Problem 33 - Chemistry: The Central Science

Question

(a) To the closest \(100 \mathrm{~nm}\), what are the largest and smallest wavelengths of visible light? (b) What is meant by the term complementary color? (c) What is the significance of complementary colors in understanding the colors of metal complexes? (d) If a complex absorbs light at \(610 \mathrm{~nm}\), what is the energy of this absorption in \(\mathrm{kJ} / \mathrm{mol}\) ?

Step-by-Step Solution

Answer

(a) The largest and smallest wavelengths of visible light are approximately 700 nm and 400 nm, respectively. (b) Complementary colors are pairs of colors that, when combined, produce a neutral color and are found oppositely across from one another in the color wheel. (c) In metal complexes, complementary colors help us deduce information about the complex's structure and electron energy levels based on the color it appears. (d) The energy of absorption for a complex at 610 nm is approximately 196 kJ/mol.

Step 1: a) Largest and smallest wavelengths of visible light

The visible light spectrum ranges from approximately 400 nm to 700 nm. The largest wavelength (closest to 700 nm) would therefore be 700 nm, and the smallest wavelength (closest to 400 nm) would be 400 nm.

Step 2: b) Definition of complementary colors

Complementary colors are pairs of colors that, when combined, cancel each other out and produce a neutral color (usually white, black, or gray). In the color wheel, complementary colors are found oppositely across from one another. These colors create a strong visual contrast when placed side by side.

Step 3: c) Significance of complementary colors in metal complexes

In the case of metal complexes, complementary colors are significant because the color observed is the result of the absorbed wavelength's complementary color. When a metal complex absorbs a certain wavelength of light, the energy of that light is used to promote electrons to a higher energy level. The remaining light is transmitted and appears as the complementary color of the absorbed wavelength. By understanding this relationship in metal complexes, we can deduce information about the complex's structure and the energy levels of its electrons based on the color it appears to the human eye.

Step 4: d) Energy of absorption at 610 nm

To find the energy of absorption, we can use the equation that relates energy (E) and wavelength (λ) for electromagnetic radiation: \[E = \frac{hc}{\lambda}\] where h is the Planck's constant (6.626 x 10^{-34} Js), c is the speed of light (2.998 x 10^8 m/s), and λ is the wavelength. First, we need to convert the wavelength from nm to meters: \[610 \mathrm{~nm} = 610 \times 10^{-9} \mathrm{~m}\] Now we can plug in the values into the equation: \[E = \frac{(6.626 \times 10^{-34} \mathrm{Js})(2.998 \times 10^8 \mathrm{m/s})}{(610 \times 10^{-9} \mathrm{m})}\] \[E = 3.260 \times 10^{-19} \mathrm{J}\] Finally, we need to convert this energy to kJ/mol. We know that 1 mol has Avogadro's number of photons (6.022 x 10^{23} photons/mol): \[3.260 \times 10^{-19} \mathrm{J/photon} \times \frac{6.022 \times 10^{23} \mathrm{photons}}{1 \mathrm{mol}} \times \frac{1 \mathrm{kJ}}{1000 \mathrm{J}} = 196 \mathrm{kJ/mol}\] Thus, the energy of absorption at 610 nm is approximately 196 kJ/mol.

Key Concepts

Complementary Colors

Complementary colors are pairs of hues situated directly opposite each other on a color wheel. When these pairs are mixed or placed next to each other, they cancel each other out by producing a grayscale color like white or black. For example, red and green are complementary, as are blue and orange.

In visual arts, complementary colors are utilized to create vibrant contrasts, making images stand out. This principle is not only artistic but also has practical applications in technology, like color printing, where colors are often layered to create the desired hue and shading.

Understanding complementary colors helps us better interpret the world around us, from art to nature. The bright colors of flowers, for instance, are often complemented by their surroundings, making them more visually appealing to pollinators.

Metal Complexes Color

The vibrant colors of metal complexes are a fascinating manifestation of the underlying physics and chemistry. A metal complex consists of a central metal ion surrounded by molecules or ions known as ligands. The color we perceive is a result of the d-electrons of the metal ion absorbing certain wavelengths of light when they transition between different energy levels within the metal ion's electron configuration.

The wavelengths that are not absorbed are reflected or transmitted, and the complex appears to be the color that complements the color of the absorbed light. This concept is crucial for chemists who decipher the structure of complexes and predict their behaviors based on the colors they exhibit.

For example, a complex that appears green does so because it absorbs light in the red portion of the spectrum. In this way, the study of the colors of metal complexes, known as coordination chemistry, intersects with physics and art, creating a bridge between the disciplines.

Absorption Energy Calculation

The energy of light absorbed by a material can be calculated using the relationship between energy (E), the speed of light (c), and the wavelength of light (λ), expressed in the formula \[E = \frac{hc}{\lambda}\]where 'h' is Planck's constant and 'c' is the speed of light in a vacuum. To find the energy absorbed by a complex at a specific wavelength, one must first convert that wavelength into meters from nanometers since the constants 'h' and 'c' are in SI units.

After obtaining the energy in joules per photon, we multiply by Avogadro's number to get the energy per mole of photons. The final step involves converting joules to kilojoules if necessary. This calculation is essential in quantum chemistry and spectroscopy, for it allows us to link a wavelength of light to the precise amount of energy required to invoke electronic transitions in molecules, and in this scenario, metal complexes.