Chapter 7: Problem 111

The UV light that is responsible for tanning the skin falls in the 320 - to 400 -nm region. Calculate the total energy (in joules) absorbed by a person exposed to this radiation for \(2.0 \mathrm{~h}\), given that there are \(2.0 \times\) \(10^{16}\) photons hitting Earth's surface per square centimeter per second over a \(80-\mathrm{nm}(320 \mathrm{nm}\) to \(400 \mathrm{nm})\) range and that the exposed body area is \(0.45 \mathrm{~m}^{2}\). Assume that only half of the radiation is absorbed and the other half is reflected by the body. (Hint: Use an average wavelength of \(360 \mathrm{nm}\) in calculating the energy of a photon.)

Short Answer

Expert verified

The total energy absorbed by the person over a 2-hour period is \(4.47\) Joules.

Step by step solution

Convert Units

Before starting calculations, it's important to convert all units to SI units for coherence. Here, the intensity of light given is per square centimeter, while the exposed body area is given in square meters. There are \(10,000\) square centimeters in a square meter, so the number of photons hitting one square meter per second is: \(2.0 \times 10^{16} \, \text{photons/cm}^2/\text{sec} \times 10,000 \, \text{cm}^2/\text{m}^2 = 2.0 \times 10^{20} \, \text{photons/m}^2/\text{sec}\).

Calculate Total Number of Photons

First, it's needed to calculate the total number of photons absorbed over the \(2.0\) hour period by the exposed body area. Note that, according to the problem, only half of the photons are absorbed. The total number of absorbed photons will be: \((2.0 \times 10^{20} \, \text{photons/m}^2/\text{sec}) \times (0.5) \times (0.45 \, \text{m}^2) \times \(2.0 \times 3600\, sec) = 8.1 \times 10^{24} \, \text{photons}\). Here, \(2\) hours has been converted to seconds as \(2 \times 60 \times 60 = 3600\) seconds.

Compute Energy per Photon

The energy per photon can be calculated using the Planck's equation, \(E = \frac{hc}{λ}\), where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{J} \cdot \text{s}\)), \(c\) is the speed of light (\(3.0 \times 10^8 \, \text{m/s}\)), and \(λ\) is the wavelength. Assuming an average wavelength of 360 nm or \(360 \times 10^{-9} \, \text{m}\), the energy per photon is: \(E = \frac{(6.626 \times 10^{-34} \text{J} \cdot \text{s}) \times (3.0 \times 10^8 \, \text{m/s})}{360 \times 10^{-9} \, \text{m}} = 5.52 \times 10^{-19} \, \text{J}\).

Calculate Total Energy Absorbed

Finally, multiply the total number of absorbed photons by the energy per photon to get the absorbed energy: \((8.1 \times 10^{24} \, \text{photons}) \times (5.52 \times 10^{-19} \, \text{J/photon}) = 4.47 \, \text{J}\).

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Short Answer

Step by step solution

Convert Units

Calculate Total Number of Photons

Compute Energy per Photon

Calculate Total Energy Absorbed

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