Question

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

Short answer

The compounds in sunscreen absorb a frequency range of approximately \(9.38 \times 10^{14} \mathrm{Hz}\) to \(1.07 \times 10^{15} \mathrm{Hz}\).

Step-by-step solution

Find the speed of light

The speed of light in a vacuum is approximately \(3.0 \times 10^8 \mathrm{m/s}\)

Convert the wavelength range to meters

The given wavelength range is in nanometers. To convert it to meters, multiply by \(10^{-9}\): Wavelength range = \(280 \times 10^{-9} \mathrm{m}\) to \(320 \times 10^{-9} \mathrm{m}\)

Calculate the frequency range

Now we can use the formula \(v = \frac{c}{λ}\) to find the frequency range. For the lower limit of the wavelength range (\(280 \times 10^{-9} \mathrm{m}\)): \(v_1 = \frac{3.0 \times 10^8 \mathrm{m/s}}{280 \times 10^{-9} \mathrm{m}}\) And for the upper limit of the wavelength range (\(320 \times 10^{-9} \mathrm{m}\)): \(v_2 = \frac{3.0 \times 10^8 \mathrm{m/s}}{320 \times 10^{-9} \mathrm{m}}\) Calculate \(v_1\) and \(v_2\).

Find the frequency range

After calculating \(v_1\) and \(v_2\), we get the frequency range of:; \(v_1 \approx 1.07 \times 10^{15} \mathrm{Hz}\) and \(v_2 \approx 9.38 \times 10^{14} \mathrm{Hz}\) So, the compounds in sunscreen absorb a frequency range of approximately \(9.38 \times 10^{14} \mathrm{Hz}\) to \(1.07 \times 10^{15} \mathrm{Hz}\).

Key concepts

UVB Light Wavelength

Ultraviolet B (UVB) light encompasses wavelengths from about 280 to 320 nanometers (nm), situated between visible light and UVC on the electromagnetic spectrum. These wavelengths are notably responsible for causing sunburn, which is why they are critical in the context of sunscreen protection. Understanding the wavelength is essential for selecting appropriate ingredients that can absorb this range of UVB light to prevent skin damage.

Sunscreen compounds like octyl methoxycinnamate and oxybenzone are specifically designed to absorb these wavelengths, thereby protecting our skin by converting UV energy into less harmful heat. The explicit focus on the UVB range is due to its high energy levels, which can lead to more severe cellular damage compared to UVA light.

Speed of Light

The speed of light, denoted as 'c' in physics equations, is a constant approximately equal to 299,792,458 meters per second (\(3.0 \times 10^8 \text{ m/s}\)) in a vacuum. This is a fundamental constant of nature, meaning it does not change and is the same across the entire universe.

In the context of calculating light frequency, knowing the speed of light is crucial. The speed allows us to understand how fast light waves travel through space and consequently helps in determining the energy and frequency of those waves. Since the speed of light is incredibly fast, it translates to high-frequency values when we look at the light in the UVB range, which implies a significant amount of energy carried by these light waves.

Frequency Range Calculation

The frequency of light is the number of waves passing a point per second, measured in Hertz (Hz). To find the frequency range that sunscreen compounds absorb, we can use the formula \(v = \frac{c}{\lambda}\), where 'v' is the frequency, 'c' is the speed of light, and '\(\lambda\)' is the wavelength.

Beginning with the conversion of nanometers to meters, as UVB light wavelengths are reported in nanometers for convenience, it is a simple yet critical step for accurate calculation. Once we have the correct form of wavelengths, inserting these values into the formula provides us with the frequency range, which is essential in understanding the energy spectrum that sunscreen ingredients are designed to protect against. The calculated frequency range of \(9.38 \times 10^{14} Hz\) to \(1.07 \times 10^{15} Hz\) shows the high-energy light that these compounds can absorb to protect our skin.