Question

An electron returns from an excited state to its ground state, emitting a photon at \(\lambda=500 \mathrm{~nm}\). What would be the magnitude of the energy change if this process were repeated such that a mole of these photons were emitted? A. \(3.98 \times 10^{-19} \mathrm{~J}\) B. \(3.98 \times 10^{-21} \mathrm{~J}\) C. \(2.39 \times 10^5 \mathrm{~J}\) D. \(2.39 \times 10^3 \mathrm{~J}\)

Short answer

Option C: \(2.39 \times 10^5 \, \text{J}\)

Step-by-step solution

Identify the Given Values

The wavelength of the emitted photon is given as \(\lambda=500 \, \mathrm{~nm}\) (which is equivalent to \(500 \, \times \, 10^{-9} \, \mathrm{~m}\)).

Use the Formula for Energy of a Photon

The energy of a single photon can be calculated using the formula \(E = \frac{hc}{\lambda}\) where h is Planck's constant \(h = 6.626 \times 10^{-34} \, \mathrm{~J \cdot s}\) and c is the speed of light \(c = 3 \times 10^8 \, \mathrm{~m/s}\).

Calculate the Energy of One Photon

Substitute the values into the formula: \[ E = \frac{(6.626 \times 10^{-34} \, \mathrm{~J \cdot s}) \times (3 \times 10^8 \, \mathrm{~m/s})}{500 \times 10^{-9} \, \mathrm{~m}} = 3.98 \times 10^{-19} \, \text{J} \]

Calculate the Energy for a Mole of Photons

One mole of photons corresponds to Avogadro's number of photons \(N_A = 6.022 \times 10^{23}\). Multiply the energy of one photon by Avogadro's number: \[ E_{\text{mole}} = (3.98 \times 10^{-19} \, \text{J/photon}) \times (6.022 \times 10^{23} \, \text{photons/mol}) = 2.39 \times 10^{5} \, \text{J} \]

Select the Correct Answer

Based on the calculations, the energy change for one mole of these photons is \(2.39 \times 10^5 \, \text{J}\). Therefore, the correct answer is Option C.

Key concepts

Planck's constant

Planck's constant, denoted as \( h \), is a fundamental constant in physics that plays a critical role in the field of quantum mechanics. It relates to the energy of a photon to its frequency through the equation: \( E = h u \), where \( E \) is the energy, \( h \) is Planck's constant, and \( u \) is the frequency.
The value of Planck's constant is approximately \( 6.626 \times 10^{-34} \, \mathrm{~J \cdot s} \).
This constant helps us understand processes at the quantum level, such as the energy changes that occur when an electron transitions between different energy states.
In the given problem, we use Planck's constant to calculate the energy of a photon emitted by an electron transitioning between energy states.

Avogadro's number

Avogadro's number, represented as \( N_A \), is a fundamental constant in chemistry that describes the number of atoms, ions, or molecules in one mole of a substance.
The value of Avogadro's number is approximately \( 6.022 \times 10^{23} \) particles per mole.
This number allows us to convert between microscopic scale (individual particles) and macroscopic scale (moles of a substance).
In the exercise, to find the total energy change for a mole of photons, we multiply the energy of a single photon by Avogadro's number, linking the microscopic energy value to a macroscopic quantity relevant in chemistry.

wavelength and energy relationship

The relationship between the wavelength of a photon and its energy is given by the equation: \( E = \frac{hc}{\lambda} \), where
• \( E \) is the energy,
• \( h \) is Planck's constant,
• \( c \) is the speed of light, and
• \( \lambda \) is the wavelength.
This equation shows that energy is inversely proportional to wavelength: shorter wavelengths correspond to higher energies, and longer wavelengths correspond to lower energies.
In the problem provided, we use this relationship to find the energy of a photon with a wavelength of 500 nm. By plugging in the values for \( h \) and \( c \), and converting the wavelength to meters, we calculate the energy of each photon.

mole concept in chemistry

The mole concept is a fundamental principle in chemistry that allows chemists to count atoms, molecules, and ions by weighing them.
One mole of any substance contains \( 6.022 \times 10^{23} \) particles (Avogadro's number).
This concept helps bridge the gap between the atomic and macroscopic scales.
In the photon energy calculation exercise, we use the mole concept to determine how much energy is involved when a mole of photons, each with a certain energy, are emitted. By multiplying the energy of one photon by the number of photons in a mole, we find the total energy change for the process, making it practical for laboratory and real-world applications.