Problem 56 - Chemistry and Chemical Reactivity

Question

Radiation in the ultraviolet region of the electromagnetic spectrum is quite energetic. It is this radiation that causes dyes to fade and your skin to develop a sunburn. If you are bombarded with 1.00 mol of photons with a wavelength of 375 nm, what amount of energy, in kilojoules per mole of photons, are you being subjected to?

Step-by-Step Solution

Answer

The energy is approximately 319 kJ/mol.

Step 1: Understand the Given Information

We are given 1.00 mole of photons with a wavelength of 375 nm. Our task is to find the energy per mole of these photons in kilojoules.

Step 2: Convert Wavelength to Meters

The wavelength is given in nanometers (nm), so convert it to meters by using the conversion: 1 nm = \(1 \times 10^{-9}\) m. Thus, \(375 \text{ nm} = 375 \times 10^{-9} \text{ m}\).

Step 3: Use the Energy Formula for One Photon

The energy \(E\) of a single photon can be calculated using the formula:\[ E = \frac{hc}{\lambda} \]where:- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{Js}\)),- \(c\) is the speed of light (\(3.00 \times 10^{8} \text{m/s}\)),- \(\lambda\) is the wavelength in meters.

Step 4: Calculate the Energy of One Photon in Joules

Substitute the known values into the photon energy equation:\[ E = \frac{(6.626 \times 10^{-34} \text{Js})(3.00 \times 10^{8} \text{m/s})}{375 \times 10^{-9} \text{m}} \]Calculate to find the energy in joules.

Step 5: Calculate Energy for a Mole of Photons

Since 1 mole of photons consists of Avogadro's number of photons (\(6.022 \times 10^{23}\)), multiply the energy of one photon by Avogadro's number to find the energy per mole:\[ E_\text{mole} = E \times 6.022 \times 10^{23} \text{ photons/mole} \]

Step 6: Convert Energy to Kilojoules per Mole

Convert the calculated energy from joules per mole to kilojoules per mole by dividing by 1,000 (since 1 kilojoule = 1,000 joules).

Key Concepts

Ultraviolet Radiation

Ultraviolet radiation is a type of electromagnetic radiation with wavelengths shorter than visible light but longer than X-rays, typically ranging from 10 nm to 400 nm. It is part of the electromagnetic spectrum that is not visible to the human eye.

• There are three types of UV radiation: UV-A, UV-B, and UV-C, with UV-A having the longest wavelength and UV-C the shortest.
• UV radiation is responsible for various effects, including skin tanning and sunburn, due to its ability to damage the DNA in skin cells.
• It's a dual-edged sword; while overexposure can be harmful, UV rays are essential for the production of Vitamin D in the skin.

Understanding the energy of ultraviolet radiation is crucial, especially because of its biological effects. The energy can cause chemical reactions, like the one leading to fading dyes and sunburn. When studying photon energy in the ultraviolet range, we often refer to its wavelength in nanometers, like the 375 nm in the exercise.

Electromagnetic Spectrum

The electromagnetic spectrum is a vast range covering all types of electromagnetic radiation, from radio waves to gamma rays. Each type of radiation within this spectrum has different wavelengths and frequencies.

• The spectrum is generally divided into regions: radio, microwave, infrared, visible light, ultraviolet, X-rays, and gamma rays.
• Radiation wavelength correlates inversely with frequency: longer wavelengths mean lower frequency and energy, while shorter wavelengths result in higher frequency and energy.

Ultraviolet radiation falls on the higher energy side of the visible spectrum. This high energy is what makes UV radiation so effective in inducing photochemical reactions. Therefore, it's important to understand where ultraviolet radiation sits within the electromagnetic spectrum, as it helps predict its potential effects and applications.

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics, essential for calculating photon energy. It defines the proportionality between the energy of a photon and the frequency of its electromagnetic wave:

• Represented by the symbol \(h\), with a value of \(6.626 \times 10^{-34} \text{ Js}\).
• It appears in the equation \(E = h \cdot f\), where \(E\) is energy and \(f\) is frequency.
• This equation indicates that energy is quantized and that each quantum of energy is directly proportional to the frequency of radiation.

In the context of the ultraviolet radiation exercise, Planck's constant is used to convert the given wavelength into an energy value. This demonstrates the dependency between a photon's energy and its wavelength, thereby illustrating one of the quintessential concepts of quantum theory.

Avogadro's Number

Avogadro's number is a key constant in chemistry and defines the number of units in one mole of a substance, typically atoms or molecules. It provides a bridge between the microscopic scale of atoms and the macroscopic scale of molar quantities.

• The value of Avogadro's number is \(6.022 \times 10^{23}\).
• It allows chemists to count large quantities of small entities by relating them to measurable macroscopic amounts.
• In the context of photon energy calculation, knowing Avogadro's number helps calculate the total energy for a mole of photons by multiplying it by the energy of a single photon.

This calculation is crucial for understanding how much energy a mole of UV photons embodies, highlighting its practical application in understanding chemical processes and reactions at a macroscopic level. Avogadro's number thus transforms the abstract world of individual photons into practical, real-world quantities of energy.