Problem 17 - Chemistry The Molecular Science

Question

Stratospheric ozone absorbs damaging UV-C radiation from the sun, preventing the radiation from reaching Earth's surface. Calculate the frequency and energy per photon of UV-C radiation that has a wavelength of \(270 . \mathrm{nm}\)

Step-by-Step Solution

Answer

Frequency is \(1.11 \times 10^{15} \text{ Hz}\) and energy per photon is \(7.36 \times 10^{-19} \text{ Joules}\).

Step 1: Convert Wavelength to Meters

The wavelength of UV-C radiation is provided as \(270 \text{ nm}\). To perform calculations, we first need to convert this wavelength into meters. We know that \(1 \text{ nm} = 10^{-9} \text{ meters}\). Thus, \(270 \text{ nm} = 270 \times 10^{-9} \text{ meters}\) or \(2.70 \times 10^{-7} \text{ meters}\).

Step 2: Calculate Frequency

We'll use the speed of light equation to find the frequency \(u\), where \(c = \lambda u\), and \(c\) is the speed of light \(3.00 \times 10^8 \text{ m/s}\). Rearranging the formula gives us \(u = \frac{c}{\lambda}\). Substituting the values, \(u = \frac{3.00 \times 10^8 \text{ m/s}}{2.70 \times 10^{-7} \text{ meters}}\), resulting in \(u \approx 1.11 \times 10^{15} \text{ Hz}\).

Step 3: Calculate Energy per Photon

Energy per photon can be found using Planck's equation: \(E = hu\), where \(h\) is Planck's constant \(6.63 \times 10^{-34} \text{ J}\cdot\text{s}\). Substituting the frequency we just found, \(E = (6.63 \times 10^{-34} \text{ J}\cdot\text{s})(1.11 \times 10^{15} \text{ Hz})\), which results in \(E \approx 7.36 \times 10^{-19} \text{ Joules}\).

Key Concepts

wavelength conversion

Converting wavelength units is crucial in physics, particularly when working with the properties of light like UV-C radiation. Wavelength is often given in nanometers (nm) but needs conversion to meters (m) for compatibility with other unit systems used in scientific equations.
To convert nanometers to meters, we utilize the conversion factor:

• 1 nm = \(10^{-9}\) meters.

Given that the wavelength of UV-C radiation is 270 nm, we apply the conversion:

• \(270 \text{ nm} = 270 \times 10^{-9} \text{ meters} = 2.70 \times 10^{-7} \text{ meters}\).

Now that we have the wavelength in meters, we can easily use it in subsequent calculations to determine properties like frequency and energy.

frequency calculation

Frequency is a fundamental characteristic of waves, describing how often the wave peaks pass a certain point per second. To calculate the frequency of UV-C radiation, we employ the relationship between speed, wavelength, and frequency:

• The speed of light (c) in a vacuum is approximately \(3.00 \times 10^8 \text{ m/s}\).
• The formula connecting speed, wavelength (\(\lambda\)), and frequency (\(u\)) is \(c = \lambda u\).

Rearranging the formula to solve for frequency:

• \(u = \frac{c}{\lambda}\)

Substitute \(\lambda = 2.70 \times 10^{-7} \text{ meters}\):

• \(u = \frac{\ 3.00 \times 10^8 \text{ m/s}}{2.70 \times 10^{-7} \text{ m}}\approx 1.11 \times 10^{15} \text{ Hz}\).

Thus, the frequency of our UV-C radiation is approximately \(1.11 \times 10^{15} \text{ Hz}\).

energy per photon

The energy per photon of a beam of light helps us understand how much energy is carried by a single particle of light. This is important in many applications, including understanding the effects of UV-C on biological tissues. To find the energy per photon, we use the frequency calculated earlier with a constant:

• Planck's constant (h) is \(6.63 \times 10^{-34} \text{ J}\cdot\text{s}\).

The equation describing the energy (E) per photon is given by:

• \(E = hu\)

Substituting the known values:

• \(E = (6.63 \times 10^{-34} \text{ J}\cdot\text{s})(1.11 \times 10^{15}\text{ Hz})\approx 7.36 \times 10^{-19} \text{ Joules}\).

This is the amount of energy associated with each photon of the 270 nm UV-C radiation.

Planck's equation

Planck's equation is a fundamental principle in quantum mechanics that describes the relationship between the energy of a photon and its frequency. Understanding this relationship helps in fields ranging from quantum physics to chemistry.
Planck's equation is:

• \(E = hu\)

Where:

• \(E\) is the energy of the photon
• \(h\) is Planck's constant (\(6.63 \times 10^{-34} \text{ J}\cdot\text{s}\))
• \(u\) is the frequency of the photon

The equation illustrates that higher frequency photons carry more energy. This principle is pivotal for understanding the behavior of light across different wavelengths. It's central to various technologies and scientific understandings, including the study of UV-C radiation's effects.