# Statistical Thermodynamics (multiple questions)

## Homework Statement

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.

## Homework Equations

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.

## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

Actually I managed to figure it out in the end, so thread can be closed.