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Epsilon in Beer’s Law and Why We Wear Sunscreen

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    Chemistry Concepts: 

    This Exemplar will teach the following concept from the ACS Examinations Institute for General Chemistry ACCM: 

    • IX. D. 5. c. Absorption and emission of light within UV-Visible wavelengths is associated with valence electronic transitions within molecules.1 

    Learning Objectives
    • Understand Beer's Law.
    • Practice Beer's Law problems in the context of sunscreen.

    Introduction

    Many people have heard about the importance of wearing sunscreen and the dangers of UV rays, which can have numerous adverse effects. Short-term overexposure may lead to sunburn, and long-term effects include premature aging and the development of skin cancer, which is the most common type of cancer in the United States.2 

     

    4-AOIf-v37RHscCJRGDoL1TgkfSC5mDKMcuZR4Obm0ZB5MpIBjdW_polGzksMYu1koEs6KYidwoHtdrvOQMrHeIVLagbxrMeUFvaSgP-TuPJ1PUf2la5trfHMnl1xftUZb_Fl6u2kzkku-9zQtMBT_pcwp491WjLREkcC7zCnrdcAlI9nFjU9v4gardpFA_3n95

    Figure 1

    Ultraviolet A rays, which are longer in wave-length, consist of approximately 90% of the sunlight reaching the earth and penetrate deep into the skin, causing indirect genetic damage that can lead to cancer. Ultraviolet B light, which is shorter in wave-length, impacts the outer layer of human skin, leading to sunburning and harming DNA.3

    UV radiation is a form of electromagnetic radiation that comes from the sun and man-made sources; its adverse effects on human skin can be averted by applying sunscreen. Using two major groups of protective substances, sunscreen prevents sun damage: physical and chemical blockers.3 Physical blockers, like titanium or zinc oxide, sit on the skin's surface and reflect UV rays away from the skin, as a mirror or shield would. Chemical blockers are organic and form a protective film around the skin that absorbs UV radiation before it can penetrate the skin.4 In this publication, we will focus on absorption, specifically in sunscreen, and how it is calculated using Beer’s Law.
     

    Beer's Law

    A=εbc

    Beer’s Law, also known as the Beer-Lambert Law, A=εbc relates the absorption of light of a solution A (unitless) to its molar absorptivity ε (M-1•cm-1), its path length b (usually in cm), and its concentration c (usually in M, which is mol / L). 

    We will now practice using Beer's law by examining different compounds in sunscreen.

     

    Note

    There are currently sixteen sun-filtering ingredients approved by the FDA, but only eight are in common use: avobenzone, homosalate, octisalate, octocrylene, octinoxate, oxybenzone, titanium dioxide, and zinc oxide. For our examples, we'll focus on three: avobenzone, homosalate, and ococrylene, which are present in Sun Bum sunscreen. 3 of the Active ingredients in Sun Bum SPF 50 sunscreen are: Avobenzone (3%), Homosalate (10%), and Octocrylene (10%).5

    Figure 2: Sun Bum SPF 50 Sunscreen

    Example 1: Avobenzone (UVA protection)6
    Problem

    Use Beer's Law to find the absorbance of avobenzone using data from an 8 ounce sunscreen bottle. If the absorbance of avobenzone calculated in the lab is 0.80 with a 1 cm cuvette, what is the molar absorptivity (ε) of avobenzone in this example?

    Solution

    Calculating c (molar concentration):
    Avobenzone composes 3% of the 8 oz sunscreen solution by mass.

    0.03 x 8 fluid oz = 0.24 fluid oz x (28.34952 grams/1 fluid oz) x (1 mol/310.39 g) = 0.0219 mol

    8 fluid oz x (0.0295735 L/1 fluid oz) = 0.236588 L

    M = m/L = 0.0219 mol/0.236588 L = 0.092566 mol/L

     

    Value for b:

    b= length of path light takes until contact w solution, 1 cm (given)

     

    Value for A: 

    0.80 (unitless)

     

    Calculating Epsilon (wavelengths dependent molar absorptivity coefficient):

    ε units are L•mol-1•cm-1

    A = εbc

    (0.80) = ε (1.00cm) (0.092566 mol/L ) 

    Answer

    ε = 8.64 L / mol•cm

    Example 2: Homosalate (UVB protection)7 
    Problem

    Similar to Example 1, now use Beer's Law to find the absorbance of homosalate using data from an 8 ounce sunscreen bottle. If the absorbance of homosalate calculated in the lab is 0.80 with a 1 cm cuvette, what is the molar absorptivity (ε) of homosalate in this example?

