Problem 79 - Chemistry The Central Science

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

Step-by-Step Solution

Answer

(a) The frequency of a 320 nm photon is approximately \(9.375 \times 10^{14}\ \text{Hz}\). (b) The energy of a mole of 320-nm photons is approximately \(3.74 \times 10^5\ \text{J}/\text{mol}\). (c) Photons of UV-B radiation are more energetic than photons of UV-A radiation. (d) The observation that UV-B radiation is a greater cause of sunburn than UV-A radiation is consistent with our energy comparison, as UV-B photons have higher energy and are more likely to cause damage to human skin.

Step 1: (a) Find the frequency of a 320 nm photon.

First, let's convert 320 nm to meters: \(320 nm = 320 \times 10^{-9} m\). Now we can use the speed of light formula to find the frequency: \(c = \lambda \nu \Rightarrow \nu = \frac{c}{\lambda}\) Plugging in the given values: \(\nu = \frac{3.0 \times 10^8\ \text{m/s}}{320 \times 10^{-9}\ \text{m}} \approx 9.375 \times 10^{14}\ \text{Hz}\)

Step 2: (b) Calculate the energy of a mole of 320 nm photons.

First, we will find the energy of a single photon using the energy of a photon formula: \(E = h\nu\) \(E = (6.626 \times 10^{-34}\ \text{J}\cdot\text{s})(9.375 \times 10^{14}\ \text{Hz}) \approx 6.210 \times 10^{-19}\ \text{J}\) Next, we will find the molar energy using the molar energy formula: \(E_{mole} = N_A \times E\) \(E_{mole} = (6.022 \times 10^{23}\ \text{mol}^{-1})(6.210 \times 10^{-19}\ \text{J}) \approx 3.74 \times 10^5\ \text{J}/\text{mol}\)

Step 3: (c) Compare the energies of UV-A and UV-B photons.

Since the wavelengths of UV-A radiation are longer than those of UV-B radiation (UV-A: 320-380 nm, UV-B: 290-320 nm), the frequencies will be lower for UV-A photons. Because energy is directly proportional to frequency (using \(E = h\nu\)), this implies that photons of UV-A radiation have lower energy than those of UV-B radiation.

Step 4: (d) Is the observed potential for sunburn consistent with our energy comparison?

Our energy comparison showed that photons of UV-B radiation have higher energy than photons of UV-A radiation. Given that higher-energy photons are more likely to cause damage to biological systems (such as human skin), the observation that UV-B radiation is a greater cause of sunburn than UV-A radiation is consistent with our answer to part (c).

Key Concepts

Electromagnetic Spectrum

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.

Photon Energy Calculation

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.

Frequency-Wavelength Relationship

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.