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.