Question

The value of \(\Delta\) for the \(\left[\mathrm{MoI}_{6}\right]^{3-}\) complex is \(198.58 \mathrm{~kJ} / \mathrm{mol}\). 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?

Short answer

The calculated wavelength of absorption for the \(\left[\text{MoI}_{6}\right]^{3-}\) complex is approximately 1.00 nm, which is not within the visible range (400 nm to 700 nm). Therefore, the complex should not absorb in the visible range.

Step-by-step solution

1. Convert energy from kJ/mol to J

Given Δ = 198.58 kJ/mol, we can convert it to J/mol by multiplying with 1000: Δ = 198.58 × 1000 = 198,580 J/mol

2. Calculate the frequency of light absorbed using Planck's equation

Planck's equation relates the energy of a photon (E) to its frequency (ν) as follows: E = hν where h is Planck's constant (6.626 × 10^{-34} Js). We know the energy needed for the electron transition in the complex is Δ. So, we can express the frequency as: ν = Δ / h Plug in the Δ value (198,580 J/mol) and Planck's constant. ν = (198,580 J/mol) / (6.626 × 10^{-34} Js) ν ≈ 2.996 × 10^{20} Hz

3. Calculate the wavelength using the speed of light

The frequency (ν) and wavelength (λ) of light are related by the speed of light (c) as follows: c = νλ where c is the speed of light (2.998 × 10^{8} m/s). Rearrange the equation to solve for λ: λ = c / ν Plug in the values of c and ν. λ = (2.998 × 10^{8} m/s) / (2.996 × 10^{20} Hz) λ ≈ 1.00 × 10^{-12} m

4. Determine if the complex absorbs in the visible range

In general, the visible range of light spans wavelengths from about 400 nm to 700 nm. Our calculated wavelength is 1.00 × 10^{-12} m, which can be converted to nm: λ = (1.00 × 10^{-12} m) × (1 × 10^9 nm/m) λ ≈ 1.00 nm Since 1.00 nm is much shorter than the wavelengths in the visible range, the complex should not absorb in the visible range.

Key concepts

d-orbital transitions

In the world of chemistry, d-orbital transitions play a crucial role in understanding how certain compounds absorb light. In transition metal complexes, like the \( \left[\mathrm{MoI}_{6}\right]^{3-} \) complex, electrons can move between different d-orbitals. This movement is known as a d-orbital transition.

D-orbital transitions occur due to changes in energy levels of electrons, which can happen when these electrons absorb specific photons of light. Such transitions are often responsible for the vibrant colors many transition metal complexes exhibit. However, not all transitions result in colors visible to the human eye.

When calculating d-orbital transitions, it's essential to measure the energy required for the electron to jump from a lower d-orbital to a higher one. This energy is often represented by \( \Delta \) (delta), and to make predictions about the absorption of light, we must convert this energy to forms that relate to frequency and wavelength.

Planck's constant

Planck's constant, denoted as \( h \), is a fundamental constant in physics that relates the energy of a photon to the frequency of the electromagnetic wave associated with that photon. With a value of \( 6.626 \times 10^{-34} \) Js, Planck's constant plays a pivotal role in quantum mechanics.

Its primary use in absorption spectroscopy is to find the frequency of light when given energy levels. The relationship is described by the formula \( E = hu \), where \( E \) is the energy of the photon and \( u \) (nu) is the frequency. This equation allows us to understand how much energy a photon has given its frequency, which is critical when examining electron transitions in microscopic chemical processes.

By using Planck's equation, scientists can convert energy level differences, like those in d-orbital transitions, to light frequencies. Knowing the frequency helps in the next step: determining the wavelength of light, which further helps in identifying which part of the light spectrum is being interacted with.

visible light spectrum

The visible light spectrum is a small part of the electromagnetic spectrum that can be perceived by the human eye. It spans wavelengths approximately ranging from 400 nanometers (nm) to 700 nm. The spectrum includes all the colors visible to us on a rainbow: red, orange, yellow, green, blue, indigo, and violet.

In absorption spectroscopy, identifying whether an absorption falls within this range determines if a compound will have a visible color. If the wavelength is outside this range, typically shorter than 400 nm or longer than 700 nm, the absorption is beyond human detection and the substance appears colorless to us.

For example, in the \( \left[\mathrm{MoI}_{6}\right]^{3-} \) complex problem, the calculated wavelength for the absorbed light was 1.00 nm. This wavelength is much shorter than the requirements for visibility and therefore, the transition would not be visible to the naked eye. Understanding the spectrum helps predict and explain the observed properties of chemical substances based on their absorption characteristics.