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 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

After calculating the energy per photon for each wavelength, one can conclude that if the energy per photon is greater than the minimum required to break a bond (the value converted from 250 kJ/mol to J/photon), the light at that wavelength has the potential to cause bond breakage.

Step-by-step solution

Convert the Given Bond Energy to Joules per Photon

First, we need to determine the energy in Joules (J) required to break the chemical bonds per photon. Since the values we are given are in kilojoules per mole (\text{kJ/mol}), we want to convert this to Joules (\text{J}) per photon. There are 1000 J in 1 kJ and Avogadro's number (\text{NA}) amount of photons in 1 mol: \(250 \text{ kJ/mol} = 250000 \text{ J/mol} \) and \(800 \text{ kJ/mol} = 800000 \text{ J/mol}\). Dividing by Avogadro's number \(6.022 \times 10^{23} \text{ photons/mol}\) gives the bond energy per photon: \(1 \text{ photon's energy} = \frac{250000 \text{ J/mol}}{6.022 \times 10^{23} \text{ photons/mol}} \) to \( \frac{800000 \text{ J/mol}}{6.022 \times 10^{23} \text{ photons/mol}}\).

Use the Energy Formula to Calculate Energy per Photon for Each Light Type

Using the photon energy formula \(E = \frac{hc}{\lambda}\), we determine the energy per photon for each type of light. Here, the constant \(h = 6.626 \times 10^{-34} \text{ J} \text{ s}\), and \(c = 3.00 \times 10^8 \text{ m/s}\).(a) For infrared light (\(\lambda = 1500 \text{ nm} = 1500 \times 10^{-9} \text{ m}\)): \(E_{ir} = \frac{(6.626 \times 10^{-34} \text{ J} \text{ s})(3.00 \times 10^8 \text{ m/s})}{1500 \times 10^{-9} \text{ m}}\).(b) For visible light (\(\lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m}\)): \(E_{vis} = \frac{(6.626 \times 10^{-34} \text{ J} \text{ s})(3.00 \times 10^8 \text{ m/s})}{500 \times 10^{-9} \text{ m}}\).(c) For ultraviolet light (\(\lambda = 150 \text{ nm} = 150 \times 10^{-9} \text{ m}\)): \(E_{uv} = \frac{(6.626 \times 10^{-34} \text{ J} \text{ s})(3.00 \times 10^8 \text{ m/s})}{150 \times 10^{-9} \text{ m}}\).

Compare Photon Energies to Bond Energies

Now compare the photon energies calculated for infrared, visible, and ultraviolet light to the converted bond energy range in J per photon calculated in Step 1.If the photon energy is within the range of 250,000 J/mol to 800,000 J/mol then it is capable of breaking the bonds.We will do this comparison for the energies calculated in Step 2 for each type of light. The bond-breaking capacity will be determined by whether the calculated photon energy exceeds the minimum required energy of 250,000 J/mol per photon.

Conclusion - Determining Which Light Can Break Bonds

With the energies computed for each type of light compared against the energy requirement to break a chemical bond, we can conclude which type(s) of light contain enough energy per photon to cause bond breakage in biological molecules.

Key concepts

Planck's constant

At the core of understanding photon energy lies Planck's constant, denoted as 'h'. This fundamental constant has value of approximately to be precise, it's exactly \(6.626 \times 10^{-34} \mathrm{J \cdot s}\). Planck's constant plays an essential role in quantum mechanics, specifically in the equation that relates the energy of a photon to its frequency, given by \(E = h u\).
Planck's constant is critical when determining the energy of photons in relation to their wavelengths. It also underscores the quantum nature of light, implying that light energy is quantized and can be emitted or absorbed in discrete units called photons. This aspect is particularly important for understanding how photons interact with chemical bonds, as only a photon with sufficient quantized energy can break a bond.
When we use Planck's constant to calculate the energy of a photon of light, we often measure the wavelength in meters and use the speed of light to convert the wavelength to frequency. This approach allows us to calculate the energy of a photon and then assess its ability to break chemical bonds, which is a fundamental concern in photochemistry and the study of light effects on biological systems.

Speed of light

The speed of light, symbolized by 'c', is a critical component in many equations across physics, including those that relate to photon energy. The speed of light is a constant at approximately \(3.00 \times 10^8 \mathrm{m/s}\) in a vacuum. This high speed is the cosmic speed limit and plays a role in relativity and the propagation of light throughout space.
For calculations involving photon energy, the speed of light allows us to relate a photon's wavelength to its frequency through the equation \(c = \lambda u\), where \(\lambda\) is the wavelength and \(u\) is the frequency.
This is crucial in calculating the energy of different types of electromagnetic radiation since we also need this value alongside Planck's constant to determine how much energy a photon carries. When considering the potential of electromagnetic radiation to cause chemical changes, like breaking chemical bonds, knowing the speed of light and how it relates to photon energy is foundational.

Avogadro's number

Avogadro's number, denoted by \(N_A\), is approximately \(6.022 \times 10^{23}\) entities per mole, and it is fundamental to chemistry as it provides a bridge between the macroscopic and microscopic worlds. It represents the number of atoms, ions, or molecules in one mole of substance and is critical when dealing with quantities in chemical reactions.
When applied to the study of photon energy and chemical bonds, Avogadro's number helps us to convert from energy per molecule to energy per mole, which is useful for understanding reactions on a scale that's more applicable to real-world situations. For instance, the exercise involves converting kJ/mol to J/photon to understand the energy per photon necessary to break chemical bonds. This conversion helps to better apprehend the magnitudes involved and to evaluate whether the energy of a photon, as dictated by its wavelength, is sufficient to disrupt the bonds in biological molecules, a concern that is especially relevant while considering the impacts of various types of light on human health.

Bond energy

Bond energy is the measure of strength of a chemical bond, defined as the amount of energy required to break one mole of bonds in a compound, measured in joules (J) or kilojoules (kJ) per mole. These values provide us an idea of how much energy is needed to disrupt the bonds and thereby induce chemical reactions.
In biological systems, bond energies are key to understanding molecular stability and reactivity under various lighting conditions. For bonds requiring approximately \(250-800 \mathrm{kJ/mol}\) of energy to break, we use equations involving the constants and principles discussed earlier to calculate the energy provided by single photons.
This information is critical in assessing the risk of electromagnetic radiation, such as ultraviolet light, where photons have enough energy to break such chemical bonds, leading to potential biological damage and implications for health, such as increased skin cancer risk due to excessive exposure to sunlight.

Ultraviolet light risks

Ultraviolet (UV) light is a form of electromagnetic radiation with wavelengths shorter than visible light but longer than X-rays. Due to its high energy, UV light poses several risks, particularly to biological organisms. The energy of UV photons is sufficient to break chemical bonds, as discussed in the preceding sections. This property makes them capable of causing damage to DNA and other critical biological molecules, leading to cellular damage and increasing the risk of conditions like skin cancer.
Understanding the energy associated with UV light, which is found in sunlight, emphasizes the importance of protective measures such as sunscreen and clothing to mitigate its risks. The potential damage of UV light is related to its photon energy, which is higher than the bond energy of certain chemical bonds in skin cells. Continuous and intense exposure to UV light can lead to the formation of harmful substances like free radicals, ultimately resulting in premature aging, immune system impact, and potentially life-threatening diseases.
By applying the principles of photon energy and calculating the specific energies related to different wavelengths of light, we can better understand the underlying reasons behind the need for sun protection and the risks associated with UV exposure.