Problem 80 - Chemistry: The Central Science

Question

The value of \(\Delta\) for the \(\left[\mathrm{CrF}_{6}\right]^{3-}\) complex is \(182 \mathrm{~kJ} / \mathrm{mol}\). Calculate the expected wavelength of the absorption corresponding to promotion of an electron from the lower-energy to the higher-energy \(d\) -orbital set in this complex. Should the complex absorb in the visible range? (You may need to review Sample Exercise 6.3; remember to divide by Avogadro's number.)

Step-by-Step Solution

Answer

The expected wavelength of absorption for the \(\left[\mathrm{CrF}_{6}\right]^{3-}\) complex is 658 nm, which falls within the visible range (400-700 nm). Therefore, the complex should absorb in the visible range.

Step 1: Convert energy into Joules per mole

First, we need to convert the given crystal field splitting energy in kJ/mol into Joules per mole. We can do this by multiplying the given value by 1000 J/kJ: ∆ = 182 kJ/mol × (1000 J/kJ) = 182,000 J/mol

Step 2: Convert energy per mole to energy per photon

We need to convert the energy per mole to energy per photon. We can do this by dividing the energy per mole by Avogadro's number (6.022 × 10^23 mol^-1): Energy per photon = (182,000 J/mol) ÷ (6.022 × 10^23 mol^-1) ≈ 3.02 × 10^-19 J

Step 3: Calculate the frequency

Next, we need to calculate the frequency corresponding to the energy per photon. We can use the formula 𝜈 = E/h where 𝜈 is the frequency, E is the energy per photon, and h is Planck's constant (6.626 × 10^-34 Js): Frequency (𝜈) = (3.02 × 10^-19 J) ÷ (6.626 × 10^-34 Js) ≈ 4.56 × 10^14 Hz

Step 4: Calculate the wavelength

Now, we need to relate the frequency to wavelength using the speed of light (c) which is 3.00 × 10^8 m/s. The formula λ = c/𝜈 can be used for this purpose: Wavelength (λ) = (3.00 × 10^8 m/s) ÷ (4.56 × 10^14 Hz) ≈ 6.58 × 10^-7 m = 658 nm

Step 5: Determine if the complex absorbs in the visible range

The visible range of light corresponds to wavelengths between 400 and 700 nm. Since our calculated wavelength is 658 nm, it falls within the visible range. Therefore, the complex will absorb in the visible range. In conclusion, the expected wavelength of absorption corresponding to promotion of an electron in the CrF6 complex ion is 658 nm, and the complex should absorb in the visible range.

Key Concepts

Electron Transition

Electron transitions are fundamental processes in the crystal field theory. When electrons absorb energy, they jump from lower-energy orbitals to higher-energy ones. This phenomenon is pivotal in understanding the colors exhibited by certain metal complexes.
In an octahedral field, such as in the \([\mathrm{CrF}_{6}]^{3-}\) complex, the presence of ligands causes the splitting of the degenerate \(d\)-orbitals. This results in two sets of orbitals: \(t_{2g}\) (lower energy) and \(e_g\) (higher energy).

• Energy absorbed during an electron transition is equal to the crystal field splitting energy \(\Delta\).
• The absorbed energy promotes an electron from the \(t_{2g}\) to the \(e_g\) orbitals.

This transition gives rise to absorption in the visible spectrum, hence explaining the color of the complex.

Wavelength Calculation

Understanding how to calculate the wavelength corresponding to an electron transition requires knowledge of both frequency and energy relationships.
First, convert the energy given for the transition into energy per photon. Divide the energy per mole by Avogadro's number, yielding the energy of a single photon.
Next, determine the frequency \(u\) using Planck’s equation \(u = \frac{E}{h}\), where \(E\) is the energy per photon and \(h = 6.626 \times 10^{-34} \, \text{Js}\) is Planck's constant.
Finally, use the frequency to calculate the wavelength \(\lambda\) with the formula \(\lambda = \frac{c}{u}\), where \(c = 3.00 \times 10^8 \, \text{m/s}\) is the speed of light. The calculated wavelength provides insight into the light absorption characteristics of the molecule.

Visible Spectrum

The visible spectrum is the portion of the electromagnetic spectrum that human eyes can detect. It ranges from approximately 400 to 700 nanometers (nm) in wavelength.
Each color corresponds to a specific range of wavelengths:

• Violet: 400-450 nm
• Blue: 450-495 nm
• Green: 495-570 nm
• Yellow: 570-590 nm
• Orange: 590-620 nm
• Red: 620-700 nm

In the case of the \([\mathrm{CrF}_{6}]^{3-}\) complex, the calculated absorption wavelength of 658 nm falls within the red spectrum. This absorption affects the color the complex appears to our eyes, often complementing the absorbed color on the pigment wheel.

Planck's Constant

Planck's constant is a critical factor in quantum mechanics, serving as the bridge between the energy and frequency of a photon. Its value \(6.626 \times 10^{-34} \, \text{Js}\) plays a standard role across calculations involving photons and light waves.
Using Planck's constant, the energy of a single photon can be directly related to its frequency by the equation \(E = hu\).
This relationship not only aids in determining the characteristic wavelength of energy absorbed through electron transitions but also in understanding fundamental physical principles governing atomic and subatomic processes such as:

• Quantum leap of electrons to higher energy states
• Emission and absorption spectra of materials

As students learn to solve wavelength calculations, they delve deeper into the world of quantum mechanics, exploring how tiny constants like Planck's shape our understanding of the universe.