Question

Use bond energies to estimate the maximum wavelength of light that will cause the reaction $$\mathrm{O}_{3} \stackrel{\mathrm{hr}}{\longrightarrow} \mathrm{O}_{2}+\mathrm{O}$$

Short answer

The maximum wavelength of light that will cause the reaction of ozone (O₃) dissociating into an oxygen molecule (O₂) and a single oxygen atom (O) is approximately \(720.4\, \mathrm{nm}\).

Step-by-step solution

Determine the bond energies involved in the reaction

First, we need to find the bond energies for the relevant molecular species involved in the reaction. We can find these in a reference source like a chemistry book or a database of bond energies. For this exercise, let's assume the average bond energies are as follows: - Ozone (O₃): 498 kJ/mol - Dioxygen (O₂): 498 kJ/mol

Calculate the energy required for the reaction using bond energies

Next, we need to calculate the energy required for the reaction to occur. This can be done using the following equation: Energy required = Energy of bonds broken - Energy of bonds formed In this case, Energy required = Energy of O₃ bonds broken - Energy of O₂ bonds formed We have the bond energies from Step 1, so we can now calculate the energy required: Energy required = 498 kJ/mol - 498 kJ/mol = 0 kJ/mol However, this calculation does not seem right as we know that energy must be required to break the bond in ozone and create an oxygen atom. So, let’s reconsider our bond energies. The energy we found for ozone is given per bond, but there are 3 bonds in ozone (forming a resonance structure). So, breaking one bond requires a third of the bond energy: Energy required to break one O-O bond in O₃ = 498 kJ/mol ÷ 3 = 166 kJ/mol Now let's redo the energy required calculation: Energy required = 166 kJ/mol - 498 kJ/mol = -332 kJ/mol Since the energy is negative, the reaction is actually exothermic. However, we were asked to calculate the energy needed to initiate the reaction. Let's assume the energy to initiate the reaction is equal to breaking one O-O bond in O₃: Energy required to initiate the reaction = 166 kJ/mol

Convert the energy to wavelength using the relationship between energy and wavelength of light

Finally, we need to convert the energy required to initiate the reaction to a wavelength of light. We can do this using Planck's equation: \(E = h * c / λ\) Where E is the energy, h is Planck's constant (6.626e-34 J∙s), c is the speed of light (3.00e8 m/s), and λ is the wavelength. First, we need to convert the energy required to initiate the reaction from kJ/mol to J/molecule: \(166\, \text{kJ/mol}\times \frac{10^3\text{ J}}{1\,\mathrm{kJ}}\times\frac{1\,\mathrm{mol}}{{6.022\times10^{23}}\,\text{molecules}} \approx 2.752\times10^{-19}\,\mathrm{J}\) Now we can solve for λ using Planck's equation: λ = h * c / E \(λ = \frac{6.626\times 10^{-34}\,\mathrm{J\,s}\times 3.00\times 10^{8}\,\mathrm{m/s}}{2.752\times 10^{-19}\,\mathrm{J}} \approx 7.204\times 10^{-7}\,\mathrm{m}\) Converting this to nanometers (nm): \(λ = 7.204\times 10^{-7}\,\mathrm{m}\times 10^9\, \frac{\mathrm{nm}}{\mathrm{m}} = 720.4\,\mathrm{nm}\) So, the maximum wavelength of light that will cause the reaction is approximately 720.4 nm.

Key concepts

Understanding Chemical Reactions

In chemistry, chemical reactions are processes where substances, known as reactants, transform into different substances called products. These transformations involve the making and breaking of chemical bonds, which requires or releases energy. The kind of energy involved in the breaking and forming of bonds is often thermal, light, or electrical energy.

During a chemical reaction, the bonds between atoms in the reactants need to be broken so that new bonds can form in the products. The strength of these bonds is quantified by bond energies, which indicate the amount of energy required to break the bond between two atoms. Bond energies are unique to each type of bond and are often provided in units of kilojoules per mole (kJ/mol). To predict whether a reaction will occur and determine the conditions needed, understanding these energies is critical.

For instance, in the example given, when ozone (O₃) decomposes to dioxygen (O₂) and an oxygen atom, bond energies guide us in estimating the energy requirement. If the energy required is positive, the reaction is endothermic, meaning it absorbs energy. If negative, it's exothermic and releases energy. In some cases, as in the provided solution, an error in the initial bond energy calculation might lead to an incorrect energy requirement. Thus, carefully reconsidering these values and understanding bond energies in the context of chemical reactions is crucial.

Planck's Equation and Energy Quantum

A cornerstone of quantum mechanics is Planck's equation, which bridges the gap between the energy of particles and the physics of waves. Simply put, it asserts that the energy (E) of a photon (a particle of light) is directly proportional to its frequency (f), and thereby inversely proportional to its wavelength (λ). The equation is given by:
\[\begin{equation} E = h \times f \end{equation}\]
where h is Planck's constant (approximately 6.626 x 10^-34 J∙s). Since frequency and wavelength are related through the speed of light (c), the equation can also be expressed in terms of wavelength: \[\begin{equation} E = \frac{h \times c}{λ} \end{equation}\]
Understanding Planck's equation allows us to convert between the energy required to initiate a chemical reaction and the corresponding wavelength of light. This conversion is necessary when estimating the maximum wavelength of light needed to stimulate a particular reaction, especially in photochemical processes where light provides the requisite energy for bond breakage. The equation shows that higher energy quanta correspond to shorter wavelengths and vice versa, linking the quantum concept with classical wave behavior of light.

The Significance of Light Wavelength

The wavelength of light is a fundamental concept in both optics and quantum chemistry. It refers to the distance between two consecutive crests (or troughs) in a wave of light and determines many of its properties, including its color and the type of chemical reaction it can induce when absorbed by a substance. Light with a short wavelength (such as ultraviolet) carries more energy and is often used to initiate photochemical reactions, whereas light with a longer wavelength (like infrared) carries less energy.

In the context of the exercise, we are looking for the 'maximum wavelength of light' that can cause a chemical reaction. This refers to the longest wavelength, hence, the lowest energy of light that is still capable of breaking a chemical bond, such as the bond in an ozone molecule. The wavelength and energy of light are inversely related—as one increases, the other decreases. Therefore, it's important for students to grasp that identifying this maximum wavelength provides insight into the minimum energy required to initiate the reaction.

Practical implications of this knowledge can be seen in designing photochemical experiments, where light is required to start a reaction. The wavelength must be at or below this calculated maximum to ensure the reaction will proceed. Moreover, it highlights the necessity of selecting appropriate light sources for specific chemical applications.