/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 The stratospheric ozone \(\left(... [FREE SOLUTION] | 91Ó°ÊÓ

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The stratospheric ozone \(\left(\mathrm{O}_{3}\right)\) layer helps to protect us from harmful ultraviolet radiation. It does so by absorbing ultraviolet light and falling apart into an \(\mathrm{O}_{2}\) molecule and an oxygen atom, a process known as photodissociation. $$\mathrm{O}_{3}(g) \longrightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)$$ Use the data in Appendix \(C\) to calculate the enthalpy change for this reaction. What is the maximum wavelength a photon can have if it is to possess sufficient energy to cause this dissociation? In what portion of the spectrum does this wavelength occur?

Short Answer

Expert verified
The enthalpy change for the photodissociation of one mole of \(\mathrm{O}_{3}\) molecules is \(106.9 \:\mathrm{kJ/mol}\). The maximum wavelength a photon can have to possess sufficient energy for this process is \(184.8 \:\mathrm{nm}\).

Step by step solution

01

Calculate the enthalpy change for the reaction.

We need to find the enthalpy change for the reaction using the given enthalpies of formation for the products and reactants. Appendix C data: \(\Delta_\mathrm{f}H^\circ (\mathrm{O}_3) = 142.3 \:\mathrm{kJ/mol}\) \(\Delta_\mathrm{f}H^\circ (\mathrm{O}_2) = 0 \:\mathrm{kJ/mol}\) (since it's the most stable form of this element) \(\Delta_\mathrm{f}H^\circ (\mathrm{O}) = 249.2 \:\mathrm{kJ/mol}\) The formula for calculating enthalpy change is: \[ \Delta H^\circ = \sum \Delta_\mathrm{f}H^\circ (\text{products}) - \sum \Delta_\mathrm{f}H^\circ (\text{reactants}) \] Substituting the given values: \[ \Delta H^\circ = [\Delta_\mathrm{f}H^\circ (\mathrm{O}_2) + \Delta_\mathrm{f}H^\circ (\mathrm{O})] - [\Delta_\mathrm{f}H^\circ (\mathrm{O}_3)] \]
02

Calculate the enthalpy change value.

Plug in the values and calculate the enthalpy change: \[ \Delta H^\circ = [0 + 249.2] - [142.3] = 106.9 \:\mathrm{kJ/mol} \] The enthalpy change for the reaction is \(106.9 \:\mathrm{kJ/mol}\).
03

Determine the maximum wavelength using Planck's equation.

Planck's equation relates the energy of a photon to its wavelength: \[ E = \dfrac{hc}{\lambda} \] Here, \(E\) is the energy of the photon, \(h\) represents Planck's constant (\(6.626 \times 10^{-34} \:\mathrm{J\:s}\)), \(c\) is the speed of light (\(2.998 \times 10^8 \:\mathrm{m/s}\)), and \(\lambda\) is the wavelength of light. We need to determine the wavelength (\(\lambda\)) that corresponds to the energy needed for the reaction: \[ \lambda = \dfrac{hc}{E} \] Remember, we want the energy needed for the process in J/mol and substitute the value: \[ \lambda = \dfrac{hc}{106.9 \times 10^3 \:\mathrm{J/mol}} \]
04

Calculate the maximum wavelength.

By plugging the values and calculating the result: \[ \lambda = \dfrac{(6.626 \times 10^{-34} \:\mathrm{J\:s})(2.998 \times 10^8 \:\mathrm{m/s})}{106.9 \times 10^3 \:\mathrm{J/mol}} = 1.848 \times 10^{-7} \:\mathrm{m} \] To express the wavelength in nanometers, multiply the result by \(10^{9}\): \[ \lambda = 1.848 \times 10^{-7} \:\mathrm{m} \times 10^9 = 184.8 \:\mathrm{nm} \] Thus, the maximum wavelength a photon can have to possess sufficient energy to dissociate one mole of \(\mathrm{O}_{3}\) molecules is \(184.8 \:\mathrm{nm}\).

