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The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature
at which the population of the first exited state will be half that of ground state.
? Related: Molecular Spectroscopy
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The vibrational frequency of a homonuclear diatomic molecule is v. Cal...
Introduction
In molecular spectroscopy, the vibrational frequency of a homonuclear diatomic molecule is an important parameter that determines the energy levels of the molecule. The temperature at which the population of the first excited state is half that of the ground state can be calculated using Boltzmann distribution law.
Boltzmann distribution law
The Boltzmann distribution law relates the population of a given energy level E to the temperature T and the energy difference ΔE between that level and the ground state.
P(E) = N exp(-ΔE/kT)
where P(E) is the population of the energy level E, N is the total number of molecules, k is the Boltzmann constant, and T is the temperature in Kelvin.
Calculation
Let us assume that the ground state has zero energy and the first excited state has energy ΔE. We want to find the temperature at which the population of the first excited state is half that of the ground state.
Using the Boltzmann distribution law, we can write:
P(0) = N exp(0) = N
P(ΔE) = N exp(-ΔE/kT)
We want to find T such that P(ΔE) = 0.5 P(0).
0.5 P(0) = N/2
N exp(-ΔE/kT) = N/2
Taking natural logarithm on both sides, we get:
ln(2) = ΔE/kT
Solving for T, we get:
T = ΔE/k ln(2)
Conclusion
In conclusion, the temperature at which the population of the first excited state is half that of the ground state can be calculated using the Boltzmann distribution law. For a homonuclear diatomic molecule with vibrational frequency v, the temperature is given by T = hv/2k ln(2), where h is the Planck constant and k is the Boltzmann constant.
In molecular spectroscopy, the vibrational frequency of a homonuclear diatomic molecule is an important parameter that determines the energy levels of the molecule. The temperature at which the population of the first excited state is half that of the ground state can be calculated using Boltzmann distribution law.
Boltzmann distribution law
The Boltzmann distribution law relates the population of a given energy level E to the temperature T and the energy difference ΔE between that level and the ground state.
P(E) = N exp(-ΔE/kT)
where P(E) is the population of the energy level E, N is the total number of molecules, k is the Boltzmann constant, and T is the temperature in Kelvin.
Calculation
Let us assume that the ground state has zero energy and the first excited state has energy ΔE. We want to find the temperature at which the population of the first excited state is half that of the ground state.
Using the Boltzmann distribution law, we can write:
P(0) = N exp(0) = N
P(ΔE) = N exp(-ΔE/kT)
We want to find T such that P(ΔE) = 0.5 P(0).
0.5 P(0) = N/2
N exp(-ΔE/kT) = N/2
Taking natural logarithm on both sides, we get:
ln(2) = ΔE/kT
Solving for T, we get:
T = ΔE/k ln(2)
Conclusion
In conclusion, the temperature at which the population of the first excited state is half that of the ground state can be calculated using the Boltzmann distribution law. For a homonuclear diatomic molecule with vibrational frequency v, the temperature is given by T = hv/2k ln(2), where h is the Planck constant and k is the Boltzmann constant.
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The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature
at which the population of the first exited state will be half that of ground state. Related: Molecular Spectroscopy?
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The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature at which the population of the first exited state will be half that of ground state. Related: Molecular Spectroscopy? for Chemistry 2024 is part of Chemistry preparation. The Question and answers have been prepared according to the Chemistry exam syllabus. Information about The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature at which the population of the first exited state will be half that of ground state. Related: Molecular Spectroscopy? covers all topics & solutions for Chemistry 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature at which the population of the first exited state will be half that of ground state. Related: Molecular Spectroscopy?.
The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature at which the population of the first exited state will be half that of ground state. Related: Molecular Spectroscopy? for Chemistry 2024 is part of Chemistry preparation. The Question and answers have been prepared according to the Chemistry exam syllabus. Information about The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature at which the population of the first exited state will be half that of ground state. Related: Molecular Spectroscopy? covers all topics & solutions for Chemistry 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The vibrational frequency of a homonuclear diatomic molecule is v. Calculate the temperature at which the population of the first exited state will be half that of ground state. Related: Molecular Spectroscopy?.
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