Question

Excessive exposure to sunlight increases the risk of skin cancer because some of the photons have enough energy to break chemical bonds in biological molecules. These bonds require approximately \(250-800 \mathrm{~kJ} / \mathrm{mol}\) of energy to break. The energy of a single photon is given by \(E=h c / \lambda\), where \(E\) is the energy of the photon in \(\mathrm{J}, h\) is Planck's constant \(\left(6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)\), and \(c\) is the speed of light \(\left(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\). Determine which kinds of light contain enough energy to break chemical bonds in biological molecules by calculating the total energy in \(1 \mathrm{~mol}\) of photons for light of each wavelength. (a) infrared light \((1500 \mathrm{~nm})\) (b) visible light ( \(500 \mathrm{~nm}\) ) (c) ultraviolet light ( \(150 \mathrm{~nm}\) )

Short answer

Using the energy formula \(E = \frac{hc}{\lambda}\), calculate the energy for a photon at each wavelength, then multiply by Avogadro's number to find the energy per mole. Wavelengths of light that provide an energy per mole in the 250-800 kJ range can break chemical bonds. Typically, ultraviolet light meets this criterion.

Step-by-step solution

Understand the Energy Requirement to Break Chemical Bonds

The energy required to break the chemical bonds in biological molecules is between 250 and 800 kJ/mol. A photon must have energy within this range to potentially cause skin cancer.

Convert kJ/mol to J/photon for Comparison

Since the energy of photons is given in J, we need to convert the bond energy from kJ/mol to J/photon. We know that 1 mol contains Avogadro's number of particles, which is approximately \(6.022 \times 10^{23}\) particles/mol. To find the energy in J/photon, we divide the energy in kJ/mol by Avogadro's number, and then convert kJ to J by multiplying by \(10^3\).

Calculate the Energy of a Single Photon

Use the formula \(E = \frac{hc}{\lambda}\) to calculate the energy of a single photon for each wavelength given. Replace \(h\) with Planck's constant \(6.626 \times 10^{-34} \text{J} \cdot \text{s}\), \(c\) with the speed of light \(3.00 \times 10^{8} \text{m}/\text{s}\), and \(\lambda\) with the respective wavelength in meters (convert nm to m by dividing by \(10^9\)).

Calculate the Energy of 1 mol of Photons

To find the energy of 1 mol of photons, multiply the energy of a single photon by Avogadro's number.

Determine If the Wavelength Can Break Bonds

Compare the energy per mol of photons calculated in Step 4 with the energy required to break chemical bonds (250-800 kJ/mol). If the energy per mol of photons falls within this range, that type of light contains enough energy to break bonds and may contribute to the risk of skin cancer.

Calculate for Infrared Light \((1500 \text{nm})\)

Convert the wavelength to meters: \(1500 \text{nm} = 1500 \times 10^{-9} \text{m}\). Then calculate the energy of a single photon using the equation from Step 3. Finally, calculate the energy of 1 mol of photons as described in Step 4.

Calculate for Visible Light \((500 \text{nm})\)

Convert the wavelength to meters: \(500 \text{nm} = 500 \times 10^{-9} \text{m}\). Then calculate the energy of a single photon using the equation from Step 3. Finally, calculate the energy of 1 mol of photons as described in Step 4.

Calculate for Ultraviolet Light \((150 \text{nm})\)

Convert the wavelength to meters: \(150 \text{nm} = 150 \times 10^{-9} \text{m}\). Then calculate the energy of a single photon using the equation from Step 3. Finally, calculate the energy of 1 mol of photons as described in Step 4. Compare this value to the bond energy range.

Key concepts

Planck's constant application

Understanding the application of Planck's constant, denoted as \(h\), is essential for explaining a myriad of physical phenomena, including the quantization of energy in photons. According to quantum theory, electromagnetic energy is not continuous, but rather quantized, existing in discrete packets called photons.

The energy \(E\) of a photon can be calculated using the very equation at the heart of quantum mechanics, \(E = \frac{hc}{\lambda}\), where \(c\) represents the speed of light, and \(\lambda\) is the photon's wavelength. This foundational equation connects Planck's constant to the quantum behavior of light, where shorter wavelengths correspond to photons with higher energy. In the context of biological damage, the energy transferred by a photon to a chemical bond can cause the bond to break if it is within a specific energy range (250-800 kJ/mol).

By applying Planck's constant, students can determine whether light of a given wavelength has sufficient energy to pose a risks, such as the potential to initiate skin cancer through bond dissociation. This illustration not only cements your understanding of the quantum description of light but also interlinks the microcosmic quantum events to macroscopic biological consequences.

Chemical bonds and photon interactions

The interaction between chemical bonds and photons is a fundamental aspect of photochemistry, which has implications in both health and technology. Chemical bonds are the glue that holds atoms together in molecules, and they possess certain discrete energy levels required for their formation or breakage.

When a photon with energy corresponding to, or exceeding, the energy of a specific bond interacts with it, the bond may absorb the photon and break - a process known as photodissociation. In biological molecules, this absorption can lead to molecular damage and initiate processes such as skin cancer development.

Energy Range for Bond Breaking

For a molecule in the cell, the photodissociation is usually in the range of 250 to 800 kJ/mol. Visible to ultraviolet wavelengths of light typically contains photons within this energy range. However, lower energy photons, such as those from infrared light, might not have sufficient energy to cause such chemical changes. Estimates of photon energies using Planck's constant enable us to predict which types of light can induce molecular damage and better understand how to protect against harmful exposures.

Avogadro's number in chemistry

Avogadro's number, approximately \(6.022 \times 10^{23}\) entities per mole, is a pillar of stoichiometry in chemistry. This number allows chemists to bridge the gap between the microscopic world of atoms and molecules and the macroscopic world we observe.

When calculating energies involved in photonic interactions, Avogadro's number transforms our understanding from individual photon energy to the molar scale used for substances in bulk. This is crucial for relating the energy of singular quantum events like photon emissions to amounts in everyday chemical reactions and biological processes.

From Photons to Moles

In the exercise at hand, by multiplying the energy of a single photon by Avogadro's number, we are able to calculate the energy contained in one mole of photons and compare this to the energies required to break chemical bonds. Effectively, this links the micro scale of quantum physics with the macro scale of chemical reaction energies, allowing us to make sense of how light influences matter in a tangible, measurable way.