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Magazine Chapter 6: Problem 28
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?
Short Answer
Expert verified
The longest wavelength of radiation that possesses the necessary energy to break the nitrogen-nitrogen bond is \(1.275 \times 10^{-7} \mathrm{m}\). This type of electromagnetic radiation is ultraviolet (UV) radiation.
Step by step solution
01
Write the Planck's Equation
Planck's equation, also known as the Planck-Einstein relation, relates the energy of a photon to its wavelength. It is given by:
\[ E = h \times f \]
where:
E - energy of a photon
h - Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\))
f - frequency of the photon
Since frequency and wavelength are related by the speed of light (c) as follows:
\[ f = \frac{c}{\lambda} \]
Planck's equation can be rewritten in terms of wavelength:
\[ E = \frac{h \times c}{\lambda} \]
In this problem, we need to find the longest wavelength (\(\lambda\)) of radiation that has the necessary energy to break the nitrogen-nitrogen bond.
02
Convert the energy given in kJ/mol to Joules per photon
We are given the energy required to break the nitrogen-nitrogen bond as \(941 \mathrm{~kJ/mol}\). We need to convert this value to Joules per photon. To do so, follow these steps:
1. Convert kJ to Joules: Multiply by \(1000\)
\[ 941 \mathrm{~kJ/mol} \times 1000 = 941000 \mathrm{~J/mol} \]
2. Divide by Avogadro's number to obtain energy per photon
\[ \frac{941000 \mathrm{~J/mol}}{6.022 \times 10^{23} \mathrm{photon/mol}} = 1.562 \times 10^{-19} \mathrm{J/photon} \]
So, the energy needed to break the bond is \(1.562 \times 10^{-19} \mathrm{J/photon}\).
03
Find the wavelength
Now, we have the energy needed to break the bond and can use Planck's equation to find the corresponding wavelength. Solve for \(\lambda\):
\[ \lambda = \frac{h \times c}{E} \]
Plug in the values:
\[ \lambda = \frac{(6.626 \times 10^{-34} \mathrm{Js}) \times (2.998 \times 10^{8} \mathrm{m/s})}{1.562 \times 10^{-19} \mathrm{J/photon}} \]
\[ \lambda = 1.275 \times 10^{-7} \mathrm{m} \]
The longest wavelength of radiation required to break the nitrogen-nitrogen bond is \(1.275 \times 10^{-7} \mathrm{m}\).
04
Identify the type of radiation
To identify the type of electromagnetic radiation that corresponds to a wavelength of \(1.275 \times 10^{-7} \mathrm{m}\), refer to the electromagnetic spectrum:
1. Radio waves: wavelengths longer than \(1 \times 10^{-3} \mathrm{m}\)
2. Microwaves: wavelengths between \(1 \times 10^{-3} \mathrm{m}\) and \(1 \times 10^{-6} \mathrm{m}\)
3. Infrared: wavelengths between \(1 \times 10^{-6} \mathrm{m}\) and \(7 \times 10^{-7} \mathrm{m}\)
4. Visible light: wavelengths between \(7 \times 10^{-7} \mathrm{m}\) and \(4 \times 10^{-7} \mathrm{m}\)
5. Ultraviolet: wavelengths between \(4 \times 10^{-7} \mathrm{m}\) and \(1 \times 10^{-8} \mathrm{m}\)
6. X-rays: wavelengths between \(1 \times 10^{-8} \mathrm{m}\) and \(1 \times 10^{-11} \mathrm{m}\)
7. Gamma rays: wavelengths shorter than \(1 \times 10^{-11} \mathrm{m}\)
Since the wavelength we found is \(1.275 \times 10^{-7} \mathrm{m}\), the corresponding type of radiation is ultraviolet (UV) radiation.
To summarize, the longest wavelength of radiation that can break the nitrogen-nitrogen bond is \(1.275 \times 10^{-7} \mathrm{m}\), and the type of electromagnetic radiation is ultraviolet.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chemical Bond Rupture
When we talk about the rupture of chemical bonds, we are referring to the process by which a bond between two atoms is broken. This process requires energy, as chemical bonds inherently possess potential energy due to the electron arrangement between the bonded atoms. The energy needed to break these bonds depends on the type of bond and the specific atoms involved. For example, a nitrogen-nitrogen bond, as mentioned in the exercise, is a strong bond that requires considerable energy to break.
