Problem 85 - Chemistry The Central Science In Si Units

Question

The rays of the Sun that cause tanning and burning are in the ultraviolet portion of the electromagnetic spectrum. These rays are categorized by wavelength. So-called UV-A radiation has wavelengths in the range of \(320-380 \mathrm{nm},\) whereas UV-B radiation has wavelengths in the range of \(290-320 \mathrm{nm}\). (a) Calculate the frequency of light that has a wavelength of \(380 \mathrm{nm} .\) (b) Calculate the energy of a mole of \(380-\mathrm{nm}\) photons. (c) Which are more energetic, photons of UV-A radiation or photons of UV-B radiation? (d) The UV-B radiation from the Sun is considered a more important cause of sunburn in humans than UV-A radiation. Is this observation consistent with your answer to part (c)?

Step-by-Step Solution

Answer

a) Frequency: \(7.89 \times 10^{14} \text{Hz}\). b) Energy per mole: \(315 \mathrm{kJ/mol}\). c) UV-B photons are more energetic than UV-A photons. d) Yes, UV-B is more energetic, consistent with it causing sunburn.

Step 1: Understanding the relationship between wavelength and frequency

The frequency (c) of light is related to its wavelength (bb) by the speed of light (c). The formula is:\[u = \frac{c}{\lambda}\]Where c (the speed of light) is approximately \(3.00 \times 10^8\) meters per second (m/s). First, convert 380 nm to meters: \(380 \mathrm{nm} = 380 \times 10^{-9} \mathrm{m}\).

Step 2: Calculating the frequency of a 380 nm wavelength

Substitute the values into the frequency formula:\[u = \frac{3.00 \times 10^8 \mathrm{m/s}}{380 \times 10^{-9} \mathrm{m}} = 7.89 \times 10^{14} \text{Hz}\]This is the frequency for the 380 nm wavelength.

Step 3: Calculating the energy of a photon with a 380 nm wavelength

The energy (E) of a single photon is given by:\[E = h \cdot u\]Where h is Planck's constant: \(6.626 \times 10^{-34} \mathrm{J \cdot s}\). Using the frequency calculated in Step 2:\[E = 6.626 \times 10^{-34} \mathrm{J \cdot s} \cdot 7.89 \times 10^{14} \mathrm{Hz} = 5.23 \times 10^{-19} \mathrm{J}\]

Step 4: Calculating the energy of a mole of 380 nm photons

A mole of photons contains Avogadro's number of photons: \(6.022 \times 10^{23} \mathrm{mol}^{-1}\). Therefore, the energy of a mole of 380 nm photons is:\[E_{\text{mole}} = 5.23 \times 10^{-19} \mathrm{J} \times 6.022 \times 10^{23} \mathrm{mol}^{-1} = 315 \mathrm{kJ/mol}\]

Step 5: Comparing energy of UV-A and UV-B photons

Since the energy of a photon is inversely proportional to its wavelength (longer wavelength corresponds to less energy), UV-B photons (290-320 nm) have shorter wavelengths and therefore more energy than UV-A photons (320-380 nm).

Step 6: Evaluating the consistency with sunburn observation

UV-B radiation is more energetic, leading to greater potential for causing sunburn compared to UV-A radiation. This aligns with the observation that UV-B is a more significant cause of sunburn.

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum encompasses a diverse range of electromagnetic waves, all traveling at the speed of light. These waves are categorized based on their varying wavelengths and frequencies. Visible light, ultraviolet rays, radio waves, and X-rays are some of the common types within this spectrum. When the wavelength changes, so does the energy and potential effects of the radiation.

Ultraviolet (UV) radiation, which is a part of this spectrum, is divided mainly into three categories: UV-A, UV-B, and UV-C. These categories differ based on their wavelengths:

• UV-A (320-400 nm): Longer wavelengths, responsible for skin aging and minor sunburns.
• UV-B (290-320 nm): Shorter wavelengths, more energetic, leading to more significant skin burns and potential damage.
• UV-C (100-290 nm): Most energetic, but usually absorbed by the Earth’s atmosphere, hence not a major concern for sunburn.

Understanding this spectrum helps us comprehend the behaviors and impacts of different types of waves in our daily lives.

Wavelength and Frequency Relationship

Light exhibits a fascinating relationship between its wavelength and frequency. The two are inversely related, meaning as one increases, the other decreases. This relationship is governed by the speed of light (c), which is a constant value in a vacuum, approximately equal to \(3.00 \times 10^8\) meters per second. The formula connecting these quantities is:

• \(u = \frac{c}{\lambda}\)

Here, \(u\) represents the frequency, \(c\) is the speed of light, and \(\lambda\) is the wavelength. Knowing this formula allows us to calculate one value if the other is provided.

For instance, a wavelength of 380 nm (where nm is nanometers, a small unit equivalent to \(10^{-9}\) meters) falls into the UV-A range. By applying the formula, you can discover its frequency to be \(7.89 \times 10^{14}\) Hz. This relationship is crucial for understanding how light and energy interact and influence each other. It underpins the calculations concerning the energy each wavelength can carry.

Photon Energy Calculation

Photons, the tiny packets of light energy, each have a distinctive energy value. This energy is directly proportional to the light's frequency and can be calculated using Planck's equation:

• \(E = h \cdot u\)

In this equation, \(E\) is the energy of a photon, \(h\) is Planck's constant \(6.626 \times 10^{-34} \mathrm{J \cdot s}\), and \(u\) is the frequency. Thus, higher frequency waves, like those in the UV-B range, carry more energy than lower frequency UV-A waves.

To determine the energy of photons at a specific wavelength, such as 380 nm, you use its calculated frequency (\(7.89 \times 10^{14}\) Hz) from previous discussions. The outcome gives a photon energy of \(5.23 \times 10^{-19}\) Joules. For larger quantities, such as a mole (\(6.022 \times 10^{23}\) photons), you simply multiply the energy of one photon by Avogadro's number, leading to an energy of \(315\) kJ/mol for 380 nm photons.

Understanding these calculations helps demonstrate why UV-B radiation, with more energetic photons due to shorter wavelengths, tends to cause more severe sunburn than UV-A radiation.