Question

The \(\left[\mathrm{Ti}(\mathrm{NCS})_{6}\right]^{3-}\) ion exhibits a single absorption band at 544 \(\mathrm{nm} .\) Calculate the crystal field splitting energy \(\Delta\) in \(\mathrm{kJ} / \mathrm{mol}\). Is NCS a stronger or weaker field ligand than water? Predict the color of \(\left[\mathrm{Ti}(\mathrm{NCS})_{6}\right]^{3-}\)

Short answer

The crystal field splitting energy \( \Delta \) is 220.6 kJ/mol. NCS is a stronger field ligand than water. The color of the ion is violet.

Step-by-step solution

Convert Wavelength to Energy

The absorption band is at a wavelength of 544 nm. To find the energy, we use the equation:\[ E = \frac{hc}{\lambda} \]where \( h = 6.626 \times 10^{-34} \text{ Js} \) is Planck's constant, \( c = 3.00 \times 10^{8} \text{ m/s} \) is the speed of light, and \( \lambda = 544 \times 10^{-9} \text{ m} \). Plugging in these values, we get:\[ E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^{8}}{544 \times 10^{-9}} = 3.663 \times 10^{-19} \text{ J} \]

Convert Energy per Photon to Energy per Mole

We convert the energy from joules per photon to joules per mole using Avogadro's number, \( N_A = 6.022 \times 10^{23} \text{ mol}^{-1} \):\[ \Delta = E \times N_A = 3.663 \times 10^{-19} \times 6.022 \times 10^{23} = 220.6 \text{ kJ/mol} \]

Compare Ligand Strength

The observed crystal field splitting energy \( \Delta \) for water is smaller than 220.6 kJ/mol. Thus, NCS is a stronger field ligand than water because it results in a higher splitting energy.

Predict Color

The ion absorbs light at 544 nm, which corresponds to the green-yellow part of the spectrum. The color observed is complementary to the absorbed color. Therefore, the complementary color to green-yellow is violet.

Key concepts

Ligand Strength

In inorganic chemistry, understanding the strength of a ligand is crucial because it affects how metal ions interact within a complex. This interaction is explained by crystal field theory, which provides a model for predicting and understanding the magnetic properties and colors of metal complexes. Ligands influence the energy levels of the incorporated metal ions, leading to a phenomenon called crystal field splitting.
Crystal field splitting refers to the difference in energy between two sets of d-orbitals when ligands approach a transition metal ion. The strength of this splitting is determined by the type of ligand involved. Ligands that cause larger splitting are known as strong field ligands, whereas those causing smaller splitting are weak field ligands. In the given context, NCS (thiocyanate) produces a splitting energy (\(\Delta \)) of 220.6 kJ/mol, which is higher compared to water as a ligand, indicating that NCS is a stronger field ligand than water.

• Strong field ligands like NCS cause significant splitting, affecting the electronic configuration and properties.
• Weaker field ligands like water result in lesser splitting, meaning different electronic configurations might be favored.

This characteristic plays a key role in determining the reactivity, magnetism, and color of the complexes.

Energy Conversion

Energy conversion in this context involves transitioning from one measure of energy to another, specifically from the energy of light (in Joules) to the energy required to split d-orbitals in a mole of ions (in kJ/mol). This involves a two-step calculation process.
Firstly, to find the energy absorbed by a single ion in the complex, we apply the widely used formula: \[E = \frac{hc}{\lambda} \]where \( h \) is Planck’s constant \( (6.626 \times 10^{-34} \text{ Js}) \) and \( c \) is the speed of light \( (3.00 \times 10^{8} \text{ m/s}) \). Using the wavelength \( \lambda = 544 \times 10^{-9} \text{ m} \), we calculate the energy per photon to be \( 3.663 \times 10^{-19} \text{ J} \).
The next step is to convert this single-photon energy to the energy applicable to a mole of these ions. Multiplying by Avogadro's number \( (6.022 \times 10^{23}\,\text{mol}^{-1}) \) provides the crystal field splitting energy in terms of \( \text{kJ/mol} \), equating to \( 220.6 \text{ kJ/mol} \).

• This conversion solidifies our understanding by scaling microscopic observations to quantifiable macroscopic values.
• It demonstrates how light energy absorption transitions into a chemical reaction or property change in a mole of a substance.

Color Prediction

Predicting colors in chemistry, especially in transition metal complexes, involves understanding absorption and reflection principles. Transition metals owe their vibrant colors to the d-d electron transitions facilitated by the d-orbital splitting described in crystal field theory. This electron movement results in the absorption of specific light wavelengths. The unabsorbed wavelengths determine the color that we observe.
For the \([\mathrm{Ti}(\mathrm{NCS})_{6}]^{3-}\) complex, absorption occurs predominantly at 544 nm, which lies within the green-yellow spectrum of light. Through the principle of complementary colors, any absorbed color dictates the complementary color we perceive.

• The color observed is complementary to the absorbed wavelength due to subtractive color mixing.
• As the complex absorbs green-yellow wavelengths, the color reflected—and thus observed—is violet, its complementary color on the color wheel.

This method of color prediction is incredibly useful and not only enriches the knowledge of the metal complexes but also finds practical applications in materials and dye industries.