Question

Octyl methoxycinnamate and oxybenzone are common ingredients in sunscreen applications. These compounds work by absorbing ultraviolet (UV) B light (wavelength 280–320 nm), the UV light most associated with sunburn symptoms. What frequency range of light do these compounds absorb?

Short answer

The compounds in sunscreen, octyl methoxycinnamate and oxybenzone, absorb light with frequencies in the range of approximately \(9.375\times10^{14}\ Hz\) to \(1.071\times10^{15}\ Hz\), corresponding to the ultraviolet (UV) B light wavelength range of 280 nm to 320 nm.

Step-by-step solution

Convert the given wavelengths to meters

To make our calculations easier, we'll convert the given wavelength range of 280 nm to 320 nm into meters. Since 1 nm = 10^-9 m, we multiply each value by 10^-9 to convert them to meters. Wavelength range: 280 nm - 320 nm \(280\ nm\times10^{-9}\frac{m}{nm}=2.8\times10^{-7} m\) \(320\ nm\times10^{-9}\frac{m}{nm}=3.2\times10^{-7} m\) The converted wavelength range is now \(2.8\times10^{-7}\) m to \(3.2\times10^{-7}\) m.

Use the speed of light formula to find the frequency range

Now, we will use the speed of light formula to determine the frequency range related to the given wavelength range. The formula for the speed of light is: \(c = \lambda \times f\) Where c is the speed of light (3 x 10^8 m/s), λ is the wavelength, and f is the frequency. To find the frequency range, we will rearrange the formula to solve for the frequency (f): \(f = \frac{c}{\lambda}\) We will calculate the frequencies corresponding to 280 nm and 320 nm wavelengths to find the range. For the lower end of the wavelength range: \(f_1 = \frac{3\times10^8\ m/s}{2.8\times10^{-7}\ m} \approx 1.071\times10^{15}\ Hz\) For the upper end of the wavelength range: \(f_2 = \frac{3\times10^8\ m/s}{3.2\times10^{-7}\ m} \approx 9.375\times10^{14}\ Hz\)

Present the frequency range

Based on our calculations, the compounds in sunscreen absorb light with frequencies in the range of: Approximately \(9.375\times10^{14}\ Hz\) to \(1.071\times10^{15}\ Hz\) This is the frequency range of the ultraviolet (UV) B light that octyl methoxycinnamate and oxybenzone absorb to protect the skin from sunburn symptoms.

Key concepts

Sunscreen Ingredients

Sunscreen is essential in protecting our skin from harmful ultraviolet light. Two common ingredients found in modern sunscreens are octyl methoxycinnamate and oxybenzone. These compounds are designed to absorb ultraviolet (UV) B light, which lies in the wavelength range of 280-320 nanometers. This range is particularly significant because UVB rays are mostly associated with causing sunburn and increasing the risk of skin cancer.

• Octyl methoxycinnamate: A chemical UV filter often used for its effectiveness in shielding skin from UVB rays. It's known for being stable in sunlight and is used widely for its protective properties.
• Oxybenzone: Another chemical filter that absorbs mainly UVB, but also some UVA rays. It's used to prevent direct damage to the skin, offering broad-spectrum protection.

These ingredients work by converting the energy from UV light into a longer, less harmful wavelength (infrared), which is then released as heat.

Ultraviolet Light

Ultraviolet (UV) light is a part of the electromagnetic spectrum that reaches Earth from the Sun. It's invisible to the human eye and has shorter wavelengths compared to visible light. UV light is categorized into UV-A, UV-B, and UV-C, each with different properties and effects.

• UV-A (320-400 nm): Long-wave UV light that can penetrate deep into the skin. It contributes to aging and long-term skin damage.
• UV-B (280-320 nm): Medium-wave UV light, affecting the outer layers of the skin and closely associated with sunburn.
• UV-C (100-280 nm): Short-wave UV light mostly absorbed by the Earth’s atmosphere and does not reach the surface. However, it's the most harmful type of UV light.

Understanding these different types of UV light helps in identifying which UV rays sunscreens are protecting against and why certain ingredients are chosen.

Frequency Calculation

Frequency refers to the number of waves that pass a point in one second, measured in hertz (Hz). When dealing with electromagnetic waves, such as UV light, it's linked to wavelength through the speed of light formula. For a given wavelength, the frequency can be calculated by:\[ f = \frac{c}{\lambda} \]Where:

• \( f \) is the frequency
• \( c \) is the speed of light, approximately \( 3 \times 10^8 \) meters per second
• \( \lambda \) is the wavelength

By rearranging the speed of light equation, we can find the frequency range for UVB light absorbed by sunscreen ingredients:

• Lower end frequency at 280 nm: \( f_1 \approx 1.071 \times 10^{15} \) Hz
• Upper end frequency at 320 nm: \( f_2 \approx 9.375 \times 10^{14} \) Hz

This calculation reveals the spectrum of UVB light that sunscreens target to protect our skin.

Wavelength Conversion

Converting the wavelength of light into meters is a crucial step in determining its frequency. Often in scientific calculations, it's easier to work with units like meters, rather than nanometers, as this matches the unit for the speed of light in equations.
For the calculation, the conversion factor used is:

• 1 nanometer (nm) = \( 10^{-9} \) meters (m)

Applying this conversion to the UVB range:

• 280 nm becomes \( 2.8 \times 10^{-7} \) m
• 320 nm becomes \( 3.2 \times 10^{-7} \) m

This adjustment allows for accurate computation of frequencies, as seen in the frequency calculation section, aligning with the standard units required for electromagnetic wave formulas.