Problem 85 - Chemistry: The Central Science in SI Units, Global Edition

Question

The rays of the Sun that cause tanning and burning are in the ultraviolet portion of the electromagnetic spectrum. These rays are categorized by wavelength. So-called UV-A radiation has wavelengths in the range of \(320-380 \mathrm{nm},\) whereas UV-B radiation has wavelengths in the range of \(290-320 \mathrm{nm}\). (a) Calculate the frequency of light that has a wavelength of \(380 \mathrm{nm} .\) (b) Calculate the energy of a mole of \(380-\mathrm{nm}\) photons. (c) Which are more energetic, photons of UV-A radiation or photons of UV-B radiation? (d) The UV-B radiation from the Sun is considered a more important cause of sunburn in humans than UV-A radiation. Is this observation consistent with your answer to part (c)?

Step-by-Step Solution

Answer

The frequency of a 380 nm light wave is \(7.89 \times 10^{14} \, Hz\). The energy of a mole of 380 nm photons is \(3.15 \times 10^5 \, J/mol\). Photons of UV-B radiation are more energetic than those of UV-A radiation, which explains why UV-B radiation is a more important cause of sunburn in humans than UV-A radiation.

Step 1: Calculate the frequency of the 380 nm light wave

Using the equation \(c = \lambda \times \nu\), we can calculate the frequency of a 380 nm light wave. Note that we need to convert the wavelength to meters first: \(380 \, nm = 380 \times 10^{-9} \, m\) Now, we can plug in the values for c and λ in the equation: \(\nu = \frac{c}{\lambda} = \frac{3 \times 10^8 \, m/s}{380 \times 10^{-9} \, m} = 7.89 \times 10^{14} \, Hz\)

Step 2: Calculate the energy of one photon of 380 nm light

Using the equation \(E = h \times \nu\) and the frequency we calculated in Step 1, we can find the energy of one photon of 380 nm light. Remember that Planck's constant is \(6.63 \times 10^{-34} Js\): \(E = (6.63 \times 10^{-34} \, Js)(7.89 \times 10^{14} \, Hz) = 5.23 \times 10^{-19} \, J\)

Step 3: Calculate the energy of a mole of 380 nm photons

To find the energy of a mole of 380 nm photons, we simply multiply the energy of one photon by Avogadro's number (\(6.022 \times 10^{23} \, mol^{-1}\)): \(E_{mole} = (5.23 \times 10^{-19} \, J)(6.022 \times 10^{23} \, mol^{-1}) = 3.15 \times 10^5 \, J/mol\)

Step 4: Determine which type of UV radiation has more energetic photons

According to Planck's equation \(E = h \times \nu\), the energy of a photon is directly proportional to its frequency. Since shorter wavelengths correspond to higher frequencies and UV-B radiation has shorter wavelengths than UV-A radiation, photons of UV-B radiation are more energetic than those of UV-A radiation.

Step 5: Compare results to the observation regarding sunburns

Given that UV-B radiation has more energetic photons than UV-A radiation, our results are consistent with the observation that UV-B radiation is a more important cause of sunburn in humans than UV-A radiation. Higher-energy photons are more likely to cause damage to the skin and thus contribute to sunburn.

Key Concepts

Wavelength and Frequency

Light travels as electromagnetic waves, characterized by their wavelength and frequency. The wavelength of light refers to the distance between consecutive peaks of the wave, usually measured in nanometers (nm) for ultraviolet light. Frequency, on the other hand, is the number of waves that pass a given point per second, typically measured in hertz (Hz).

The speed of light ( \(c\)) in a vacuum is constant at approximately \(3 \times 10^8 \, m/s\). The relationship between wavelength ( \(\lambda\)) and frequency ( \(u\)) can be described by the equation:

• \(c = \lambda \times u\)

When we know the wavelength of a light wave, we can rearrange this equation to find its frequency:

• \(u = \frac{c}{\lambda}\)

For instance, ultraviolet light with a wavelength of 380 nm can have its frequency calculated by first converting the wavelength into meters ( \(380 \,\text{nm} = 380 \times 10^{-9} \,\text{m}\)) and using the above formula to find \(u = 7.89 \times 10^{14} \,\text{Hz}\).

Photon Energy

A photon is the fundamental particle of light, with each photon carrying a specific amount of energy. This energy is determined by the frequency of the light wave and can be calculated using the equation:

• \(E = h \times u\)

where \(h\) is Planck's constant ( \(6.63 \times 10^{-34} \,J\cdot s\)) and \(u\) is the frequency. From Step 1, for a photon with a frequency of \(7.89 \times 10^{14} \,Hz\), the energy of one photon is calculated as \(5.23 \times 10^{-19} \,J\).

While individual photon energies seem small, when considering a large number of photons such as Avogadro's number ( \(6.022 \times 10^{23}\) photons per mole), the energy becomes substantial. For instance, the energy of a mole of 380 nm photons is calculated as \(3.15 \times 10^{5} \,J/mol\), illustrating how these tiny energy packets together pack a significant punch.

Electromagnetic Spectrum

The electromagnetic spectrum is a broad range of all possible electromagnetic radiation wavelengths and frequencies. It extends from radio waves with long wavelengths to gamma rays with very short wavelengths. Among this range lies ultraviolet (UV) radiation, closely associated with the visible spectrum but with shorter wavelengths.

Ultraviolet radiation itself is divided into several subtypes based on wavelength, primarily UV-A and UV-B. Each type of UV radiation carries different levels of energy, largely because of their varying wavelengths. UV-A ranges from 320-380 nm, while UV-B spans from 290-320 nm. Since shorter wavelengths translate to higher frequencies according to the equation \(c = \lambda \times u\), UV-B photons are more energetic than those of UV-A. This means UV-B radiation has more potential to cause biological effects, such as sunburns.

• UV-A: Longer wavelength, lower energy
• UV-B: Shorter wavelength, higher energy

Understanding this helps us grasp why UV-B is more potent in causing skin damage than UV-A, aligning with observed phenomena where UV-B is recognized as a primary contributor to sunburn.