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Prove the claim that a photon with an energy of 2230 cm corresponds to ~4.5 μm light.

Prove the claim that a photon with an energy of 2230 cm corresponds to ~4.5 μm light.

Chemistry
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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Spectroscopy involves the use of laser systems (light amplification by stimulated emission of emitted radiation) to understand the properties of chemical systems. Said another way, spectroscopists direct laser light into a chemical system and then observe the light that comes out of the system. Spectropsocoptics compare the light before it interacts with the chemical system to the light after the interaction to be able to decipher how the light interacts with the system. To help with the task of deciphering what happened, spectroscopists have developed what are known as spectroscopic selection rules which are rules dictating the allowable transitions for a system. We will talk more about such selection rules later in the class. Within the harmonic approximation, a well-known selection rule is that chemical vibrations can only gain one quanta of vibrational energy at a time. Vibrational quanta are often noted by the greek letter nu, v, so another way to write the selection rule would be transitions like v -> (absorption of a photon) or v₁ -> v₂, and v₁ -> (emission of a photon) are allowed but transitions like 1 -> 1/₂ or 1/₂ -> vare not. You will be asked later to connect this idea to raising and lowering operators. Commonly, the most intense transition (the easiest transition to detect) is the v -> 01 for the various vibrational modes of the molecule being studied. One such mode is the nitrile stretching mode that was referenced in problem 4 of "2-1 - The Classical and Quantum Harmonic Oscillator. The v->transition for this mode occurs at an energy of 2230 cm (infrared light with a ~4.5μm wavelength). With this setup, we will explore vibrational modes and the relationship of raising and lowering operators to spectroscopic selection rules.
Transcribed Image Text:Spectroscopy involves the use of laser systems (light amplification by stimulated emission of emitted radiation) to understand the properties of chemical systems. Said another way, spectroscopists direct laser light into a chemical system and then observe the light that comes out of the system. Spectropsocoptics compare the light before it interacts with the chemical system to the light after the interaction to be able to decipher how the light interacts with the system. To help with the task of deciphering what happened, spectroscopists have developed what are known as spectroscopic selection rules which are rules dictating the allowable transitions for a system. We will talk more about such selection rules later in the class. Within the harmonic approximation, a well-known selection rule is that chemical vibrations can only gain one quanta of vibrational energy at a time. Vibrational quanta are often noted by the greek letter nu, v, so another way to write the selection rule would be transitions like v -> (absorption of a photon) or v₁ -> v₂, and v₁ -> (emission of a photon) are allowed but transitions like 1 -> 1/₂ or 1/₂ -> vare not. You will be asked later to connect this idea to raising and lowering operators. Commonly, the most intense transition (the easiest transition to detect) is the v -> 01 for the various vibrational modes of the molecule being studied. One such mode is the nitrile stretching mode that was referenced in problem 4 of "2-1 - The Classical and Quantum Harmonic Oscillator. The v->transition for this mode occurs at an energy of 2230 cm (infrared light with a ~4.5μm wavelength). With this setup, we will explore vibrational modes and the relationship of raising and lowering operators to spectroscopic selection rules.
-1 1. Prove the claim that a photon with an energy of 2230 cm corresponds to ~4.5 μm light. 2. PIPES Problem: Assume the number of states for vand vare the same for the nitrile vibrations. With this assumption, calculate how many benzonitrile molecules are in the state for every 1 benzonitrile in the v₁ state at room temperature (~27C°) using the Boltzmann distribution. (hint: you'll have to manipulate the population ratio to the obtain population relative to a v population of 1). 3. Using the result of problem 2, justify, in plain language, why you can expect (at room temperature) for the v->transition to be more intense than the v₁ -> ₂ transition (hint: a spectroscopist only sees the superimposed, collective signal from the system). 4. Explain, in plain language, how the spectroscopic selection rule that vibrations can only change one quanta of vibrational energy at a time is directly a result of raising and lowering operators. Important Note: This problem is impossible to solve with your given knowledge. However, that doesn't mean the problem is impossible. I have stealthily given the answer in the reading before these problems when talking about the allowed vibrational quanta (see v -> v being allowed but ->not being allowed in the harmonic approximation). If you still cannot answer the question, the internet surely knows the answer. You just have to go and find it.
Transcribed Image Text:-1 1. Prove the claim that a photon with an energy of 2230 cm corresponds to ~4.5 μm light. 2. PIPES Problem: Assume the number of states for vand vare the same for the nitrile vibrations. With this assumption, calculate how many benzonitrile molecules are in the state for every 1 benzonitrile in the v₁ state at room temperature (~27C°) using the Boltzmann distribution. (hint: you'll have to manipulate the population ratio to the obtain population relative to a v population of 1). 3. Using the result of problem 2, justify, in plain language, why you can expect (at room temperature) for the v->transition to be more intense than the v₁ -> ₂ transition (hint: a spectroscopist only sees the superimposed, collective signal from the system). 4. Explain, in plain language, how the spectroscopic selection rule that vibrations can only change one quanta of vibrational energy at a time is directly a result of raising and lowering operators. Important Note: This problem is impossible to solve with your given knowledge. However, that doesn't mean the problem is impossible. I have stealthily given the answer in the reading before these problems when talking about the allowed vibrational quanta (see v -> v being allowed but ->not being allowed in the harmonic approximation). If you still cannot answer the question, the internet surely knows the answer. You just have to go and find it.
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