Question

For a photochemical reaction, \(\mathrm{A} \rightarrow \mathrm{B}\), \(1 \times 10^{-5}\) moles of 'B' were formed on absorption of \(6.626 \times 10^{7}\) erg at \(360 \mathrm{~nm}\). The quantum efficiency (molecules of ' \(\mathrm{B}\) ' formed per photon) is (a) \(1.0\) (b) \(0.25\) (c) \(0.5\) (d) \(2.0\)

Short answer

The quantum efficiency is 1.0 (Option (a)).

Step-by-step solution

Calculate the energy of a photon at 360 nm

Use the equation for energy of a photon, which is given by Planck's equation: \(E = h u\), where \(h\) is Planck's constant and \(u\) is the frequency of the photon. The frequency \(u\) can be found using the speed of light \(c\) and the wavelength \(\lambda\), \(u = \frac{c}{\lambda}\). Given Planck's constant \(h = 6.626 \times 10^{-27}\) erg s and the speed of light \(c = 3 \times 10^{10}\) cm/s, calculate the energy of one photon.

Calculate the total number of photons absorbed

Using the total energy absorbed, calculate the number of photons by dividing the total energy by the energy of one photon: \(\text{Number of photons} = \frac{\text{Total Energy absorbed}}{\text{Energy per photon}}\).

Calculate quantum efficiency

Quantum efficiency is given by the number of molecules of 'B' formed divided by the number of photons absorbed. Use the values calculated in step 1 and step 2 to find the quantum efficiency.

Key concepts

Planck's Equation

Understanding the fundamental aspects of photochemical reactions involves appreciating the concept of Planck's equation. This pivotal equation establishes a key relationship between the energy of a photon and its frequency. It can be expressed as:
\[ E = h u \]
Here, \( E \) represents the energy of the photon, \( h \) is Planck's constant (which has a value of \( 6.626 \times 10^{-34} \) Joules per second), and \( u \) (nu) is the frequency of the photon. Typically, since energy in photonic reactions is often considered in ergs, Planck's constant would accordingly be used as \( 6.626 \times 10^{-27} \) erg s. Employing Planck's equation allows us to determine the precise energy of a single photon, which is a critical step in understanding how much energy is needed to facilitate a given photochemical reaction.

Energy of a Photon

A photon is a fundamental particle of light that carries energy which is directly related to its electromagnetic frequency. According to Planck's equation, the energy contained in a single photon can be calculated by multiplying the photon's frequency by Planck's constant.
If the frequency is not directly provided, it's often derived through the relationship between frequency (\( u \)) and wavelength (\( \lambda \)), since \( u = \frac{c}{\lambda} \) where \( c \) is the speed of light. Knowing that the energy of photons is the key to enabling photochemical processes—where light causes or affects a chemical change—the accurate calculation of this energy is crucial in quantifying and predicting the efficiency of such reactions.

Quantum Efficiency Calculation

Quantum efficiency quantifies the efficiency of a photochemical reaction in terms of the number of photochemical events per photon absorbed. To calculate this efficiency, one must first know the total number of molecules of product formed and the total number of photons absorbed during the reaction. The formula is given by:
\[ \text{Quantum efficiency} = \frac{\text{Number of molecules of 'B' formed}}{\text{Number of photons absorbed}} \]
Applying this calculation to the mentioned photochemical reaction, \(A \rightarrow B\), helps us understand how efficiently the absorbed photons are being utilized to produce the product 'B'. A high quantum efficiency indicates an effective use of photons in the reaction, which is a desirable characteristic in processes like solar energy conversion and photo-synthesis.

Frequency and Wavelength Relationship

The frequency (\( u \)) of a photon is inversely proportional to its wavelength (\( \lambda \)). This relationship is imperative to comprehend as it allows us to connect the measurable wavelength of light to the less-tangible concept of frequency, which is crucial for calculating energy via Planck's equation. The mathematical expression of this relationship is:
\[ u = \frac{c}{\lambda} \]
where \( c \) represents the constant speed of light, approximately \( 3 \times 10^{10} \) cm/s in a vacuum. Since light of different wavelengths carries photons with different energies, this relationship helps us establish that shorter wavelengths correspond to higher frequencies and, therefore, higher photon energies. This concept is foundational when studying photochemistry, as it influences the outcomes of photon-material interactions substantially.