Question

Photo-dissociation of water \(\mathrm{H}_{2} \mathrm{O}(\mathrm{I})+h v\) \(\rightarrow \mathrm{H}_{2}(\mathrm{~g})+1 / 2 \mathrm{O}_{2}(\mathrm{~g})\) has been suggested as a source of hydrogen. The heat absorbed in this reaction is \(289.5 \mathrm{~kJ} / \mathrm{mole}\) of water decomposed. The maximum wavelength that would provide the necessary energy assuming that one photon causes the dissociation of one water molecule is \((1 \mathrm{eV}=96.5 \mathrm{~kJ} / \mathrm{mol})\) (a) \(413.33 \mathrm{~nm}\) (b) \(826.67 \mathrm{~nm}\) (c) \(206.67 \mathrm{~nm}\) (d) \(4.3 \mathrm{~nm}\)

Short answer

The maximum wavelength that would provide the necessary energy is about 413.33 nm, so the correct option is (a) 413.33 nm.

Step-by-step solution

Convert energy required per mole to energy per photon

First, convert the heat absorbed from kJ/mole to Joules per mole. 1 kJ = 1000 J. Then use the energy of one mole of photons (Avogadro's number) to find the energy per photon.

Use the Energy-Photon Wavelength Equation

Use the equation that relates the energy of a photon to its wavelength, which is given by the equation E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon.

Solve for Maximum Wavelength

Rearrange the equation to solve for λ and substitute the values for h, c, and the energy per photon calculated in Step 1 to find the maximum wavelength.

Check the Possible Answers

Check the calculated wavelength against the multiple-choice options to find the correct answer.

Key concepts

Energy Per Photon

The concept of energy per photon is crucial in understanding photo-dissociation and many other photochemical reactions. In physics, a photon is a particle representing a quantum of light or other electromagnetic radiation. The energy of a single photon is determined by its frequency (u), which is related to the speed of light (c) and its wavelength (lambda) through the equation c = lambda u.

When we talk about the energy per photon in the context of photo-dissociation of water, we refer to the individual energy packets that photons must carry to break the chemical bonds in water molecules. This energy can be calculated by using the equation E = h u, where h is Planck's constant, and u is the frequency of the photon. However, given that photon frequency and wavelength are inversely proportional (the higher the frequency, the shorter the wavelength), this relationship can also be shown as E = hc/lambda.

Understanding energy per photon is vital because it determines whether a photon has sufficient energy to cause photo-dissociation of water molecules. If the energy is too low, the photon cannot break the bonds, and no reaction will occur.

Photon Wavelength Equation

The photon wavelength equation lies at the heart of understanding how light interacts with matter. It is expressed as E = hc/lambda, where E is the energy of the photon, h is Planck's constant, c is the speed of light in a vacuum, and lambda is the wavelength. This equation effectively bridges the gap between the classical wave description of light and its quantum particle perspective.

In the exercise, we use this foundational equation to calculate the maximum wavelength of light required to induce photo-dissociation in water molecules. As the wavelength lambda increases, the energy E of each photon decreases, so there is a maximum wavelength that can still produce the necessary energy to trigger the reaction. Solving for lambda in the equation, we get lambda = hc/E, and we can see that for a larger lambda, we have less energy per photon, based on the fact that both h and c are constants.

Using this equation, students can better understand how high-energy photons with shorter wavelengths can break chemical bonds, while photons with wavelengths longer than the maximum will not have enough energy to cause photo-dissociation.

Planck's Constant

Planck's constant, denoted as h, is a fundamental constant in quantum mechanics. It has a value of approximately 6.626 x 10^-34 joules-seconds. Planck's constant relates the energy of a photon to its frequency, as shown in the equation E = hu. It represents the proportionality constant between the minimum energy increments that can be exchanged and the frequency of the electromagnetic wave.

In our exercise involving photo-dissociation, Planck's constant serves as the bridge between the quantum world of photons and the macroscopic world we can measure, for example in the energy required to break the bonds in water molecules. The immutable nature of Planck's constant allows us to perform precise calculations when dealing with photon energies, frequencies, and wavelengths. It's fundamental to our understanding of quantum phenomena and plays a decisive role when calculating the maximum wavelength capable of inducing the photo-dissociation of water.