Thermodynamics and Statistical Mechanics I - Home Exercise 12

Thermodynamics and Statistical Mechanics I - Home Exercise 12
1. molecules in a box
A vessel of volume V contains N gas molecules. Let n be the number of molecules in
a part of the vessel of volume v. Considering that the probability of finding a certain
molecule in v is v/V in the thermal equilibrium state of the system.
(a) Find the probability distribution of the number n, f (n).
(b) Calculate hni and hni2 − hn2 i.
(c) Use stirlings formula to show that if N and n are large, f (n) is a Gaussian.
(d) Sow that if v/V → 0, or V → ∞ while N/V = constant, f (n) approaches a Poisson
distribution f (n) =
1 −hni
e
hnin .
n!
2. Bead on a wire
A bead is threaded on a thin and very long wire at a temperature τ .
(a) Calculate the partition function and the internal energy, keeping the spatial part
as an integral.
(b) The wire is now bounded between two walls. What is the force exerted by the
walls on the wire (both separately and together)?
3. Adiabatic atmosphere model
In this problem we improve the atmospheric equations received under the assumption
of total (mechanical, thermal and chemical) equilibrium. We assume that:
• The pressure changes responsible for mechanical equilibrium are immediate
(they are at speeds of the order of the speed of sound).
• The height changes responsible for the process in which a layer of atmosphere
absorbs heat from the ground, expands to equate pressure, rises upwards because
of pressure differences and contracts due to hydrostatic pressures are adiabatic
(almost no heat is exchanged).
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Consider a layer of atmosphere of width dh at height h.
(a) Use mechanical considerations to acquire an expression for
dp
.
dh
Check that under
the assumption of thermal equilibrium this reduces to the known exponential
barometric equation.
(b) Use the adiabatic nature of the process to obtain a relation for
dT
.
dh
(c) Validate your result using the following: air is 70% N2 & 30% O2 → M ≈
29 g/mole, Cp ≈ 7/2N , CV ≈ 5/2N , R = 8.3J/mole/K.
4. Gibbs sum for a two level system
(a) Consider a system that may be unoccupied with energy zero or occupied by one
particle in either of two states, one of energy zero and one of energy ε. Show
that the Gibbs sum (or grand partition function) for the system is ξ = 1 + λ +
λe−βε , where λ ≡ eβµ is the absolute activity also known as the fugacity. Our
assumption excludes the possibility of one particle in each state at the same time.
Notice that we include in the sum a term for N = 0 as a particular state of a
system of a variable number of particles.
(b) Show that the thermal average occupancy of the system is hN i =
λ
ξ
1 + e−βε
(c) Show that the thermal average occupancy of the state at energy ε is hN (ε)i =
λ −βε
e .
ξ
(d) Find an expression for the thermal average energy of the system.
(e) Allow the possibility that the orbital at 0 and at ε may be occupied each by one
particle at the same time. Show that ξ = 1+λ+λe−βε +λ2 e−βε = (1+λ)()1+λe−βε .
Because ξ can be factored as shown, we have in effect two independent systems.
5. Adsorbsion Consider a gas in a container with volume V and chemical potential µ.
There are M sites that a gas molecule can ”stick” to. Each site can attach either 0 or
1 gas molecule. Adsorbsion energy is ε.
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(a) Calculate the number of molecules that are ”stuck” using the grand-canonical
ensemble.
(b) Calculate the number of molecules that are ”stuck” using the canonical ensemble.
(c) Calculate the number of molecules that are ”stuck” using the microcanonical ensemble. Compare with previous result. Discuss.
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