Problem 5 - Chemistry and Chemical Reactivity

Question

Green light has a wavelength of \(5.0 \times 10^{2} \mathrm{nm}\) What is the energy, in joules, of one photon of green light? What is the energy, in joules, of 1.0 mol of photons of green light?

Step-by-Step Solution

Answer

Energy of one photon: \(3.9756 \times 10^{-19}\) J; Energy of 1 mole of photons: \(2.39 \times 10^{5}\) J.

Step 1: Understand the relationship between wavelength and energy

The energy of a photon can be calculated using the relationship between wavelength \( \lambda \) and energy \( E \). The formula to use is \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J s} \) and \( c \) is the speed of light \( 3.00 \times 10^{8} \text{ m/s} \).

Step 2: Convert the wavelength from nanometers to meters

The wavelength provided is \( 5.0 \times 10^{2} \text{ nm} \). We convert this to meters by using the conversion factor \( 1 \text{ nm} = 1 \times 10^{-9} \text{ m} \), thus \( 5.0 \times 10^{2} \text{ nm} \) is equivalent to \( 5.0 \times 10^{-7} \text{ m} \).

Step 3: Calculate the energy of one photon

Using the converted wavelength, plug the values into the equation: \( E = \frac{6.626 \times 10^{-34} \text{ J s} \times 3.00 \times 10^{8} \text{ m/s}}{5.0 \times 10^{-7} \text{ m}} \). This results in \( E \approx 3.9756 \times 10^{-19} \text{ J} \) per photon.

Step 4: Calculate the energy for 1 mole of photons

There are Avogadro's number of photons in a mole, \( 6.022 \times 10^{23} \text{ photons/mol} \). Multiply the energy of one photon by Avogadro's number: \( 3.9756 \times 10^{-19} \text{ J/photon} \times 6.022 \times 10^{23} \text{ photons/mol} \approx 2.39 \times 10^{5} \text{ J/mol} \).

Key Concepts

Wavelength to Energy Conversion

The concept of wavelength to energy conversion is fundamental in understanding the behavior of light and photons. Light, including visible light like green light, can be described by its wavelength, which is the distance between successive peaks of the wave. Photons, which are particles of light, carry energy. You can determine the energy of a photon if you know its wavelength. The formula that connects these two properties is:\[E = \frac{hc}{\lambda}\]Where:

• E is the energy of the photon in joules ( \(\text{J}\))
• h is Planck's constant ( \(6.626 \times 10^{-34} \text{ J s}\))
• c is the speed of light ( \(3.00 \times 10^{8} \text{ m/s}\))
• \(\lambda\) is the wavelength of the photon in meters ( \(\text{m}\))

To find the energy of a photon, it is crucial to ensure the wavelength is in meters. If given in nanometers, as is common in many problems, convert it using the factor ( \(1 \text{ nm} = 1 \times 10^{-9} \text{ m}\)). This conversion is necessary because the units must align with the constants used in the formula. Once you have the wavelength in meters, substitute it into the equation to find the photon's energy.

Planck's Constant

Planck's constant is named after Max Planck, the physicist who discovered it. This constant is one of the key fundamentals in quantum mechanics. It is symbolized by h, and has a value of \(6.626 \times 10^{-34} \text{ J s}\). This constant represents the proportionality factor between the energy ( \(E\)) of a photon and the frequency ( \(f\)) of its electromagnetic wave:\[E = hf\]Planck's constant signifies that energy is quantized, meaning it is not continuous but comes in "chunks".

• In practical terms, Planck's constant is used to calculate energy from wavelength or frequency using Einstein's energy relationship.
• In the given exercise, it helps in determining the energy of a single photon based on its wavelength.

The central role of Planck's constant in modern physics cannot be overstated. It is fundamental to understanding not just photons, but also the mechanics of atoms and subatomic particles in quantum mechanics.

Avogadro's Number

Avogadro's number \(6.022 \times 10^{23}\) is a fundamental constant that serves as a bridge between the microscopic world of atoms and the macroscopic world of grams and moles. Named after the Italian scientist Amedeo Avogadro, this number represents the number of atoms, ions, or molecules contained in one mole of a substance.- In the context of photons, Avogadro's number tells us how many photons are in one mole. - This is valuable when calculating the energy of a mole of photons, a task common in chemistry and physics problems.To find the total energy for a mole of photons, multiply the energy of a single photon by Avogadro's number. For example, if you previously calculated the energy of a single photon, \(3.9756 \times 10^{-19}\text{ J/photon}\), then the energy of one mole of these photons is:\[3.9756 \times 10^{-19} \text{ J/photon} \times 6.022 \times 10^{23} \text{ photons/mol} \approx 2.39 \times 10^{5} \text{ J/mol}\]This calculation illustrates how Avogadro's number is utilized to relate individual particles to quantities large enough to be measured in laboratory settings. Understanding and using Avogadro's number effectively allows scientists and students alike to transition between atomic scale and tangible quantities.