1.) For N particles in a gravity field, the Hamiltonian has a contribution of external potential only (-mgh). Show that the particle density follows the barometric height equation (1).
2.) For N particles in a open system at constant pressure p and temperature T, let there be an interface with area A in the system (eg. air-water), that is (μ, p, A, T). N, V and U fluctuate in this sytem. Identify the characteristic function X.(2)
3.) Same as above, only for a system where V instead of p is fixed.
1.) ρ(h) = ρ(0)·exp(-mgh/kT), with m mass of particle, g gravitational constant, h barometric height, k boltzmann's constant and T temperature.
2.) X = - kT·lnΔ, where Δ is the partition function for the grand canonical ensemble.
3.) Same as above.
The Attempt at a Solution
1.) From: <N> = kT ∂lnΞ/∂μ = Lx·Ly·∫ρ(h)dh
I’m supposed to arrive at: ρ(h) = ∫[exp(μ/kT)/Λ3]·exp(-mgh/kT) dh
And finally to: ρ(h) = ρ(0) ·exp(-mgh/kT), where ρ(0) = exp(μ/kT)/Λ3
I understand the setup with the expectation value of N, however the transition to the integral and getting the ρ(0) out of the integral is not quite clear to me.
Also, Ξ is the grand-canonical partition function in this case, while Lx·Ly is neglected since they're both uniform and only the barometric height h is considered.
2.) By using the equation X = - kT·lnΔ, with Δ = exp(S/k)·exp(-U/kT)·exp(μN/kT)·exp(-pV/kT), I arrive at
X = -ST + U - μN + pV
X = [ (U + pV) - ST ] - μN
X = (H-TS) -μN = G - μN
However, I know the characteristic function is supposed to be X = γ·A, so the energy stored in the interface. I do not understand how I arrive at that conclusion.
3.) Same procedure as above, only here the volume is fixed instead of the pressure, so the derivation should go through the helmholtz energy. I use the same formula X = - kT·lnΔ, however I'm also supposed to implement dF = -SdT - pdV + μdN + γdA, so to my guess the partition function may look something like this:
Δ = exp(S/k)·exp(-U/kT)·exp(μN/kT)·exp(-pV/kT)·exp(γA/kT)
which then gives:
X = (H-TS) -μN + γA = G - μN + γA
As with the previous question, I don't know how I should conclude from this what the characteristic function is.