Question

Photodissociation of water $$ \mathrm{H}_{2} \mathrm{O}(l)+h \nu \longrightarrow \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) $$ has been suggested as a source of hydrogen. The \(\Delta H_{\mathrm{rxn}}^{\circ}\) for the reaction, calculated from thermochemical data, is \(285.8 \mathrm{~kJ}\) per mole of water decomposed. Calculate the maximum wavelength (in \(\mathrm{nm}\) ) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?

Short answer

The maximum wavelength is 419 nm; sunlight is feasible for this process.

Step-by-step solution

Understanding the Problem

We are given the reaction for the photodissociation of water into hydrogen and oxygen. The enthalpy change for this reaction is \( \Delta H_{\mathrm{rxn}}^{\circ} = 285.8 \mathrm{~kJ/mol} \). The task is to calculate the maximum wavelength of light that can provide this energy and determine if sunlight could be a feasible energy source for this reaction.

Convert Energy Units

The energy for one mole of photon is given in kJ, but we need to convert this to Joules because the energy of a photon is usually expressed in Joules. In this case, 1 kJ = 1000 J, thus:\[ 285.8 \mathrm{~kJ/mol} = 285800 \mathrm{~J/mol} \]

Calculate Energy per Photon

The energy required to dissociate one molecule is the energy per mole divided by Avogadro's Number (\(6.022 \times 10^{23} \) mol\(^{-1}\)). Therefore, the energy per photon, \(E\), is calculated as follows: \[ E = \frac{285800 \mathrm{~J/mol}}{6.022 \times 10^{23} \text{ mol}^{-1}} = 4.747 \times 10^{-19} \mathrm{~J} \]

Use the Energy-Wavelength Relationship

To find the wavelength, use the equation for the energy of a photon: \( E = \frac{hc}{\lambda} \), where \(h\) is Planck's constant \( (6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}) \) and \(c\) is the speed of light \( (3.00 \times 10^8 \mathrm{~m/s}) \). Rearranging for \(\lambda\), we have:\[ \lambda = \frac{hc}{E} \]

Calculate Maximum Wavelength

Substitute the known values into the rearranged equation to compute the maximum wavelength. \[ \lambda = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^{8}}{4.747 \times 10^{-19}} = 418.5 \times 10^{-9} \mathrm{~m} = 418.5 \mathrm{~nm} \] Thus, the maximum wavelength of light that can provide the necessary energy is approximately 419 nm.

Assess Feasibility Using Sunlight

Sunlight contains significant amounts of ultraviolet, visible, and infrared radiation. The wavelength 419 nm falls into the blue-violet region of the visible spectrum. Since sunlight includes these wavelengths, using sunlight for this reaction is feasible in principle.

Key concepts

Hydrogen production

Hydrogen production through photodissociation involves breaking down water into hydrogen and oxygen using light energy. This process is considered sustainable because it uses water, a renewable resource, and produces hydrogen, a clean fuel. When water absorbs the necessary energy from photons, the bonds between hydrogen and oxygen can be broken, resulting in the release of hydrogen gas. This hydrogen can then be used in various applications, such as in fuel cells for generating electricity, powering vehicles, or heating.
The photodissociation of water, represented by the chemical equation $$ \mathrm{H}_{2} \mathrm{O}(l) + h u \longrightarrow \mathrm{H}_{2}(g) + \frac{1}{2} \mathrm{O}_{2}(g) $$ establishes a direct link between light energy and hydrogen production. Photodissociation offers a potential method for hydrogen generation but requires careful consideration of the energy requirements and the efficiency of various light sources.

Enthalpy change

The enthalpy change, denoted as \( \Delta H_{\mathrm{rxn}}^{\circ} \), is a crucial factor in understanding the energy dynamics of any chemical reaction. In the context of water photodissociation, the enthalpy change is positive (\(+285.8 \mathrm{~kJ/mol}\)), indicating that the process is endothermic. This means that energy must be absorbed from the surroundings for the reaction to occur.
A positive enthalpy change translates into a requirement for an external source of energy, such as light, to drive the decomposition of water into hydrogen and oxygen. The magnitude of \( \Delta H_{\mathrm{rxn}}^{\circ} \) provides insight into the total amount of energy necessary to break the chemical bonds in water molecules and to facilitate the release of hydrogen gas. Understanding this concept helps in assessing the energy inputs needed for efficient hydrogen production.

Energy-wavelength relationship

The energy-wavelength relationship describes the inverse connection between a photon's energy and the wavelength of light. According to Planck's equation, the energy of a photon \( E \) is given by:\[ E = \frac{hc}{\lambda} \] where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \mathrm{~J \cdot s}) \), and \( c \) is the speed of light \((3.0 \times 10^8 \mathrm{~m/s}) \). Rearranging the formula to find the wavelength \( \lambda \) gives:\[ \lambda = \frac{hc}{E} \] This equation illustrates that shorter wavelengths correspond to higher energy photons, while longer wavelengths correspond to lower energy photons. In the exercise, we calculated that the maximum wavelength required to supply enough energy for the photodissociation of water is \( 419 \mathrm{~nm} \). This wavelength is within the visible spectrum, which confirms that certain visible light photons possess sufficient energy to promote this chemical reaction.

Feasibility of using sunlight

The feasibility of using sunlight as a source of energy for water photodissociation depends on the wavelengths present in sunlight. Sunlight provides a spectrum of light, including ultraviolet (UV), visible, and infrared (IR) light, each with varying energies.
Since our calculated maximum wavelength for effective photodissociation is \( 419 \mathrm{~nm} \), which falls in the visible range (blue-violet light), it overlaps with the spectrum of sunlight. This means that sunlight potentially contains enough energy to drive the reaction naturally.
Photodissociation using sunlight is appealing due to its renewability and abundance. However, practical implementation may face challenges such as capturing and efficiently using photons of the right wavelength, as well as managing the intermittent nature of sunlight. Overcoming these hurdles could make sunlight a sustainable and viable energy source for hydrogen production.