Question

Calculate the energy in joules of a photon of red light having a frequency of \(4.0 \times 10^{14} \mathrm{~Hz}\). What is the energy of one mole of these photons?

Short answer

The energy of one photon of red light with a frequency of 4.0 x 10^14 Hz is 2.6504 x 10^-19 Joules. The energy of one mole of these photons is 1.596 x 10^5 Joules.

Step-by-step solution

Identify the given frequency and Planck's constant

Identify the given frequency of the red light photon, which is stated to be 4.0 x 10^14 Hz. Recall that the energy (E) of a photon can be calculated using the formula E = hν, where h is Planck's constant (h = 6.626 x 10^-34 J·s) and ν is the frequency of the photon.

Calculate the energy of one photon

Substitute the given frequency and Planck's constant into the equation E = hν to find the energy of one photon. E = (6.626 x 10^-34 J·s) x (4.0 x 10^14 Hz).

Calculate energy per photon

Multiply Planck's constant by the frequency to find the energy of a single photon. E = 6.626 x 10^-34 J·s x 4.0 x 10^14 Hz = 2.6504 x 10^-19 Joules.

Find the Avogadro's number

Recall Avogadro’s number, which is the number of particles in one mole. Avogadro's number (NA) is 6.022 x 10^23 particles per mole.

Calculate the energy of one mole of photons

Multiply the energy of one photon by Avogadro's number to calculate the energy of one mole of these photons. E(mole) = 2.6504 x 10^-19 Joules/photon x 6.022 x 10^23 photons/mole.

Compute the energy for a mole of photons

Perform the multiplication to find the total energy for one mole of photons. E(mole) = (2.6504 x 10^-19) x (6.022 x 10^23) = 1.596 x 10^5 Joules/mole.

Key concepts

Planck's constant

Understanding Planck's constant is crucial when studying the quantum aspects of physics. It is denoted by the symbol 'h' and has a value of approximately 6.626 x 10^-34 joules per second (J·s). This constant is fundamental in the quantum theory of light and matter, introducing the concept that energy is quantized. Essentially, it reveals that energy can be exchanged only in discrete units called 'quanta'.

When dealing with photon energy, Planck's constant connects the frequency of the photon to its energy. The relationship indicates that as the frequency increases, so does the energy, meaning that photons of light with higher frequencies carry more energy. This is a cornerstone in understanding phenomena such as the photoelectric effect, where electrons are ejected from materials after the absorption of photons with sufficient energy.

Frequency of photon

The frequency of a photon, represented by the Greek letter nu (ν), is the number of wave cycles—or peaks and troughs—that pass a given point per second. It's measured in hertz (Hz), where one Hz equates to one cycle per second.

A photon's frequency is directly related to its energy, and as per the energy formula for a photon, higher frequencies mean higher energies. When looking at different types of light, red light typically has a lower frequency compared to, say, violet light, which indicates that red light photons carry less energy. To visualize frequency, one could imagine waves on the ocean—waves that are closer together have a higher frequency than those further apart.

Avogadro's number

In chemistry, Avogadro's number is a key constant that reveals how many constituent particles, such as molecules, atoms, or in this case, photons, are contained in one mole of substance. The numerical value is approximately 6.022 x 10^23 particles per mole.

This number isn't arbitrary; it was chosen to give atomic and molecular weights in grams per mole, which generally correlates with the mass of an individual atom or molecule in atomic mass units. Therefore, knowing Avogadro's number allows chemists and physicists to count particles by weighing them. When we talk about the energy of a mole of photons, we're using Avogadro's number to scale up from the energy of one photon to the collective energy of a mole's worth of them.

Energy formula for photon

The energy formula for a photon is a fundamental equation in quantum physics and can be expressed as \( E = hu \), where \( E \) is the photon's energy, \( h \) is Planck's constant, and \( u \) is the frequency of the photon. This formula derives from the quantum theory proposition that light has properties of both particles and waves.

To apply this formula, simply multiply Planck's constant by the frequency of the photon. The result is the energy in joules for a single photon. For example, the solution steps provided show how to calculate the energy of a photon of red light given its frequency, using the known value of Planck's constant.

Mole concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we experience daily. It allows chemists to count out a specific number of molecules or atoms by simply weighing them. One mole corresponds to Avogadro's number of particles, and the mass of one mole of a substance is its molecular or atomic mass expressed in grams.

To calculate the amount of energy contained in a mole of photons, you multiply the energy of one photon by Avogadro's number. This total energy is often an essential calculation in chemical reactions involving bond energies and in physical processes like the generation of light by lasers or LEDs.