Question

One type of sunburn occurs on exposure to UV light of wavelength in the vicinity of 325 \(\mathrm{nm} .\) (a) What is the energy of a photon of this wavelength? (b) What is the energy of a mole of these photons? (c) How many photons are in a 1.00 \(\mathrm{mJ}\) burst of this radiation? (d) These UV photons can break chemical bonds in your skin to cause sunburn-a form of radiation damage. If the 325 -nm radiation provides exactly the energy to break an average chemical bond in the skin, estimate the average energy of these bonds in kJ/mol.

Short answer

(a) The energy of a photon with a wavelength of 325 nm is approximately \(3.84 \times 10^{-19}\, \text{J}\) or \(2.4\, \text{eV}\). (b) The energy of a mole of these photons is approximately \(231\, \text{kJ/mol}\). (c) There are approximately \(2.6 \times 10^{18}\) photons in a 1.00 mJ burst of this radiation. (d) The average energy of chemical bonds in the skin is estimated to be \(231\, \text{kJ/mol}\).

Step-by-step solution

(Identify given information and unknowns)

We are given: 1. Wavelength (\(\lambda\)) of the UV light: 325 nm 2. Energy of the radiation burst: 1.00 mJ We need to find: (a) The energy of a photon of the given wavelength. (b) The energy of a mole of these photons. (c) The number of photons in a 1.00 mJ burst of this radiation. (d) The average energy of chemical bonds in the skin in kJ/mol, assuming that the 325-nm radiation provides exactly the energy to break these bonds.

(Step 1: Calculate the energy of photon)

We can find the energy of a photon using Planck's equation: \(E = \frac{hc}{\lambda}\) Where: \(E\) = Energy of the photon, \(\lambda\) = Wavelength of light, \(h\) = Planck's constant (\(6.63 \times 10^{-34}\, \text{Js}\)), \(c\) = Speed of light (\(3.0 \times 10^8\, \text{m/s}\)). Given, \(\lambda = 325\, \text{nm} = 325 \times 10^{-9}\, \text{m}\) Now, calculating the energy: \(E = \frac{6.63 \times 10^{-34}\, \text{Js} \times 3.0 \times 10^8\, \text{m/s}}{325 \times 10^{-9}\, \text{m}}\)

(Step 2: Convert energy of photon to eV)

The energy of the photon obtained in Step 1 is in Joules. To express it in electronvolts (eV), we can use the conversion factor of \(1\, \text{eV} = 1.6 \times 10^{-19}\, \text{J}\): \(E\,(\text{eV}) = \frac{E\,(\text{J})}{1.6 \times 10^{-19}\, \text{J/eV}}\)

(Step 3: Calculate the energy of a mole of photons)

To find the energy of a mole of photons, we can multiply the energy of a single photon by Avogadro's number (\(N_A = 6.022 \times 10^{23}\, \text{mol}^{-1}\)): \(E_{\text{mole}} = E_{\text{photon}} \times N_A\)

(Step 4: Calculate the number of photons in the 1.00 mJ energy burst)

We can find the number of photons in the 1.00 mJ energy burst by dividing the energy of the burst by the energy of a single photon: \(N = \frac{E_{\text{burst}}}{E_{\text{photon}}}\) Given, \(E_{\text{burst}} = 1.00\, \text{mJ} = 1.00 \times 10^{-3}\, \text{J}\)

(Step 5: Calculate the average energy of skin chemical bonds in kJ/mol)

Since the 325 nm radiation provides exactly the energy to break an average chemical bond in the skin, we can assume the energy of one photon is equal to the energy needed to break one chemical bond. To find the energy in kJ/mol, we can multiply the energy of one photon (in J) by Avogadro's number and convert it to kJ: \(E_{\text{bond}}\, (\text{kJ/mol}) = E_{\text{photon}}\, (\text{J}) \times N_A \times \frac{1\, \text{kJ}}{1000\, \text{J}}\)

Key concepts

Energy of a Photon

When we talk about the energy of a photon, we refer to the quantum of electromagnetic energy that is carried by a single particle of light, known as a photon. This energy is crucial for understanding phenomena such as UV radiation's effect on chemical bonds, which can lead to sunburn.

According to quantum theory, the energy associated with a photon is directly proportional to its frequency and inversely proportional to its wavelength. This relationship means that photons with short wavelengths, like those in UV light, have higher energies and therefore, the potential to cause more damage to materials they encounter, such as the DNA in our skin cells resulting in sunburns.

In the context of the given problem, calculating the energy of a photon at a specific wavelength allows us to determine the threshold of energy required to break chemical bonds within our skin. This is a fundamental concept in photochemistry and other scientific fields that examine the effects of light on substances.

Planck's Equation

Planck's equation is a fundamental principle in quantum mechanics that provides a way to calculate the energy of a photon. The equation is denoted as \(E = \frac{hc}{\text{\(lambda\)}}\), where \(E\) represents the energy of the photon, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\text{\(lambda\)}}\) is the wavelength of the photon.

Understanding Planck's Constant

Planck's constant, represented by \(h\), is a unique value in physics that connects the frequency of light with its energy. Its value, \(6.63\times10^{-34}\text{Js}\), is a critical factor in our exercise as it is used in computations to find the energy of a photon.

When solving the exercise, applying Planck's equation to the given wavelength of UV light allows us to determine the energy of a single photon of that wavelength. This energy is then used to infer the impact of UV light on skin cells, emphasizing the importance of understanding this equation for evaluating photonic interactions with matter.

Avogadro's Number

Avogadro's number is a constant that represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. This number is remarkably large, \(6.022 \times 10^{23} \text{mol}^{-1}\), signifying the vast number of atoms or molecules present even in seemingly minuscule quantities of material.

In chemistry and physics, Avogadro's number enables us to bridge the gap between the microcosmic world of atoms and molecules and the macroscopic world we interact with daily. When considering the energy of a mole of photons, for instance, we multiply the energy of a single photon by Avogadro's number to shift from the energy of an individual particle to the collective energy contained in a macroscopic quantity.

This approach is applied in the provided exercise, where the calculation using Avogadro's number allows students to grasp the scale of energy changes when transitioning from a single photon to an amount relevant in everyday experiences, such as a burst of UV radiation and its subsequent effect on the skin in terms of sunburn.