Problem 29 - Chemistry The Central Science In Si Units

Question

One type of sunburn occurs on exposure to UV light of wavelength in the vicinity of \(325 \mathrm{nm}\). (a) What is the energy of a photon of this wavelength? (b) What is the energy of a mole of these photons? (c) How many photons are in a \(1.00 \mathrm{~mJ}\) burst of this radiation? (d) These UV photons can break chemical bonds in your skin to cause sunburn-a form of radiation damage. If the \(325-\mathrm{nm}\) radiation provides exactly the energy to break an average chemical bond in the skin, estimate the average energy of these bonds in \(\mathrm{kJ} / \mathrm{mol}\).

Step-by-Step Solution

Answer

(a) 6.114 x 10^-19 J per photon; (b) 368.1 kJ/mol; (c) 1.636 x 10^15 photons; (d) 368.1 kJ/mol.

Step 1: Calculate photon energy

Use the formula \( E = \frac{h c}{\lambda} \) where \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J s} \), \( c \) is the speed of light \(3.00 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength \(325 \times 10^{-9} \text{ m} \). \[ E = \frac{(6.626 \times 10^{-34}) (3.00 \times 10^8)}{325 \times 10^{-9}} \approx 6.114 \times 10^{-19} \text{ J} \] This is the energy of a single photon.

Step 2: Calculate energy per mole of photons

Avogadro's number \(N_A\) is \(6.022 \times 10^{23} \text{ mol}^{-1} \). Multiply the energy of one photon by Avogadro's number to find the energy per mole: \[ E_{mol} = 6.114 \times 10^{-19} \text{ J} \times 6.022 \times 10^{23} \text{ mol}^{-1} \approx 368.1 \text{ kJ/mol} \] This is the energy of a mole of these photons.

Step 3: Determine number of photons in a 1.00 mJ burst

Convert \(1.00 \text{ mJ} \) to Joules: \( 1.00 \text{ mJ} = 1.00 \times 10^{-3} \text{ J} \). Divide this energy by the energy of a single photon to find the number of photons: \[ n = \frac{1.00 \times 10^{-3}}{6.114 \times 10^{-19}} \approx 1.636 \times 10^{15} \] This is the number of photons in the burst.

Step 4: Estimate average energy of chemical bonds

Since the energy per mole is equivalent to the energy required to break a mole of bonds, simply use the energy per mole calculated previously: The average energy of these bonds is approximately \( 368.1 \text{ kJ/mol} \).

Key Concepts

Understanding UV Light

Ultraviolet (UV) light is a form of electromagnetic radiation, positioned next to visible light on the electromagnetic spectrum. Its wavelength ranges between 10 nm and 400 nm, and it's divided into three types: UVA, UVB, and UVC, each with varying wavelengths and energy levels. In the context of the exercise, UV light with a wavelength of around 325 nm is significant.

• UVA has longer wavelengths (320-400 nm), and is the least harmful but can penetrate deep into the skin.
• UVB, which includes 325 nm, is responsible for sunburns and can directly damage the DNA in our skin cells.
• UVC is almost entirely absorbed by the Earth's atmosphere and does not reach the surface.

Understanding UV light is crucial because its energy can be sufficient to break chemical bonds, leading to skin damage such as sunburns.

Planck's Constant: The Bridge Between Wavelength and Energy

Planck's constant is a fundamental constant in physics, denoted as \( h \), with a value of \(6.626 \times 10^{-34} \text{ J s} \). It serves as a critical link between the energy of a photon and its frequency or wavelength.In the formula for photon energy, \( E = \frac{h c}{\lambda} \), Planck's constant allows us to calculate the energy of photons when we know their wavelength. Here, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the light.

• The smaller the wavelength, the higher the energy of the photons.
• It's a key constant in quantum mechanics that explains the quantization of energy.

Hence, Planck's constant is not just a number; it's a vital component in understanding how energy and light interact on a quantum level.

Avogadro's Number and Its Role in Chemistry

Avogadro's number, \( N_A = 6.022 \times 10^{23} \text{ mol}^{-1} \), is another essential constant in chemistry. It defines the number of atoms, molecules, or particles in one mole of a substance, providing a bridge between the microscopic world and the macroscopic world.For instance, when calculating the energy of a mole of photons, Avogadro's number helps us scale up from the energy of a single photon to the energy contained in a mole (a large group) of photons.

• It allows chemists to measure quantities in lab-scale amounts using moles.
• Helps in expressing macroscopic amounts of a chemical element or compound.

Avogadro's number makes it feasible to apply quantum-level measurements to chemical reactions, like those involving photons and chemical bonds.

Breaking Down Chemical Bonds with Energy

Chemical bonds are the forces holding atoms together within molecules. The energy needed to break these bonds determines the stability of compounds and how they react with other substances. When UV photons hit the skin, they can have enough energy to break these bonds, which might lead to damage like sunburn. This energetic interaction can lead to significant changes at the molecular level.

• Bond breaking requires energy, often supplied by heat or light.
• Each type of bond (ionic, covalent) requires a different amount of energy to break.

Understanding the energy involved in breaking chemical bonds helps explain how radiation can damage cells and tissues, highlighting the importance of protection from UV exposure.

Calculating Wavelength and Its Effect

Wavelength is a fundamental concept in physics, particularly when discussing light and electromagnetic radiation. It is the distance between successive peaks of a wave and is typically measured in nanometers (nm).Wavelength determination is crucial because it directly affects the energy of the radiation; shorter wavelengths correlate with higher energy.

• Through the equation \( E = \frac{h c}{\lambda} \), you can see that energy is inversely proportional to wavelength.
• This means that as the wavelength decreases, the energy of each photon increases.

Understanding and calculating wavelengths enable us to predict the behavior of different types of radiation, such as UV light, and how they interact with matter.