Statistical Thermodynamics (multiple questions)

In summary, the conversation discusses the relationship between particle density and external potential in a gravity field. It also explores the characteristic function for a system with an interface and fixed pressure or volume. The final equations show that the characteristic function is related to the energy stored in the interface.
  • #1
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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.
 
  • #3
Actually I managed to figure it out in the end, so thread can be closed.
 

1. What is statistical thermodynamics?

Statistical thermodynamics is a branch of physics that combines the principles of thermodynamics with statistical mechanics to explain and predict the behavior of thermodynamic systems on a microscopic level.

2. How is statistical thermodynamics different from classical thermodynamics?

Classical thermodynamics deals with macroscopic systems and their overall properties, while statistical thermodynamics focuses on the behavior and properties of individual particles within a system.

3. What are the key principles of statistical thermodynamics?

The key principles of statistical thermodynamics include the Boltzmann distribution, which describes the distribution of particles in a system, and the concept of entropy, which measures the disorder of a system.

4. What are the main applications of statistical thermodynamics?

Statistical thermodynamics has numerous applications in fields such as chemistry, biology, and materials science. It is used to understand and predict the behavior of gases, liquids, and solids, as well as chemical reactions and phase transitions.

5. How does statistical thermodynamics relate to quantum mechanics?

Statistical thermodynamics is based on the principles of quantum mechanics, which describes the behavior of particles on a subatomic level. It uses concepts such as energy levels and wave functions to explain the behavior of particles within a system.

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