Problem 34 - Chemistry The Central Science In Si Units

Question

An important reaction in the formation of photochemical smog is the photodissociation of \(\mathrm{NO}_{2}\) : $$\mathrm{NO}_{2}+h \nu \longrightarrow \mathrm{NO}(g)+\mathrm{O}(g)$$ The maximum wavelength of light that can cause this reaction is \(420 \mathrm{nm} .\) (a) In what part of the electromagnetic spectrum is light with this wavelength found? (b) What is the maximum strength of a bond, in \(\mathrm{kJ} / \mathrm{mol}\), that can be broken by absorption of a photon of \(420-\mathrm{nm}\) light? \((\mathbf{c})\) Write out the photodissociation reaction showing Lewis-dot structures.

Step-by-Step Solution

Answer

(a) 420 nm light is in the visible spectrum. (b) Maximum bond strength broken is 285.35 kJ/mol. (c) Lewis structure: \( \text{NO}_2 \) is O=N-O; \( \text{NO} \) is N≡O.

Step 1: Identify the electromagnetic spectrum region

To determine the part of the electromagnetic spectrum in which light with a wavelength of 420 nm is found, we refer to the electromagnetic spectrum range. Visible light ranges from approximately 380 nm to 750 nm. Since 420 nm falls within this range, the light is part of the visible spectrum.

Step 2: Calculate the energy of the photon

We use the equation for the energy of a photon: \( E = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \) J·s is Planck's constant, \( c = 3.00 \times 10^8 \) m/s is the speed of light, and \( \lambda = 420 \times 10^{-9} \) m is the wavelength. Substituting these values into the equation gives:\[ E = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{420 \times 10^{-9}} \approx 4.74 \times 10^{-19} \text{ J} \]

Step 3: Convert the energy to kJ/mol

We convert the energy of one photon to kilojoules per mole by using Avogadro's number, \( 6.022 \times 10^{23} \text{ mol}^{-1} \). Thus, the energy per mole is:\[ E_{\text{mol}} = 4.74 \times 10^{-19} \text{ J} \times 6.022 \times 10^{23} \text{ mol}^{-1} = 285.35 \text{ kJ/mol} \] This is the maximum bond energy that can be broken by a photon of 420 nm light.

Step 4: Draw the Lewis-dot structure for NO2 and NO

The Lewis-dot structure for \( \text{NO}_2 \) shows nitrogen in the middle, with one single bond to an oxygen atom and one double bond to another oxygen. The structure is: O=N-O (with two dots on the single-bonded oxygen and two unpaired electrons on nitrogen). For \( \text{NO} \), nitrogen is bonded to oxygen with a double bond and has an unpaired electron: N\(\cdot\)≡O

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum is a vast range of all types of electromagnetic radiation. Electromagnetic radiation is energy that travels and spreads out as it goes. It includes everything from radio waves to gamma rays. Each type has a different wavelength and frequency, which ultimately links to how much energy it carries.
Light with a wavelength of 420 nm, as mentioned in the exercise, falls within the visible spectrum. This part of the spectrum is where we perceive light as colors—spanning roughly from 380 nm (violet) to 750 nm (red). It's fascinating to note that different wavelengths of visible light correspond to different colors. For instance, a wavelength of 420 nm is seen as blue or violet.
Whenever we deal with wavelengths and their energies, it’s useful to understand which part of the spectrum we are referring to. This knowledge helps in applications like photodissociation, where specific photon energies are needed to break chemical bonds.

Photon Energy Calculation

Calculating the energy of a photon is crucial for understanding how light interacts with matter. The energy of a photon can be calculated using the formula: \[ E = \frac{hc}{\lambda} \]where:

• \( E \) = energy of the photon in joules (J)
• \( h \) = Planck's constant, approximately \( 6.626 \times 10^{-34} \) J·s
• \( c \) = speed of light, approximately \( 3.00 \times 10^8 \) m/s
• \( \lambda \) = wavelength in meters (m)

For the 420 nm light wavelength used in the exercise, you'd first convert nm to m, resulting in \( 420 \times 10^{-9} \) m. Substituting the values, we calculated the photon's energy to be approximately \( 4.74 \times 10^{-19} \) J.
This energy represents the maximum that this photon can transfer, sufficient to break bonds, such as in the photodissociation of \( \text{NO}_2 \). To connect this to chemistry, we convert this energy to kilojoules per mole, aligning with how we measure bond energies in chemistry. By multiplying by Avogadro’s number, we find the energy per mole, giving valuable information about the strength of the molecular bonds that can be broken.

Lewis-dot Structures

Lewis-dot structures are a simple way to show the arrangement of electrons around an atom. They are visual representations that help predict how atoms will bond, sharing or transferring electrons.
For the molecule \( \text{NO}_2 \), the Lewis-dot structure is drawn with nitrogen bonded to two oxygen atoms. Nitrogen typically forms three bonds, so one oxygen is double-bonded, while a single bond connects nitrogen to the other oxygen. Additionally, an unpaired electron is depicted on the nitrogen, reflecting its high reactivity.In \( \text{NO} \), nitrogen is double-bonded to oxygen and retains an unpaired electron. This makes nitric oxide a radical species, meaning it has an odd number of electrons and is likely to engage in further reactions.
Understanding these structures not only provides insight into the chemical properties of a molecule but also helps to predict its behavior in reactions, such as photodissociation shown in the exercise.

Visible Light

Visible light is the portion of the electromagnetic spectrum that can be detected by the human eye. It is roughly situated between 380 nm to 750 nm wavelengths. Each wavelength within this range corresponds to a different color, forming a spectrum that includes all the colors we can perceive.
In the given exercise, light of 420 nm is specifically used for photodissociation. This wavelength is perceived as blue-violet by our eyes, indicating high energy compared to longer wavelengths like red.
This part of the spectrum is extremely important for life on Earth as it is involved in photosynthesis, the process by which plants convert light energy into chemical energy. Visible light's ability to initiate photochemical reactions is also pivotal in nature and industrial applications. The ability of visible light to cause photodissociation of \( \text{NO}_2 \) highlights its potential to influence atmospheric chemistry and environmental conditions.