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Chapter 20: Problem 53

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

Short Answer

Expert verified
The maximum wavelength of light that will cause the reaction of ozone dissociating into molecular oxygen and an oxygen atom is approximately 498 nm. This is calculated using bond energies, Planck's equation, and the energy change during the reaction.

Step by step solution

01

Determine the change in bond energies for the reaction

To determine the change in bond energies, we need to calculate the energy required to break the ozone bond and the energy gained from forming the oxygen molecule. According to the bond energies, the energy needed to break O3 into O2 and O is: 螖H = Energy(O3) - (Energy(O2) + Energy(O)) Keep in mind that the energy change during a reaction is determined by the energy of the bonds broken minus the energy of the bonds formed.
02

Planck鈥檚 equation and energy conversion

Planck's equation relates the energy (E) of a photon to its frequency (v) and wavelength (位) as: \(E = h \nu = \dfrac{hc}{\lambda}\), where h is the Planck's constant, c is the speed of light, and 位 is the maximum wavelength of light that can cause the reaction. The energy calculated in the first step will be equal to the energy of the photon absorbed to cause the reaction. So, 螖H = \(\dfrac{hc}{\lambda}\) To solve for 位, we need the values for Planck's constant (h) and the speed of light (c): - Planck's constant (h) = 6.626 脳 10鈦宦斥伌 J路s - The speed of light (c) = 2.998 脳 10鈦� m/s
03

Calculate the value of 螖H

According to the bond energies, we have the following values: - Energy of O3 (O=O bond): 498 kJ/mol - Energy of O2 (O=O bond): 494 kJ/mol - Energy of O (no bond, so 0 kJ/mol) Now, plug in these values in the equation we wrote in Step 1: 螖H = 498 kJ/mol - 494 kJ/mol 螖H = 4 kJ/mol Since we're using Planck's equation, we need to convert 螖H into joules (J): 螖H = 4 脳 10鈦宦� J/mol
04

Solve for the maximum wavelength (位)

Using Planck's equation, plug in the values for h, c, and 螖H, and solve for 位: \(4 脳 10^{-3} \text{ J/mol}= \dfrac{(6.626\times 10^{-34} \text{ J}\cdot \text{s})(2.998\times10^{8} \text{ m/s})}{\lambda}\) \(4 脳 10^{-3}J/mol \cdot \lambda = (6.626\times 10^{-34} J \cdot s)(2.998\times10^{8} m/s)\) \(\lambda = \dfrac{(6.626\times 10^{-34} J \cdot s\times 2.998\times10^{8} m/s)}{4 脳 10^{-3}J/mol} \approx 4.98 \times 10^{-7} m\) The maximum wavelength of light that will cause the reaction is approximately 4.98 脳 10鈦烩伔 m or 498 nm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Planck's equation
Understanding Planck's equation is crucial to grasp how energy is quantized in the realm of quantum mechanics. This fundamental theory connects the energy of a photon, the most elementary particle of light, to its frequency and wavelength.

Planck's equation is mathematically expressed as:
\[E = h u = \frac{hc}{\lambda}\]
where:
  • \(E\) stands for the energy of a photon,
  • \(h\) represents Planck's constant, a crucial universal constant in quantum physics,
  • \(u\) is the frequency of the photon,
  • \(\lambda\) is the wavelength of the light,
  • and \(c\) is the speed of light in a vacuum.
By using this equation, you can calculate the energy of a photon if you know its wavelength or frequency, and vice versa. This concept is pivotal in the study of not just theoretical physics but practical applications, including reaction wavelength estimation in chemistry.
Chemical bond energy
Chemical bond energy refers to the amount of energy required to break a bond between two atoms in a molecule or the energy released when a bond is formed. It's a critical factor that influences how chemical reactions occur and is measured in kilojoules per mole (kJ/mol).

In chemical reactions, breaking bonds requires energy (endothermic process) and forming bonds releases energy (exothermic process). The overall energy change in a reaction is the difference between the energy needed to break the reactant bonds and the energy released upon forming product bonds.

For example, in the exercise provided, the energy change for the reaction involving ozone (\(O_3\)) is calculated by subtracting the energy of the bonds formed in the oxygen molecule (\(O_2\)) and atomic oxygen (\(O\)) from the energy required to break bonds in the ozone molecule. Understanding this helps one predict if a reaction will release heat to the surroundings or absorb it.
Reaction wavelength estimation
Reaction wavelength estimation is a practical application of Planck's equation in chemistry. It allows us to estimate the maximum wavelength of light needed to induce a chemical reaction. By calculating the difference in bond energies, which gives us the energy change (\(\Delta H\)) of the reaction, we can use Planck鈥檚 equation to find the exact wavelength capable of providing that amount of energy.

