T}_{1\text{g}}\) (P)
• \(\bar{\nu}_3 = \frac{10^7 \text{ cm}^{-1}}{291} \approx 34364.26 \text{ cm}^{-1}\)

Relating Transition Energies to Crystal Field and Racah Parameters

For a d³ octahedral complex, the first transition ( \(^4\text{A}_{2\text{g}} \rightarrow ^4\text{T}_{2\text{g}}\) ) corresponds directly to the crystal field splitting energy, \(\Delta_\text{o}\).

\(\bar{\nu}_1 = \Delta_\text{o}\)

So, \(\Delta_\text{o} \approx 14903.13 \text{ cm}^{-1}\).

The energies of the transitions to the \(^4\text{T}_{1\text{g}}\) states are related to \(\Delta_\text{o}\) and the Racah parameters B and C. Using approximations often applied to transition metal complexes, a useful relationship between the two higher energy quartet transitions (\(\bar{\nu}_2\) and \(\bar{\nu}_3\)) and the parameters is the sum rule:

\(\bar{\nu}_2 + \bar{\nu}_3 = \Delta_\text{o} + 15\text{B'}\)

where B' is the Racah parameter in the complex.

Calculating Racah Parameter B'

We can rearrange the sum rule equation to solve for B':

\(15\text{B'} = \bar{\nu}_2 + \bar{\nu}_3 - \Delta_\text{o}\)

Substitute the calculated wavenumbers and \(\Delta_\text{o}\) value:

\(15\text{B'} \approx 22675.74 \text{ cm}^{-1} + 34364.26 \text{ cm}^{-1} - 14903.13 \text{ cm}^{-1}\)

\(15\text{B'} \approx 57040.00 \text{ cm}^{-1} - 14903.13 \text{ cm}^{-1}\)

\(15\text{B'} \approx 42136.87 \text{ cm}^{-1}\)

Now, divide by 15 to find B':

\(\text{B'} \approx \frac{42136.87 \text{ cm}^{-1}}{15}\)

\(\text{B'} \approx 2809.12 \text{ cm}^{-1}\)

Comparing Calculated B' with Options

The calculated value for the Racah parameter B' is approximately 2809.12 cm⁻¹. We compare this value to the given options:

• Option 1: 2813 cm⁻¹
• Option 2: 1986 cm⁻¹
• Option 3: 827 cm⁻¹
• Option 4: 213 cm⁻¹

The calculated value of 2809.12 cm⁻¹ is closest to 2813 cm⁻¹.