Chapter 2, Problem 43E

Octyl methoxycinoamate 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?

Interpretation Introduction

Interpretation: The frequency range of the UV light that Octyl methoxycinnamate and oxybenzone compounds absorb is to be calculated.

Concept introduction: Frequency of light is defined as the number of waves passes in space during a given interval of time of one second. The frequency of light is measured in the unit of hertz that is also called as cycles per second.

To determine: The frequency range of the UV light that Octyl methoxycinnamate and oxybenzone compounds absorb.

Answer to Problem 43E

Answer

The frequency range of the UV light that Octyl methoxycinnamate and oxybenzone compounds absorb is ${\underset{\_}{1.6\times 10^{14}s}}^{-1}$.

Explanation of Solution

Explanation

The relation between the speed, wavelength and frequency of light is shown below.

The speed of the wave is defined as the product of its wavelength and its frequency. The expression which gives the relationship between the speed, wavelength and frequency of the electromagnetic radiation is as follows,

$\lambda \nu =c$

Where,

• $\lambda$ is the wavelength of light.
• $c$ is the speed of light $\left(2.9979\times {10}^8\text{ m/s }\right)$.
• $\nu$ is the frequency of light.

The frequency of $28\text{0 nm}$ light is ${\underset{\_}{1.1\times 10^{15}s}}^{-1}$.

Given

The wavelength $\left(\lambda \right)$ range of UV light is $28\text{0 nm}-32\text{0 nm}$.

The conversion of nm to m is done as,

$\text{1nm}={10}^{-9}\text{ m}$.

Therefore, the conversion of $280\text{ nm to m}$ is done as,

$\begin{array}{c}280\text{ nm}=\left(280\times {10}^{-9}\right)\text{m}\\ =2.8\times {10}^{-7}\text{ m}\end{array}$.

The speed of light is $\left(2.9979\times {10}^8\text{ m/s }\right)$.

The frequency of $28\text{0 nm}$ light is calculated by using the formula,

$\nu =\frac{c}{\lambda }$

Substitute the value of $c$ and $\lambda$ in the above expression to calculate the frequency of UV light.

$\begin{array}{c}\nu =\frac{c}{\lambda }\\ =\frac{2.9979\times {10}^8\text{ m/s}}{2.8\times {10}^{-7}\text{ m}}\\ ={\underset{\_}{1.1\times 10^{15}s}}^{-1}\end{array}$

Therefore, the upper limit of frequency of the UV light that compound absorbs is ${\underset{\_}{1.1\times 10^{15}s}}^{-1}$

The frequency of $\text{320 nm}$ light is ${\underset{\_}{9.4\times 10^{14}s}}^{-1}$.

The wavelength $\left(\lambda \right)$ range of UV light is $28\text{0 nm}-32\text{0 nm}$.

The conversion of nm to m is done as,

$\text{1nm}={10}^{-9}\text{ m}$.

Therefore, the conversion of $\text{320 nm to m}$ is done as,

$\begin{array}{c}320\text{ nm}=\left(320\times {10}^{-9}\right)\text{m}\\ =3.2\times {10}^{-7}\text{ m}\end{array}$.

The speed of light is $\left(2.9979\times {10}^8\text{ m/s }\right)$.

The frequency of $\text{320 nm}$ light is calculated by using the formula,

$\nu =\frac{c}{\lambda }$

Substitute the value of $c$ and $\lambda$ in the above expression to calculate the frequency of UV light.

$\begin{array}{c}\nu =\frac{c}{\lambda }\\ =\frac{2.9979\times {10}^8\text{ m/s}}{3.2\times {10}^{-7}\text{ m}}\\ ={\underset{\_}{9.4\times 10^{14}s}}^{-1}\end{array}$

Therefore, the lower limit of frequency of the UV light that compound absorbs is ${\underset{\_}{9.4\times 10^{14}s}}^{-1}$.

Therefore, the frequency range of the UV light is calculated by the formula,

$\text{Frequency range }=\text{ Upper limitof frequency }-\text{lower limit of frequency}$

Substitute the value of upper and lower limit of frequency in the above expression,

$\begin{array}{c}\text{Frequency range }=\text{ Upper limitof frequency }-\text{Lower limit of frequency}\\ =1.1\times {10}^{15}{\text{s}}^{-1}-9.4\times {10}^{14}{\text{s}}^{-1}\\ =\underset{\_}{1.6\times 10^{14}{s}^{-1}}\end{array}$

Hence, the frequency range of the UV light that Octyl methoxycinnamate and oxybenzone compounds absorb is ${\underset{\_}{1.6\times 10^{14}s}}^{-1}$.

Conclusion

Conclusion

The frequency range of the UV light that Octyl methoxycinnamate and oxybenzone compounds absorb is ${\underset{\_}{1.6\times 10^{14}s}}^{-1}$.