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Chapter 20: Problem 59

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

Expert verified
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

01

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.
02

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.
03

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: \[ \lambda = \dfrac{hc}{E} \]
04

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: \[ E_{J} = \dfrac{447\, kJ/mol \times 1000\, J/kJ}{6.022 \times 10^{23}\, photons/mol} = 7.42 \times 10^{-19}\, J \]
05

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.

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