Question

(a) If a complex absorbs light at \(610 \mathrm{nm},\) what color would you expect the complex to be? (b) What is the energy in joules of a photon with a wavelength of \(610 \mathrm{nm}\) ? (c) What is the energy of this absorption in \(\mathrm{kJ} / \mathrm{mol} ?\)

Short answer

(a) The complex will appear cyan because it absorbs light at 610 nm, which is in the red wavelength range. (b) The energy of a photon with a wavelength of 610 nm is \(3.26 \times 10^{-19} \text{J}\). (c) The energy of this absorption in kilojoules per mole is 196.4 kJ/mol.

Step-by-step solution

1. Determine the color based on the absorbed wavelength

**To determine the color we would expect the complex to be, we need to understand the color wheel and the complementary color theory. When a complex absorbs light at a specific wavelength (color), it reflects the complementary color, which is what we perceive as its color. The complementary colors are as follows: - Red (650-750 nm) ↔ Cyan - Green (495-570 nm) ↔ Magenta - Blue (450-495 nm) ↔ Yellow - Yellow (570-590 nm) ↔ Blue - Magenta (380-450 nm) ↔ Green - Cyan (500-520 nm) ↔ Red Since the complex absorbs light at 610 nm, which falls within the red wavelength range, its complementary color is cyan. So, the complex will appear cyan. **

2. Calculate the energy of a photon with a wavelength of 610 nm

**To calculate the energy of a photon with a wavelength of 610 nm, we will use the Planck's equation: \( E = hf \) where E is the energy, h is the Planck's constant (6.626 x 10^{-34} Js), and f is frequency. However, we are given the wavelength (λ), not frequency, so we need to use the speed of light equation: \(c = f\cdot \lambda\) where c is the speed of light (3.00 x 10^8 m/s) and λ is the wavelength. First, we need to convert the wavelength from nanometers to meters: \(610 \ \text{nm} = 610 \times 10^{-9} \ \text{m}\) Now, we calculate the frequency: \(f = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \ \text{m/s}}{610 \times 10^{-9} \ \text{m}} = 4.92 \times 10^{14} \ \text{Hz}\) Finally, use the Planck's equation to calculate the energy of a photon: \(E = (6.626 \times 10^{-34} \ \text{Js})\times(4.92 \times 10^{14}\ \text{Hz}) = 3.26 \times 10^{-19} \text{J}\) So, the energy of a photon with a wavelength of 610 nm is \(3.26 \times 10^{-19} \text{J}\). **

3. Convert the energy to kilojoules per mole

**To convert the energy to kilojoules per mole, we will use Avogadro's number (6.022 x 10^{23} mol^{-1}): \( E_{\text{mol}} = 3.26 \times 10^{-19} \text{J} \times \frac{6.022 \times 10^{23}\ \text{mol}^{-1}}{1\ \text{mol}} \times \frac{1\ \text{kJ}}{10^3 \text{J}} \) \(E_{\text{mol}} = 196.4\ \text{kJ/mol}\) So, the energy of this absorption in kilojoules per mole is 196.4 kJ/mol.

Key concepts

Wavelength and Color

When a chemical complex absorbs light at a certain wavelength, it reflects a specific color that our eyes perceive. This concept is crucial in understanding the visible spectrum of light and how color perception works. The visible spectrum ranges from approximately 380 nm to 750 nm and includes colors like violet, blue, green, yellow, orange, and red.

Violet: Around 380-450 nm

Blue: Approximately 450-495 nm

Green: About 495-570 nm

Yellow: Around 570-590 nm

Orange: Approximately 590-620 nm

Red: About 620-750 nm

Color perception is determined by which wavelengths are absorbed and which are reflected. If a compound absorbs a specific wavelength, the color we see is the complementary color to the one absorbed.

Photon Energy Calculation

The energy of a photon is directly related to its frequency or inversely to its wavelength. This progression is expressed through the formula:\[ E = h\cdot f \]
where \(E\) is the energy of the photon, \(h\) is Planck's constant \( (6.626 \times 10^{-34}\ \text{Js}) \), and \(f\) is the frequency of the light.
However, it's often practical to use a related formula utilizing the wavelength \(\lambda\) of the light as we often measure light in terms of wavelength:\[ f = \frac{c}{\lambda} \]
In this equation, \(c\) is the speed of light \( (3.00 \times 10^8\ \text{m/s}) \) and \(\lambda\) is the wavelength. Once you determine \(f\), substitute it back into Planck's equation to find the energy.

Planck's Equation

Planck's Equation is fundamental for calculating the energy associated with a photon. It highlights the relationship between energy and frequency, and by extension, wavelength.For photons, the equation is:\[ E = h\cdot f \]
This formula revolutionized physics, helping us understand how energy is quantized.
This quantization indicates that energy is only absorbed or emitted in discrete units or "quanta," making Planck's Equation central to quantum mechanics.
It forms the basis for understanding interactions at the molecular and atomic levels, particularly in quantum chemistry, where the behavior of electrons and light is often evaluated.

Avogadro's Number

Avogadro's Number is a constant \( (6.022 \times 10^{23}\ \text{mol}^{-1}) \) used to describe the number of atoms, molecules, or particles in one mole of a substance. This concept is vital in bridging the gap between the microscopic world of atoms and the macroscopic measurements we observe.
For instance, when calculating the energy per mole of photons, we multiply the energy of one photon by Avogadro's Number. This way, we obtain a value that is more meaningful on a macroscopic level while considering the energy involved in chemical reactions and light absorption.
By knowing this constant, chemists can predict quantities and reactions at the macroscopic scale with precision.

Complementary Colors

In the world of chemistry and physics, complementary colors play a significant role in understanding how substances interact with light. Each color of light has a complementary color positioned opposite to it on the color wheel.
When a solution absorbs one color, the complementary color is the one that our eyes perceive.
Here's a simple pairing:

Red and Cyan

Green and Magenta

Blue and Yellow

If a chemical absorbs light at 610 nm, which lies in the red region, its complementary color cyan is what is reflected and seen.
This principle is widely applied in analyzing the absorption spectra of compounds, helping scientists to deduce the concentration and properties of a substance based on the color exhibited.