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? \((\mathbf{d})\) These \(\mathrm{UV}\) photons can break chemical bonds in your skin to cause sunburn-a form of radiation damage. If the \(325-\mathrm{nm}\) radiation provides exactly the energy to break an average chemical bond in the skin, estimate the average energy of these bonds in \(\mathrm{kJ} / \mathrm{mol}\).

Short answer

(a) The frequency of the photon is \(f = \frac{3.0 \times 10^8 m/s}{325 \times 10^{-9} m} = 9.23 \times 10^{14} Hz\). (b) The energy of a single photon is \(E = 6.63 \times 10^{-34} Js \cdot 9.23 \times 10^{14} Hz = 6.12 \times 10^{-19} J\). (c) The energy of one mole of photons is \(E_{mole} = 6.12 \times 10^{-19} J \cdot 6.022 \times 10^{23} entities/mol = 3.68 \times 10^5 J/mol\). (d) The number of photons in a 1.00 mJ burst of radiation is \(N_{photons} = \frac{1.00 \times 10^{-3} J}{6.12 \times 10^{-19} J} = 1.63 \times 10^{15} photons\). (e) The average energy of chemical bonds broken by the UV photons is \(E_{bond} = \frac{3.68 \times 10^{5} J/mol}{10^3} = 368\, kJ/mol\).

Step-by-step solution

1. Calculate the frequency of a photon with a wavelength of 325 nm

First, we need to determine the frequency of the photon with the given wavelength, 325 nm. We can use the speed of light equation for this, which is \(c = \lambda \cdot f \), where c is the speed of light (approximately \(3.0 \times 10^{8} m/s \)), \(\lambda\) is the wavelength (325 nm, but we must convert it to meters: \( 325 \times 10^{-9} m\)), and f is the frequency we want to find. By rearranging the equation, we get: \( f = \frac{c}{\lambda} = \frac{3.0 \times 10^8 m/s}{325 \times 10^{-9} m} \)

2. Calculate the energy of a single photon

Now we can use Planck's equation, \( E = h \cdot f \), to find the energy of a single photon. In this equation, E represents energy, h is Planck's constant (approximately \(6.63 \times 10^{-34} Js\)), and f is the frequency we found in step 1. Simply plug in the numbers: \( E = 6.63 \times 10^{-34} Js \cdot f \)

3. Calculate the energy of one mole of photons

Next, to find the energy of one mole of photons, we just need to multiply the energy of a single photon (found in step 2) by Avogadro's number, \( N_{A} \) (approximately \(6.022 \times 10^{23} entities/mol\)): \( E_{mole} = E \cdot N_{A} \)

4. Calculate the number of photons in a 1.00 mJ burst of radiation

In this step, we need to find out how many photons are in a 1.00 mJ burst of radiation. We can do this by dividing the total energy of the burst (1.00 mJ) by the energy of a single photon (found in step 2). First, we need to convert 1.00 mJ to Joules: \( 1.00 \, mJ = 1.00 \times 10^{-3} J \) Now, divide the total energy by the energy of a single photon: \( N_{photons} = \frac{1.00 \times 10^{-3} J}{E} \)

5. Estimate the average energy of chemical bonds in kJ/mol

Finally, to estimate the average energy of chemical bonds in the skin broken by the UV photons, we assume that each photon provides exactly the amount of energy needed to break one bond. We can use the energy of one mole of photons (found in step 3) and convert it from Joules to kJ/mol: \( E_{bond} = \frac{E_{mole}} { 10^3 } \, kJ/mol \) Now we can plug in the numbers for each step and solve the exercise.

Key concepts

UV Light Wavelength

UV light is a part of the electromagnetic spectrum with wavelengths ranging between 10 nm to 400 nm. In this exercise, we're focusing on a wavelength of 325 nm, which falls into the UV-A category, known to cause some sunburn without tanning. Understanding the wavelength helps us to calculate other properties like frequency and photon energy, which are essential for studying light interactions with matter. When a UV photon hits the skin, it may have enough energy to break chemical bonds in the skin, leading to sunburn and potential DNA damage.

Planck's Equation

Planck's equation, denoted as \( E = h \cdot f \), is a fundamental principle that connects a photon's energy to its frequency. Here, \( E \) represents energy, \( h \) is Planck's constant (\( 6.63 \times 10^{-34} \, Js \)), and \( f \) is the frequency of the photon. This equation was pivotal in establishing quantum mechanics, as it helped describe how light behaves on a microscopic scale. By knowing the wavelength of light (325 nm), we can find the frequency using the speed of light equation \( c = \lambda \cdot f \), where \( c \) is the speed of light (approximately \( 3.0 \times 10^8 \, m/s \)). Rearranging, we have \( f = \frac{c}{\lambda} \), which then lets us find the energy \( E \).

Avogadro's Number

Avogadro's Number \( N_{A} \), approximately \( 6.022 \times 10^{23} \) entities per mole, is a key concept in chemistry that allows us to translate between the macroscopic and microscopic worlds. It represents the number of atoms or molecules in one mole of a substance, providing a bridge to quantify large quantities of tiny entities like photons. In this exercise, determining the energy of one mole of photons involves multiplying the energy of a single photon by Avogadro's Number. This gives us the total energy in a more chemical-relevant context, useful for considering reactions and processes on a mole scale.

Chemical Bonds Energy

Chemical bonds hold atoms together within a molecule and have specific energy values needed to break them. In this context, the energy to break these bonds can match the energy supplied by a UV photon. To determine this, we use the energy per mole of photons calculated earlier. When you estimate the energy required to break chemical bonds in kJ/mol, you help to consider the energetic requirements of reactions and damages at a molecular level. This estimation plays a crucial role in fields like photochemistry and dermatology, where understanding and managing UV damage is essential for protecting skin health.