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 \(\mathrm{UV}-\mathrm{B}\) 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 \((c)\) ?

The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. It encompasses a variety of waves including, but not limited to, radio waves, microwaves, infrared, visible light, ultraviolet (UV), X-rays, and gamma rays. These waves are classified based on their wavelength and frequency, which are inversely proportional.

UV radiation, which is the focal point of our exercise, lies between visible light and X-rays on the spectrum, characterized by wavelengths ranging from about 10 nm to 400 nm. UV radiation itself is divided into several categories, including UV-A and UV-B, as highlighted in the provided exercise. It plays a crucial role in numerous biological and chemical processes. For instance, UV light can cause the production or breakdown of vitamin D in the skin or lead to photochemical reactions.

Every photon, the elementary particle of light, carries a specific amount of energy. The energy of a photon can be calculated using the Planck's equation: \(E = hu\), where \(E\) represents the energy of the photon, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} J \bullet s \)), and \(u\) is the frequency of the radiation.

This equation connects the seemingly abstract concept of light with the tangible quality of energy. When dealing with moles of photons, as we often do in chemistry, the energy per mole can be found by multiplying the energy of a single photon by Avogadro's number, \(N_A = 6.022 \times 10^{23} mol^{-1}\). This relationship becomes a powerful tool for understanding how light energy can influence matter, particularly in the photochemical reactions crucial to disciplines like biochemistry and photophysics.

The frequency-wavelength relationship is fundamental to understanding the electromagnetic spectrum. The speed of light \(c\), approximately \(3.0 \times 10^{8} m/s\), is a constant in a vacuum for all electromagnetic waves, which sets a simple relationship between frequency \(u\) and wavelength \(\lambda\): \(c = \lambdau\). This means that as the wavelength increases, the frequency decreases, and vice versa.

This inverse relationship is illustrated in our UV radiation exercise, showing how to convert between wavelength and frequency for photons. Knowing this relationship enables us to deduce properties of electromagnetic radiation across the spectrum and the implications on their energy. It's particularly significant in chemistry where electromagnetic radiation often initiates or accelerates chemical reactions, and the type of reaction can depend on the specific frequency or wavelength of light involved.