Hello
for a diatomic gas in thermodynamics equilibrium, what is the vibrational level n probability?
is below equation correct?
P(vibration level n probability)=(1/Total Partition Function(Translation*Vibration*Rotation*electronics))*Exp[-hf(n+1/2)/kT]
is this approach correct for rotational and electronics levels?
Thank you
The equilibrium probability distribution function is the normalised Boltzmann factor, determined by the energies of states in the canonical ensemble, and by the energies of states and the numbers of particles in the grand-canonical ensemble.
For calculations, it is convenient to work with the density operator ρ = exp(-βK)/Z, where β = 1/κΒΤ, and K = H in the canonical ensemble, and K = H - μΝ in the grand-canonical ensemble; Z = Tr[exp(-βK)] is the partition function. Here H denotes the Hamiltonian operator of the system, and N the number operator. Given ρ, one can calculate the ensemble average of the operator A from the expression = Tr[ρA]. In practice, it is convenient to calculate Tr[ρA] in the canonical ensemble of N particles as the sum over the expectation values with respect to the N-particle eigenstates of H, and for H commuting with N, it is convenient to calculate Tr[ρA] in the grand-canonical ensemble as the sum over the expectation values with respect to the simultaneous eigenstates of H and N, where i refers to the i-th energy eigenstate, and ν to the number of particles; one has
H |i,ν> = Ei(ν) |i,ν>,
N |i,ν> = ν |i,ν>.
Because of the latter equalities,
- In the canonical ensemble of N particles:
= Σi exp(-βEi(N)) /Z, where
Z = Σi exp(-βEi(N)).
- In the grand-canonical ensemble:
= Σi,ν exp(-β{Ei(ν)- μν}) /Z, where
Z = Σi,ν exp(-β{Ei(ν)-μν}).
Lastly, the expression that you have presented above contains elements specific to a one-dimensional harmonic oscillator. It is therefore very specific and far from general (not all oscillators are one-dimensional and harmonic).
The equilibrium probability distribution function is the normalised Boltzmann factor, determined by the energies of states in the canonical ensemble, and by the energies of states and the numbers of particles in the grand-canonical ensemble.
For calculations, it is convenient to work with the density operator ρ = exp(-βK)/Z, where β = 1/κΒΤ, and K = H in the canonical ensemble, and K = H - μΝ in the grand-canonical ensemble; Z = Tr[exp(-βK)] is the partition function. Here H denotes the Hamiltonian operator of the system, and N the number operator. Given ρ, one can calculate the ensemble average of the operator A from the expression = Tr[ρA]. In practice, it is convenient to calculate Tr[ρA] in the canonical ensemble of N particles as the sum over the expectation values with respect to the N-particle eigenstates of H, and for H commuting with N, it is convenient to calculate Tr[ρA] in the grand-canonical ensemble as the sum over the expectation values with respect to the simultaneous eigenstates of H and N, where i refers to the i-th energy eigenstate, and ν to the number of particles; one has
H |i,ν> = Ei(ν) |i,ν>,
N |i,ν> = ν |i,ν>.
Because of the latter equalities,
- In the canonical ensemble of N particles:
= Σi exp(-βEi(N)) /Z, where
Z = Σi exp(-βEi(N)).
- In the grand-canonical ensemble:
= Σi,ν exp(-β{Ei(ν)- μν}) /Z, where
Z = Σi,ν exp(-β{Ei(ν)-μν}).
Lastly, the expression that you have presented above contains elements specific to a one-dimensional harmonic oscillator. It is therefore very specific and far from general (not all oscillators are one-dimensional and harmonic).
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Thermodynamics