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 320 \(\mathrm{nm}\) . (b) Calculate the energy of a mole of 320 -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 greater cause of sunburn in humans than is UV-A radiation. Is this observation consistent with your answer to part (c)?

First, we will calculate the frequency of light having a wavelength of 320 nm. We can use the formula: \(c = \lambda v\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(v\) is the frequency. We have the value of the speed of light \(c = 3.00 \times 10^8 \, m/s\) and the wavelength, \(\lambda = 320 \, nm = 320 \times 10^{-9} \, m\). Now we can calculate the frequency (v): \(v = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \, m/s}{320 \times 10^{-9} \, m}\) \(v = 9.375 \times 10^{14} \, Hz\) The frequency of 320 nm light is \(9.375 \times 10^{14} \, Hz\).

To calculate the energy of a mole of 320 nm photons, first, we need to find out the energy of a single photon using the formula: \(E = h v\), where \(E\) is the energy, \(h\) is the Planck's constant (\(6.63 \times 10^{-34} \, Js\)), and \(v\) is the frequency. From part (a), we have the frequency: \(v = 9.375 \times 10^{14} \, Hz\). Thus, the energy of a single photon can be calculated: \(E = (6.63 \times 10^{-34} \, Js)(9.375 \times 10^{14} \, Hz)\) \(E = 6.21 \times 10^{-19} \, J\) Now, we can calculate the energy of a mole of photons using the formula: \(E_{mole} = E \times N_A\), where \(N_A\) is the Avogadro's number (\(6.02 \times 10^{23} \, mol^{-1}\)). \(E_{mole} = (6.21 \times 10^{-19} \, J)(6.02 \times 10^{23} \, mol^{-1})\) \(E_{mole} = 3.74 \times 10^5 \, J/mol\) The energy of a mole of 320 nm photons is \(3.74 \times 10^5 \, J/mol\).

We know that shorter wavelengths have higher frequencies and thus higher energy. Since UV-A radiation has wavelengths in the range of \(320-380 \, nm\), and UV-B radiation has wavelengths in the range of \(290-320 \, nm\), we can conclude that photons of UV-B radiation are more energetic than photons of UV-A radiation.

The observation states that UV-B radiation is considered a greater cause of sunburn in humans than UV-A radiation. In part (c), we concluded that UV-B radiation photons are more energetic than UV-A radiation photons. Thus, this observation is consistent with the answer found in part (c), as higher energy photons in UV-B radiation cause more damage to the skin, leading to sunburn.