    Solution

    Calculating c (molar concentration):
    10% of total solution = 0.10 x 8 fluid oz = 0.80 fluid oz x (28.34952 grams/1 fluid oz) x (1 mol/262.36 g) = 0.086445 mol

    8 fluid oz x (0.0295735 L/1 fluid oz) = 0.236588 L

    M = m/L = 0.086445 mol/0.236588 L = 0.365380 mol/L

     

    Value for b:

    b= length of path light takes until contact w solution, 1 cm (given)

     

    Value for A: 

    0.80 (unitless)

     

    Calculating Epsilon (wavelengths dependent molar absorptivity coefficient):

    ε units are L•mol-1•cm-1

    A = εbc

    (0.80) = ε (1.00cm) (0.365380 mol/L ) 

    Answer

    ε = 2.19 L / mol•cm 

     

    Example 3: Octocrylene8
    Problem

    Lastly, use Beer's Law to find the absorbance of octocrylene using data from an 8 ounce sunscreen bottle. If the absorbance of octocrylene calculated in the lab is 0.80 with a 1 cm cuvette, what is the molar absorptivity (ε) of octocrylene in this example?

    Solution

    Calculating c (molar concentration):
    10% of total solution = 0.10 x 8 fluid oz = 0.80 fluid oz x (28.34952 grams/1 fluid oz) x (1 mol/361.48 g) = 0.062741 mol

    8 fluid oz x (0.0295735 L/1 fluid oz) = 0.236588 L

    M = m/L = 0.062741 mol/0.236588 L = 0.265191 mol/L

     

    Value for b:

    b= length of path light takes until contact w solution, 1 cm (given)

     

    Value for A: 

    0.80 (unitless)

     

    Calculating Epsilon (wavelengths dependent molar absorptivity coefficient):

    ε units are L•mol-1•cm-1

    A = εbc

    (0.80) = ε (1.00cm) (0.265191 mol/L ) 

    Answer

    ε = 3.02 L / mol•cm 

     

    Conclusion

    Hopefully you now have a better understanding of both Beer's Law and of the importance of sunscreen! Check out our further reading section for more practice, and hopefully you are encouraged to explore these topics more. 

    References 

    1. Anchoring Concepts Content Map (ACCM) Project: Inorganic Chemistry. University of Wisconsin-Milwaukee, 2016. https://pubs.acs.org/doi/suppl/10.10...498_si_001.pdf (accessed Dec 7, 2022). 

    2. National Cancer Institute. Skin cancer (including melanoma)-patient version. https://www.cancer.gov/types/skin (accessed Dec 7, 2022).

    3. Boerner, L. K. What’s in sunscreen, and how does it protect your skin from the sun’s rays? https://cen.acs.org/business/consume...un-rays/99/i27 (accessed Dec 7, 2022).

    4. Young, A R. PubChem. Chromophores in human skin. https://pubmed.ncbi.nlm.nih.gov/9172259/ (accessed Dec 7, 2022).

    5. Sun Bum. Original SPF 50 Sunscreen Lotion. https://www.sunbum.com/products/original-spf-50-sunscreen-lotion (accessed Dec 7, 2022).

    6. PubChem. Compound Summary: Avobenzone. https://pubchem.ncbi.nlm.nih.gov/compound/Avobenzone (accessed Dec 7, 2022).

    7. PubChem. Compound Summary: Homosalate. https://pubchem.ncbi.nlm.nih.gov/compound/Homosalate (accessed Dec 7, 2022).

    8. PubChem. Compound Summary: Octocrylene. https://pubchem.ncbi.nlm.nih.gov/compound/Octocrylene (accessed Dec 7, 2022).

    Figures 

    1. Boerner, L. K. What’s in sunscreen, and how does it protect your skin from the sun’s rays? https://cen.acs.org/business/consume...un-rays/99/i27 (accessed Dec 7, 2022).

    2. Sun Bum. Original SPF 50 Sunscreen Lotion. https://www.sunbum.com/products/original-spf-50-sunscreen-lotion (accessed Dec 7, 2022).

    Further Reading 

    1. Beer–Lambert Law (practice). https://www.khanacademy.org/science/...er-lambert-law (accessed Dec 7,  2022). 

     

    Epsilon in Beer’s Law and Why We Wear Sunscreen is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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