Problem 41 - Chemistry The Science in Context

Question

Solutions of \(\mathrm{NiCl}_{4}^{2-}\) and \(\mathrm{NiBr}_{4}^{2-}\) absorb light at \(702 \mathrm{nm}\) and \(756 \mathrm{nm},\) respectively. In which ion is the split of \(d\) -orbital energies greater?

Step-by-Step Solution

Answer

Answer: The split of d-orbital energies is greater in the NiCl₄²⁻ ion.

Step 1: Understanding the energy difference and wavelengths relation

The energy difference, \(\Delta E\), and the wavelength of absorbed light, \(\lambda\), are related by Planck's equation: \[E = h\nu = \frac{hc}{\lambda},\] where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\nu\) is the frequency of the light. Since \(\Delta E \propto -1/\lambda\), as \(\lambda\) decreases, \(\Delta E\) increases, meaning that the split of \(d\)-orbital energies would be greater for a complex which absorbs light at a shorter wavelength.

Step 2: Compare the absorbed wavelengths

We are given that \(\mathrm{NiCl}_{4}^{2-}\) absorbs light at 702 nm and \(\mathrm{NiBr}_{4}^{2-}\) absorbs it at 756 nm. Since 702 nm < 756 nm, \(\mathrm{NiCl}_{4}^{2-}\) absorbs light at a shorter wavelength.

Step 3: Determine which ion has a greater split of d-orbital energies

Since \(\mathrm{NiCl}_{4}^{2-}\) absorbs light at a shorter wavelength, it has a larger energy difference between the split d-orbital energies compared to \(\mathrm{NiBr}_{4}^{2-}\). So, the split of \(d\)-orbital energies is greater in the \(\mathrm{NiCl}_{4}^{2-}\) ion.

Key Concepts

Transition Metal Complexes

Transition metal complexes are fascinating chemical entities that consist of a central metal atom or ion, often from the transition metal series in the periodic table, bonded to surrounding molecules or ions called ligands. These complexes are important in many chemical processes and have unique chemical and physical properties due to the d-orbitals in the metal ions.

When these ligands interact with transition metal ions, they cause the d-orbitals of the metal to split into different energy levels. This phenomenon is known as d-orbital splitting or crystal field splitting, and it gives rise to many of the colorful properties observed in these complexes.

The d-orbital splitting is influenced by several factors:

• The nature of the metal ion (size, charge, and electronic configuration).
• The type and geometry of the ligands attached to the metal center.
• The oxidation state of the metal.

The degree of splitting in the d-orbitals can affect the color of the complexes, their absorption properties, and overall stability, making their study valuable for various applications such as catalysis, material science, and bioinorganic chemistry.

Absorption Wavelength

The absorption wavelength is a crucial concept in understanding the interaction between light and matter, especially in transition metal complexes. When a complex absorbs light, electrons can be excited from lower-energy d-orbitals to higher-energy ones, resulting in distinct wavelengths of absorbed light.

This absorption corresponds to the energy difference between the split d-orbitals, and the specific wavelength at which absorption occurs can provide insight into the extent of d-orbital splitting. If a complex absorbs light at a shorter wavelength, it means the energy difference between the d-orbital levels is larger.

In the given example,

• NiCl_4^{2-} absorbs light at 702 nm, indicating greater d-orbital splitting compared to
• NiBr_4^{2-} which absorbs at a longer wavelength of 756 nm.

This relationship between wavelength and energy gap is pivotal in determining the properties of complex ions and helps in understanding their behavior in different chemical environments. Understanding this principle can guide chemists in designing new complexes with desirable properties.

Planck's Equation

Planck's equation is a fundamental formula that connects the properties of light with the energy transitions of electrons within atoms and molecules. It is a cornerstone in quantum mechanics and is utilized to calculate the energy associated with absorbed or emitted light: \[ E = hu = \frac{hc}{\lambda} \] Where:

• h is Planck's constant \( (6.626 \times 10^{-34} \; \,Js) \)
• u is the frequency of the light
• c is the speed of light \( (3.00 \times 10^{8} \; \,m/s) \)
• \lambda is the wavelength of the light

This equation shows that the energy of light is inversely proportional to its wavelength. Therefore, higher energy light corresponds to a shorter wavelength. This principle is critical in analyzing how light interacts with transition metal complexes.

By using Planck's equation, scientists can determine the energy differences that drive electronic transitions in complexes, allowing them to infer the magnitude of d-orbital splitting. Understanding this equation enables chemists to predict the properties and reactivities of various complexes.