In the textbook exercise, we solve for the maximum wavelength (\(\lambda\)) by rearranging Planck鈥檚 equation to get:\[\Delta H = \frac{hc}{\lambda}\]
By substituting the calculated change in bond energy (\(\Delta H\)) and known constants for Planck's constant (\(h\)) and the speed of light (\(c\)), we find the wavelength that corresponds to the energy absorbed to cause the reaction. This is fundamental in fields like photochemistry, where light is used to initiate chemical reactions.

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Most popular questions from this chapter

Nitrous oxide \(\left(\mathrm{N}_{2} \mathrm{O}\right)\) can be produced by thermal decomposition of ammonium nitrate: $$ \mathrm{NH}_{4} \mathrm{NO}_{3}(s) \stackrel{\text { heat }}{\longrightarrow} \mathrm{N}_{2} \mathrm{O}(g)+2 \mathrm{H}_{2} \mathrm{O}(l) $$ What volume of \(\mathrm{N}_{2} \mathrm{O}(g)\) collected over water at a total pressure of \(94.0 \mathrm{kPa}\) and \(22^{\circ} \mathrm{C}\) can be produced from thermal decomposition of \(8.68 \mathrm{~g} \mathrm{NH}_{4} \mathrm{NO}_{3} ?\) The vapor pressure of water at \(22^{\circ} \mathrm{C}\) is 21 torr.

A proposed two-step mechanism for the destruction of ozone in the upper atmosphere is a. What is the overall balanced equation for the ozone destruction reaction? b. Which species is a catalyst? c. Which species is an intermediate? d. What is the rate law derived from this mechanism if the first step in the mechanism is slow and the second step is fast? e. One of the concerns about the use of Freons is that they will migrate to the upper atmosphere, where chlorine atoms can be generated by the reaction $$ \mathrm{CCl}_{2} \mathrm{~F}_{2} \stackrel{\mathrm{H}}{\longrightarrow} \mathrm{CF}_{2} \mathrm{Cl}+\mathrm{Cl} $$ Freon-12 Chlorine atoms also can act as a catalyst for the destruction of ozone. The first step of a proposed mechanism for chlorinecatalyzed ozone destruction is \(\mathrm{Cl}(g)+\mathrm{O}_{3}(g) \longrightarrow \mathrm{ClO}(g)+\mathrm{O}_{2}(g)\)

While selenic acid has the formula \(\mathrm{H}_{2} \mathrm{SeO}_{4}\) and thus is directly related to sulfuric acid, telluric acid is best visualized as \(\mathrm{H}_{6} \mathrm{TeO}_{6}\) or \(\mathrm{Te}(\mathrm{OH})_{6}\) a. What is the oxidation state of tellurium in \(\mathrm{Te}(\mathrm{OH})_{6}\) ? b. Despite its structural differences with sulfuric and selenic acid, telluric acid is a diprotic acid with \(\mathrm{p} K_{\mathrm{a}_{1}}=7.68\) and \(\mathrm{P} K_{a_{2}}=11.29 .\) Telluric acid can be prepared by hydrolysis of tellurium hexafluoride according to the equation $$ \mathrm{TeF}_{6}(g)+6 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Te}(\mathrm{OH})_{6}(a q)+6 \mathrm{HF}(a q) $$ Tellurium hexafluoride can be prepared by the reaction of elemental tellurium with fluorine gas: $$ \mathrm{Te}(s)+3 \mathrm{~F}_{2}(g) \longrightarrow \mathrm{TeF}_{6}(g) $$ If a cubic block of tellurium (density \(\left.=6.240 \mathrm{~g} / \mathrm{cm}^{3}\right)\) measuring \(0.545 \mathrm{~cm}\) on edge is allowed to react with \(2.34 \mathrm{~L}\) fluorine gas at \(1.06 \mathrm{~atm}\) and \(25^{\circ} \mathrm{C}\), what is the \(\mathrm{pH}\) of a solution of \(\mathrm{Te}(\mathrm{OH})_{6}\) formed by dissolving the isolated \(\mathrm{TeF}_{6}(g)\) in \(115 \mathrm{~mL}\) solution? Assume \(100 \%\) yield in all reactions.

\(\mathrm{Al}_{2} \mathrm{O}_{3}\) is amphoteric. What does this mean?

How can the paramagnetism of \(\mathrm{O}_{2}\) be explained using the molecular orbital model?
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