Question

\(\mathrm{MnF}_{6}^{2-}\) has a crystal field splitting energy, \(\Delta_{\mathrm{o}},\) of \(2.60 \times 10^{2} \mathrm{~kJ} / \mathrm{mol}\). What is the wavelength responsible for this energy?

Short answer

The wavelength responsible for the crystal field splitting energy, \(\Delta_{\mathrm{o}}\), in \(\mathrm{MnF}_{6}^{2-}\) is 768 nm.

Step-by-step solution

Convert the energy to joules

First, we need to convert the given crystal field splitting energy, \(\Delta_{\mathrm{o}}\) from kJ/mol to J/mol, using the conversion factor (1 kJ = 1000 J). \(\Delta_{\mathrm{o}} = 2.60 \times 10^{2} \frac{\mathrm{kJ}}{\mathrm{mol}} \times \frac{1000 \ \mathrm{J}}{1 \ \mathrm{kJ}} = 2.60 \times 10^{5} \frac{\mathrm{J}}{\mathrm{mol}}\).

Use Planck's equation to solve for wavelength

Now, we can use Planck's equation to find the wavelength responsible for this energy. Recall that the equation is \(E = h \cdot c / \lambda\), where \(E\) is energy, \(h\) is Planck's constant (\(6.63 \times 10^{-34} \mathrm{J \cdot s}\)), \(c\) is the speed of light (\(2.998 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is wavelength. Rearrange the equation to solve for \(\lambda\): \(\lambda = \frac{h \cdot c}{E}\).

Plug in the values and calculate the wavelength

Next, plug in the values for the constants and the energy we found in Step 1: \(\lambda = \frac{6.6372 \times 10^{-34} \mathrm{J \cdot s} \times 2.998 \times 10^{8} \mathrm{m/s}}{2.60 \times 10^{5} \frac{\mathrm{J}}{\mathrm{mol}}} = 7.679 \times 10^{-31} \frac{\mathrm{J \cdot m}}{\mathrm{mol}}\).

Convert to nm and round to the nearest whole number

Finally, convert the wavelength from meters to nanometers (1 m = \(10^9\) nm) and round to the nearest whole number: \(\lambda = 7.679 \times 10^{-31} \frac{\mathrm{J \cdot m}}{\mathrm{mol}} \times \frac{10^9 \ \mathrm{nm}}{1 \ \mathrm{m}} = 768 \ \mathrm{nm}\) (rounded to the nearest whole number). Thus, the wavelength responsible for the crystal field splitting energy, \(\Delta_{\mathrm{o}}\), in \(\mathrm{MnF}_{6}^{2-}\) is \(\boxed{768 \ \mathrm{nm}}\).

Key concepts

Planck's Equation

Planck's equation is a fundamental concept in quantum mechanics that relates the energy of a photon to its frequency. The equation is represented as
\( E = h \cdot u \),
where \( E \) is the energy of the photon, \( h \) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{J \cdot s}\)), and \( u \) is the frequency of the photon. Since frequency and wavelength \( (\lambda) \) are inversely related by the speed of light \( (c) \) with the equation \( u = \frac{c}{\lambda} \), Planck's equation can also be written as
\( E = \frac{h \cdot c}{\lambda} \).
This form is useful for solving problems related to light's interaction with matter, particularly in scenarios involving energy levels such as electronic transitions or the crystal field splitting energy in inorganic complexes like \( \mathrm{MnF}_{6}^{2-} \).

Wavelength Calculation

The calculation of wavelength is essential in understanding the properties of light and its interaction with matter. Wavelength \( (\lambda) \) is the distance over which the wave's shape repeats and is typically measured in meters. It is related to the energy and frequency of light by the equations mentioned earlier.

The rearranged Planck's equation \( \lambda = \frac{h \cdot c}{E} \) allows us to solve for the wavelength when the energy is known, as in the problem at hand. This problem requires us to find the wavelength of light that corresponds to the crystal field splitting energy. To find this wavelength, we use the values for the energy we have, Planck's constant, and the speed of light to get the result in meters, which can then be converted to more convenient units like nanometers.

This calculation is crucial for spectroscopy and other techniques used to analyze the electrical structure of molecules and materials.

Conversion Factor

A conversion factor is a numerical factor used to express the same quantity in different units, facilitating calculations and comparisons. Conversion factors are of the utmost importance when working with units that are not immediately compatible. In our exercise, we have to covert from the unit 'joules per mole' to 'nanometers' for wavelength.

The most common conversion factors related to energy and wavelength include:

• 1 kilojoule (kJ) = 1000 joules (J)
• 1 meter (m) = \(10^9\) nanometers (nm)

Applying these conversions properly is key to arriving at the correct answer, as seen in our exercise, where the initial energy was given in kilojoules per mole and the final wavelength had to be stated in nanometers. The proper use and understanding of conversion factors help ensure accuracy and comprehensibility in scientific and mathematical calculations.