Transcript Ch 10

Chapter 10: Boltzmann
Distribution Law (BDL)
From Microscopic Energy Levels
to Energy Probability Distributions
to Macroscopic Properties
I. