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 is expected to be blue. (b) The energy of a photon with a wavelength of 610 nm is \(3.260\times10^{-19} \, \mathrm{J}\). (c) The energy of this absorption in kJ/mol is 196.3 kJ/mol.

Step-by-step solution

Finding the color of the complex

To find the color of the complex, we must first determine which color is complementary to the color it absorbs. Visible light has a wavelength range of about 380 nm to 750 nm. We can use the following approximate ranges for colors: - Violet: 380 - 450 nm - Blue: 450 - 495 nm - Green: 495 - 570 nm - Yellow: 570 - 590 nm - Orange: 590 - 620 nm - Red: 620 - 750 nm Since the complex absorbs light at 610 nm, it falls within the orange color range. This means that the complex will likely appear blue because blue is the complementary color to orange.

Calculate the energy of a single photon with a wavelength of 610 nm in Joules

To calculate the energy of a photon, we can use Planck's equation: \( E = h\cdot\nu \), where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.626\times10^{-34} \, \mathrm{Js}\)), and \(\nu\) is the frequency of light. Note that we are given the wavelength of light (\(\lambda\)), rather than its frequency. We can convert wavelength to frequency using the speed of light (\(c\)) and the equation: \( \nu = \dfrac{c}{\lambda} \). First, we need to convert the wavelength to meters. Since 1 nm = \(1\times10^{-9} \, \mathrm{m}\), 610 nm = 610×\(1\times10^{-9} \, \mathrm{m}\) = \(6.10\times10^{-7} \, \mathrm{m}\). Next, we can find the frequency: \( \nu = \dfrac{c}{\lambda} = \dfrac{3\times10^{8} \, \mathrm{m/s}}{6.10\times10^{-7} \, \mathrm{m}} = 4.918\times10^{14} \, \mathrm{s^{-1}} \). Finally, we can plug the frequency into Planck's equation to find the energy of a photon: \( E = h\cdot\nu = (6.626\times10^{-34} \, \mathrm{Js})(4.918\times10^{14} \, \mathrm{s^{-1}}) = 3.260\times10^{-19} \, \mathrm{J} \).

Calculate the energy of this absorption in kJ/mol

To find the energy per mole, we need to multiply the energy of a single photon by Avogadro's number (\(6.022\times10^{23}\, \mathrm{mol^{-1}}\)). Then, we need to convert the energy from Joules to kilojoules. \( E_\mathrm{mol} = E \cdot N_A = (3.260\times10^{-19} \, \mathrm{J})(6.022\times10^{23}\, \mathrm{mol^{-1}}) = 1.9630\times10^{5} \, \mathrm{J/mol} \) Now, convert to kJ/mol: \( \dfrac{1.9630\times10^{5} \, \mathrm{J/mol}}{1\times10^{3} \, \mathrm{J/kJ}} = 196.3 \, \mathrm{kJ/mol} \). To summarize our results: (a) The complex is expected to be blue. (b) The energy of a photon with a wavelength of 610 nm is \(3.260\times10^{-19} \, \mathrm{J}\). (c) The energy of this absorption in kJ/mol is 196.3 kJ/mol.

Key concepts

Planck's Equation

Understanding the energy of a photon is crucial in both physics and chemistry. The energy of a photon can be calculated using Planck's equation, which is a fundamental relation in quantum mechanics.

Planck's equation is expressed as: \( E = h \cdot u \), where \( E \) is the energy, \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)), and \( u \) is the frequency of the photon. To find this frequency, one can use the speed of light, \( c \), and the wavelength, \( \lambda \), using the formula: \( u = \dfrac{c}{\lambda} \). Consequently, by knowing the wavelength of the emitted or absorbed light, we can calculate the photon's energy.

This concept is particularly important in the study of light-matter interactions such as the absorption of light by a chemical complex, as demonstrated in the provided exercise. When photons are absorbed, they can bring about various effects, including electronic transitions in molecules that relate directly to the color observed.

Complementary Colors in Chemistry

When a substance absorbs light within the visible spectrum, it generally appears as the complementary color to the one absorbed. This concept comes into play in chemistry, particularly in the study of transition metal complexes, which can exhibit vibrant colors.

The color wheel is often used to identify complementary colors. Colors opposite each other on this wheel are considered complementary. For instance, when a complex absorbs orange light (around 610 nm), it generally appears blue, as seen in this exercise.

Importance of Complementary Colors

• Identification of unknown substances by color comparison.
• Understanding the electronic transitions that occur during absorption.
• Applications in industries such as dyes and pigments.

Avogadro's Number

In chemistry, Avogadro's number, symbolized as \( N_A \), is a constant that indicates the number of constituent particles, commonly atoms or molecules, in one mole of a substance. The currently accepted value of Avogadro's number is \( 6.022 \times 10^{23} \) entities per mole.

This value allows chemists and physicists to count particles by weighing them. Avogadro's number is pivotal when converting from microscopic to macroscopic quantities, and vice versa. For example, when the energy of a single photon is calculated, as with Planck's equation, it often must be multiplied by Avogadro's number to obtain energy per mole for the substance involved, which relates to the amount of substance in practical, measurable terms.