Question

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

Short answer

The maximum wavelength of light that will cause the reaction can be calculated using the dissociation energy of O3, which is approximately 447 kJ/mol, and Planck's equation. First, convert the bond energy to J/photon by dividing by Avogadro's number and multiplying by 1000, yielding 7.42 x 10^-19 J. Then, use the equation \(\lambda = \frac{hc}{E}\) with Planck's constant (h) and the speed of light (c) to find the wavelength \(\lambda\). The result is approximately 268 nm.

Step-by-step solution

Identify the bond energies in the reaction

We need to find the bond energies between O3, O2, and O. The bond energy to break O3 into O2 and O is known as the dissociation energy, which is approximately 106.7 kcal/mol, or 447 kJ/mol.

Calculate the energy of the incoming light

We have just identified the energy required to cause the reaction to occur (which is the bond energy of breaking O3 to O2 and O). According to the conservation of energy principle, this energy should be equal to the energy of the incoming light. Therefore, the energy of the light should be equal to 447 kJ/mol.

Relate the energy of light to its wavelength

We can now relate the energy of light to its wavelength using Planck's equation: \[ E = h\nu = \dfrac{hc}{\lambda} \] where E is the energy of the light, h is Planck's constant (6.626 x 10^-34 Js), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength of light. Rearrange the equation to find the wavelength: \[ \lambda = \dfrac{hc}{E} \]

Convert energy from kJ/mol to J

Since the bond energy is given in kJ/mol, we need to convert it to joules (J) by multiplying the value by 1000 (1 kJ = 1000 J) and dividing by Avogadro's number (6.022 x 10^23) to obtain the energy value in the unit of J per photon: \[ E_{J} = \dfrac{447\, kJ/mol \times 1000\, J/kJ}{6.022 \times 10^{23}\, photons/mol} = 7.42 \times 10^{-19}\, J \]

Calculate the wavelength

Now that we have the energy in the units of Joules, we can plug this value into the equation we found in Step 3: \[ \lambda = \dfrac{hc}{E} = \dfrac{(6.626 \times 10^{-34}\, Js)(3.00 \times 10^8\, m/s)}{7.42 \times 10^{-19}\, J} = 2.68 \times 10^{-7}\, m \] Finally, we can convert the wavelength to nanometers to get the final answer: \[ \lambda = 2.68 \times 10^{-7}\, m \times \dfrac{10^9\, nm}{1\, m} = 268\, nm \] So, the maximum wavelength of light that will cause the reaction is approximately 268 nm.

Key concepts

Wavelength Calculation

Understanding wavelength calculation involves relating the energy of light to its wavelength. Since energy and wavelength are connected through Planck's equation, which we'll explore later, determining wavelength means calculating the specific distance over which wave patterns repeat. This is crucial in deciding how much energy a photon must have to make a chemical reaction happen, like breaking down ozone. Calculation begins once energy values are known, as shown in chemical reactions requiring precise energy input. This allows for determining the maximum wavelength by converting energy units and using the correct formula to find this distance.

Planck's Equation

Planck's equation forms the basis for linking light energy to frequency and wavelength. The equation is expressed as:

• \( E = hu = \dfrac{hc}{\lambda} \)

Here, \( E \) represents energy, \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ Js})\), \( c \) is the speed of light \((3.00 \times 10^8 \text{ m/s})\), and \( \lambda \) is wavelength.

This equation shows that energy and wavelength are inversely related. When energy rises, the wavelength shortens. By rearranging the formula to \( \lambda = \dfrac{hc}{E} \), the wavelength can be determined once energy is calculated, guiding the transformation of energy levels into their corresponding wavelengths.

Energy Conversion

Energy conversion is key in adjusting energy units for calculations. To convert energy from kJ/mol to J/photon, it's essential to multiply by 1000 (since 1 kJ = 1000 J) and then divide by Avogadro's number \((6.022 \times 10^{23} \text{ photons/mol})\). This allows us to express bond energy in terms of energy per photon:

• \( E_{J} = \dfrac{447 \, \text{kJ/mol} \times 1000 \, \text{J/kJ}}{6.022 \times 10^{23} \, \text{photons/mol}} \approx 7.42 \times 10^{-19} \, \text{J} \)

Converting energy to joules per photon simplifies the use in Planck's equation, enhancing the ease of calculating wavelength. This conversion process underscores understanding the nature of energy in reactions and its application to real-world scenarios.

Dissociation Energy

Dissociation energy refers to the specific energy required to break bonds in a molecule, such as converting \( \text{O}_3 \) to \( \text{O}_2 \) and \( \text{O} \). In chemical reactions, knowing this energy is crucial since it is the threshold needed to drive the reaction forward. For ozone, this dissociation energy is approximately 447 kJ/mol.

• It highlights how strong bonds hold atoms together and the exact energy needed to overcome these forces.

This concept connects directly with how energy and wavelength interact, as dissociation energy signifies the minimum quantity that incoming light must possess to cause bond breakage. Understanding dissociation enables clearer insight into reaction dynamics, energy needs, and consequent effects of light on molecules.