Question

A sunburn is caused primarily by sunlight in what is known as the UVB band, or the wavelength range from 290 to \(320 \mathrm{nm} .\) The minimum dose of radiation needed to create a sunburn (erythema) is known as a MED (minimum erythema dose \() .\) The MED for a person of average resistance to burning is \(50.0 \mathrm{mJ} \mathrm{cm}^{-2}\) a. Determine the number of \(290 .\) nm photons corresponding to the MED, assuming each photon is absorbed. Repeat this calculation for \(320 .\) nm photons. b. \(\mathrm{At} 20^{\circ}\) latitude, the solar flux in the UVB band at the surface of the earth is \(1.45 \mathrm{mW} \mathrm{cm}^{-2} .\) Assuming that each photon is absorbed, how long would a person with unprotected skin be able to stand in the sun before acquiring one MED?

Short answer

Answer: The formula to calculate the energy of a photon is E = h×c/λ, where E is the energy of the photon, h is the Planck constant, c is the speed of light, and λ is the wavelength of the photon. 2. What are the units for the minimum erythema dose (MED) before and after conversion? Answer: Before conversion, the units for MED are mJ/cm². After conversion, the units are J/m². 3. How do you obtain the time required for a person to acquire one MED? Answer: To obtain the time required for a person to acquire one MED, divide the MED by the solar flux: Time = MED / Solar flux.

Step-by-step solution

Find the energy per photon for both wavelengths

To calculate the energy of a photon, we will use the equation E = h×c/λ, where E is the energy of the photon, h is the Planck constant (\(6.63 \times 10^{-34} \mathrm{Js}\)), c is the speed of light (\(3.00 \times 10^8 \mathrm{m/s}\)), and λ is the wavelength of the photon. For 290 nm: \(E_{290} = \frac{(6.63 \times 10^{-34} \mathrm{Js})(3.00 \times 10^8 \mathrm{m/s})}{290 \times 10^{-9} \mathrm{m}}\) For 320 nm: \(E_{320} = \frac{(6.63 \times 10^{-34} \mathrm{Js})(3.00 \times 10^8 \mathrm{m/s})}{320 \times 10^{-9} \mathrm{m}}\)

Calculate the number of required photons

The MED is given in mJ/cm², so we first need to convert it to joules per square meter (J/m²) to be consistent with the photon energy calculation. The conversion factor is: 1 mJ = \(10^{-3} \mathrm{J}\) and 1 cm² = \(10^{-4} \mathrm{m}^2.\) MED = \(50.0 \mathrm{mJ/cm^{2}} = 50.0 \times 10^{-3} \mathrm{J} \times \frac{1 \mathrm{m^2}}{10^{-4} \mathrm{cm^{2}}} = 5.0 \times 10^3 \mathrm{J/m^{2}}\) Now, divide the MED by the energy per photon for both wavelengths to find the number of photons: For 290 nm, \(N_{290}=\frac{5.0 \times 10^{3} \mathrm{J/m^2}}{E_{290}}\) For 320 nm, \(N_{320}=\frac{5.0 \times 10^{3} \mathrm{J/m^2}}{E_{320}}\) #b. Calculate the time to acquire one MED#

Convert solar flux into energy per square meter per second

The solar flux is given in mW/cm². To convert it to energy per square meter per second (J/m²s), use the following conversion factors: 1 mW = \(10^{-3} \mathrm{W}\), and 1 W = 1 J/s. Solar flux = \(1.45 \mathrm{mW/cm^{2}} = 1.45 \times 10^{-3} \mathrm{W} \times \frac{1 \mathrm{m^2}}{10^{-4} \mathrm{cm^{2}}} = 1.45 \times 10^{1} \mathrm{W/m^{2}} = 1.45 \times 10^{1} \mathrm{J/m^{2}s}\)

Divide the MED by the solar flux

To find the time required for a person to acquire one MED, divide the MED by the solar flux: Time = \(\frac{5.0 \times 10^3 \mathrm{J/m^2}}{1.45 \times 10^{1} \mathrm{J/m^2s}}\) Calculate the time and express it in the desired units (e.g., minutes).

Key concepts

Minimum Erythema Dose

Understanding the concept of minimum erythema dose (MED) is crucial when discussing the impact of ultraviolet radiation on the skin. MED is the smallest amount of UVB radiation that results in a noticeable reddening of the skin, known as sunburn, in a particular individual after a single exposure. This measurement is significant because it helps to quantify the ability of UV radiation to cause skin damage, serving as a gauge for sun protection strategies.

For example, if a person of average sensitivity to UVB radiation has an MED of 50.0 mJ/cm², it means that this amount of energy per square centimeter is needed to initiate sunburn. Factors like skin type, age, and the presence of protective agents like sunscreen can influence a person's MED, rendering it an important factor in the study and management of skin health related to sun exposure. In the given task, understanding MED assists in calculating the number of photons necessary to cause erythema, thereby helping to highlight the potency of UVB photons in causing skin damage.

Photon Energy Calculation

The process of calculating photon energy is fundamental in understanding how electromagnetic radiation interacts with matter. The energy of a photon can be determined using the equation \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of the single photon, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the photon's wavelength.

In the wavelength range responsible for sunburns (290-320 nm), it is insightful to look at how energy varies with wavelength. By performing these calculations for 290 nm and 320 nm photons, one can see that shorter wavelengths carry more energy per photon. This concept is key to understanding why UVB radiation is more harmful compared to UVA radiation, which has longer wavelengths and, therefore, less energy per photon. Through photon energy calculation, the precise number of these high-energy photons required to reach the MED and cause skin damage can be estimated.

Solar UV Exposure

The amount of time it takes for an individual to experience sunburn is intimately connected to the concept of solar UV exposure. This is defined as the rate at which solar UVB radiation reaches the Earth's surface at a given time and location. Factors such as geographical position, time of day, and atmospheric conditions can all affect the UV exposure level.

Understanding solar UV exposure helps individuals manage their time in the sun wisely to prevent skin damage. In the task, the solar flux, which is a measure of the power per unit area received from the sun, was used to determine how long a person can stay under direct sunlight before reaching the MED. With a solar flux of 1.45 mW/cm², it is possible to calculate the duration after which UVB radiation reaches harmful levels for the skin. As an educational concept, solar UV exposure is a crucial element in fields like dermatology, environmental science, and is also imperative for public health awareness concerning sun-related risks.