Question

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

Short answer

The energy required for the reaction is 8 kJ/mol, which is equal to 8000 J/mol. Using Planck's equation \(\lambda = \dfrac{h\cdot c}{E}\) and plugging in the values, we find the maximum wavelength to be approximately \(4.117 \times 10^{-7}\) meters or 411.7 nm.

Step-by-step solution

Calculate the bond energy for the given reaction

We need to calculate the energy required to cause the given reaction, which can be estimated using bond energies. The reaction is: \( O_3 \longrightarrow O_2 + O \) First, we need to look up the bond energies for O3 and O2. According to reference data, the average of the double and single bond in O3 is approximately 490 kJ/mol, while the bond energy for the O=O double bond in O2 is 498 kJ/mol.

Calculate the energy required for the reaction

To find the energy required for the reaction, subtract the bond energy of the products from the bond energy of the reactants. Energy required = Bond energy of O2 + Bond energy of O - Bond energy of O3 Energy required = 498 kJ/mol - 490 kJ/mol = 8 kJ/mol

Convert the energy to Joules

To use the Planck's equation, we need the energy in Joules. We can convert the energy from kJ/mol to J/mol by using the conversion factor: 1 kJ/mol = 1000 J/mol So, Energy required = 8 kJ/mol * 1000 J/mol = 8000 J/mol

Calculate the maximum wavelength using Planck's equation

The Planck's equation is: \( E = \dfrac{h\cdot c}{\lambda} \) Where E is the energy, h is the Planck's constant, c is the speed of light, and λ is the wavelength. Rearranging the equation to find λ: \( \lambda = \dfrac{h\cdot c}{E}\) Using the Planck's constant value of approximately 6.626 x 10^-34 Js and the speed of light value of approximately 3 x 10^8 m/s, we plug in the values: \( \lambda = \dfrac{(6.626 \times 10^{-34} \text{ Js})(3 \times 10^8 \text{ m/s})}{8000 \text{ J/mol}}\)

Calculate and present the answer

Now, we can calculate the maximum wavelength: \( \lambda = \dfrac{(6.626 \times 10^{-34})(3 \times 10^8)}{8000} = 2.47825 \times 10^{-29} \text{ m/mol} \) To convert it to meters per molecule, we need to divide by Avogadro's number (approximately 6.022 x 10^23): \( \lambda = \dfrac{2.47825 \times 10^{-29}}{6.022 \times 10^{23}} = 4.117 \times 10^{-7} \text{ m} \) Therefore, the maximum wavelength of light that will cause the given reaction is approximately \(4.117 \times 10^{-7}\) meters or 411.7 nm.

Key concepts

Chemical Reaction Energy Estimation

Understanding the amount of energy involved in a chemical reaction is crucial for predicting the behavior of reactants and products. Energy estimation involves calculating the energy required to break bonds in reactants and form new bonds in products. Each chemical bond has a certain energy associated with it, known as 'bond energy', which is specific to the types of atoms involved and the nature of the bond (single, double, triple, etc.).

To estimate the energy of a reaction, you'll need to consider the overall energy change, which involves subtracting the total bond energies of the reactants from the total bond energies of the products. If the result is positive, the reaction is endothermic, meaning it absorbs energy. Conversely, if the result is negative, the reaction is exothermic, releasing energy. This calculation gives us an insight into whether a reaction will require an input of energy to occur or if it will release energy that could possibly be harvested for other uses.

Planck's Equation

Planck's equation is a fundamental concept in quantum mechanics, bridging the gap between the energy of photons and the frequency of light. The equation is often represented as:
\[ E = h u \]
Here, \(E\) is the energy of a photon, \(h\) is Planck's constant (approximately \(6.626 \times 10^{-34} Js\)), and \(u\) (or \(f\) for frequency) is the frequency of the radiation. If we express frequency in terms of wavelength (\(\lambda\)), the speed of light (\(c\)), we can rewrite the equation as:
\[ E = \frac{h \cdot c}{\lambda} \]
Planck's equation is essential for understanding how much energy is associated with photons of different wavelengths. This is particularly relevant for photochemical reactions where light is used to initiate or alter chemical bonds.

Wavelength of Light

The wavelength of light is a physical property that describes the distance over which the wave's shape repeats. It is usually denoted by \(\lambda\) and is measured in meters. Light waves can have wavelengths across a vast spectrum, known as the electromagnetic spectrum. This spectrum includes everything from gamma rays, which have very short wavelengths, to radio waves, which have very long wavelengths.

Visible light, which is the part of the spectrum detectable by the human eye, ranges from violet (with the shortest wavelength of around \(380\) nm) to red (with the longest wavelength of around \(750\) nm). The particular wavelength of light can affect how it interacts with matter, including how it can induce chemical reactions as in photochemical processes.

Avogadro's Number

Avogadro's number, approximately \(6.022 \times 10^{23}\), is a fundamental constant in chemistry representing the number of constituent particles (usually atoms or molecules) in one mole of a substance. A mole is one of the base units in the International System of Units (SI) for the amount of substance.

Avogadro's number allows chemists to count particles by weighing, as it is impractical to count each particle individually due to their inconceivably small size and vast quantity. This constant is essential when converting between the number of moles and the number of particles, especially when dealing with reactions at the molecular level, such as calculating the energy per photon needed to initiate a reaction, as seen in the original exercise.