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Key Concepts

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

Stratospheric Ozone Layer
The stratospheric ozone layer acts like a protective shield around Earth. It absorbs harmful ultraviolet (UV) radiation from the sun. Without this layer, life on Earth would be at risk from increased UV exposure.
Ozone in the stratosphere undergoes a process called photodissociation. In this process, ozone molecules (\(\mathrm{O}_3\)) break down into oxygen molecules (\(\mathrm{O}_2\)) and oxygen atoms (\(\mathrm{O}\)).
This absorption of UV radiation helps prevent skin cancer, cataracts, and other health issues in humans. It also protects crops and marine ecosystems, essential for life on Earth.
The stratospheric ozone layer is situated between 10 to 50 kilometers above Earth’s surface. Its critical role could be compromised by pollutants like chlorofluorocarbons (CFCs), hence the importance of environmental conservation efforts.
Enthalpy Change
Enthalpy change, represented by \(\Delta H\), measures the heat change during a chemical reaction at constant pressure. It can indicate whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).
To determine the enthalpy change for the ozone decomposition, assess the energy needed to break ozone into \(\mathrm{O}_2\) and \(\mathrm{O}\). Use the formula:
\[\Delta H^\circ = \sum \Delta_\mathrm{f}H^\circ (\text{products}) - \sum \Delta_\mathrm{f}H^\circ (\text{reactants})\]
In this case, the values for ozone (\(\mathrm{O}_3\)), oxygen molecule (\(\mathrm{O}_2\)), and oxygen atom (\(\mathrm{O}\)) are used.
The enthalpy change for the reaction \(\Delta H^\circ\) is \(106.9 \:\mathrm{kJ/mol}\), meaning the process is endothermic as it requires energy input to break the bonds.
Maximum Wavelength
The maximum wavelength of light sufficient for a chemical reaction refers to the longest wavelength that can provide enough energy to cause the reaction. For photodissociation, it's crucial to know this to understand what kind of light breaks down substances like ozone.
To find out, use the calculated enthalpy change of the reaction to help determine the corresponding photon energy. The formula derived from Planck’s equation is:
\[\lambda = \dfrac{hc}{E}\]\(\lambda\) is the wavelength of the photon, \(h\) is Planck’s constant, \(c\) is the speed of light, and \(E\) is the energy in Joules required to cause the reaction.
In the given exercise, the maximum wavelength calculated was \(184.8 \:\mathrm{nm}\), placing it in the ultraviolet part of the spectrum. This demonstrates which regions of light are capable of causing photodissociation.
Planck's Equation
Planck's equation helps in understanding the relationship between a photon's energy and its wavelength. It is essential for calculating the maximum wavelength that can drive a photochemical reaction like ozone photodissociation.
The formula for Planck's equation is:
\[E = \dfrac{hc}{\lambda}\]
Where \(E\) is energy in Joules, \(h\) (Planck's constant) is \(6.626 \times 10^{-34} \:\mathrm{J\:s}\), \(c\) (speed of light) is \(2.998 \times 10^8 \:\mathrm{m/s}\), and \(\lambda\) is the wavelength in meters.
This equation is crucial for determining not only the energy but also the potential effects of light on compounds, especially in the stratospheric ozone layer. Knowing how to calculate and apply these values helps in studying environmental and chemical processes.

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Most popular questions from this chapter

The energy from radiation can be used to cause the rupture of chemical bonds. A minimum energy of \(941 \mathrm{~kJ} / \mathrm{mol}\) is required to break the nitrogen-nitrogen bond in \(\mathrm{N}_{2}\). What is the longest wavelength of radiation that possesses the necessary energy to break the bond? What type of electromagnetic radiation is this?

What is the maximum number of electrons that can occupy each of the following subshells: (a) \(3 p\), (b) \(5 d\), (c) \(2 s\), (d) \(4 f ?\)

Using only a periodic table as a guide, write the condensed electron configurations for the following atoms: (a) Se, (b) \(\mathrm{Rh}\), (c) \(\mathrm{Si}\), (d) \(\mathrm{Hg}\), (e) Hf.

List the following types of electromagnetic radiation in order of increasing wavelength: (a) the gamma rays produced by a radioactive nuclide used in medical imaging; (b) radiation from an FM radio station at \(93.1 \mathrm{MHz}\) on the dial; (c) a radio signal from an AM radio station at \(680 \mathrm{kHz}\) on the dial; \((\mathrm{d})\) the yellow light from sodium vapor streetlights; (e) the red light of a light-emitting diode, such as in a calculator display.

Determine which of the following statements are false, and correct them. (a) Electromagnetic radiation is incapable of passing through water. (b) Electromagnetic radiation travels through a vacuum at a constant speed, regardless of wavelength. (c) Infrared light has higher frequencies than visible light. (d) The glow from a fireplace, the energy within a microwave oven, and a foghorn blast are all forms of electromagnetic radiation.

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