In practical terms, this energy can be imparted through collisions with other particles or the absorption of radiation. When radiation with sufficient energy hits the bond, it can transfer energy to the electrons, exciting them to the point where they overcome the bond energy, resulting in bond rupture. This is why understanding the energy associated with radiation is crucial in processes such as chemical reactions, material processing, and even biological effects like DNA damage caused by ultraviolet light.
In practical terms, this energy can be imparted through collisions with other particles or the absorption of radiation. When radiation with sufficient energy hits the bond, it can transfer energy to the electrons, exciting them to the point where they overcome the bond energy, resulting in bond rupture. This is why understanding the energy associated with radiation is crucial in processes such as chemical reactions, material processing, and even biological effects like DNA damage caused by ultraviolet light.
Planck's Equation
Planck's equation, a fundamental concept in quantum mechanics, describes the quantized nature of energy carried by photons. The equation, which is indeed the Planck-Einstein relation, is articulated as:
\[ E = h \times f \]
where \(E\) is the energy of a single photon, \(h\) is Planck's constant鈥� a very important number in quantum physics, and \(f\) is the frequency of the photon. Planck's constant has a value of about \(6.626 \times 10^{-34} \mathrm{Js}\), which might seem extremely small, but it's crucial to the behavior of particles at the quantum level. This equation encapsulates the idea that electromagnetic radiation can be absorbed or emitted in discrete units, called quanta, and that the energy of these quanta is proportional to the frequency of the radiation.
\[ E = h \times f \]
where \(E\) is the energy of a single photon, \(h\) is Planck's constant鈥� a very important number in quantum physics, and \(f\) is the frequency of the photon. Planck's constant has a value of about \(6.626 \times 10^{-34} \mathrm{Js}\), which might seem extremely small, but it's crucial to the behavior of particles at the quantum level. This equation encapsulates the idea that electromagnetic radiation can be absorbed or emitted in discrete units, called quanta, and that the energy of these quanta is proportional to the frequency of the radiation.
Wavelength and Frequency Relationship
The relationship between wavelength and frequency is an inverse one, as they are inversely proportional to each other. This is expressed through the formula:
\[ f = \frac{c}{\lambda} \]
where \(f\) represents frequency, \(c\) stands for the speed of light in a vacuum (approximately \(3.00 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) (lambda) symbolizes the wavelength. When the frequency of radiation increases, the wavelength decreases, and vice versa. This relationship is critical in determining the energy of photons, as previously discussed with Planck's equation. Essentially, it's what links the energy of a photon to its position in the electromagnetic spectrum and allows us to predict the kind of interactions it might have with matter, such as the ability to induce chemical bond rupture.
\[ f = \frac{c}{\lambda} \]
where \(f\) represents frequency, \(c\) stands for the speed of light in a vacuum (approximately \(3.00 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) (lambda) symbolizes the wavelength. When the frequency of radiation increases, the wavelength decreases, and vice versa. This relationship is critical in determining the energy of photons, as previously discussed with Planck's equation. Essentially, it's what links the energy of a photon to its position in the electromagnetic spectrum and allows us to predict the kind of interactions it might have with matter, such as the ability to induce chemical bond rupture.
Electromagnetic Spectrum
The electromagnetic spectrum is a comprehensive range of all possible frequencies of electromagnetic radiation, from very long radio waves to very short gamma rays. It's important to understand this spectrum because different types of radiation within it have vastly different properties and uses. Here is a short overview of the main categories:
The type of radiation that can break a nitrogen-nitrogen bond, as in our exercise, falls in the ultraviolet range. It's this ability to categorize the energy of radiation that allows scientists and engineers to employ specific wavelengths for particular purposes, including breaking chemical bonds as required in various chemical processes.
- Radio waves: These have the longest wavelengths and lowest frequencies. Used in communications and broadcasting.
- Microwaves: Used for cooking and certain types of radars.
- Infrared: Experienced as heat and used in heating, night-vision devices, and remote controls.
- Visible light: The only part of the spectrum visible to the human eye.
- Ultraviolet light: Has higher energy than visible light and can cause sunburns but is also used for sterilization.
- X-rays: Used in medical imaging and security scanners.
- Gamma rays: The highest energy radiation, used in cancer therapy and emitted by nuclear reactions and certain astronomical phenomena.
The type of radiation that can break a nitrogen-nitrogen bond, as in our exercise, falls in the ultraviolet range. It's this ability to categorize the energy of radiation that allows scientists and engineers to employ specific wavelengths for particular purposes, including breaking chemical bonds as required in various chemical processes.
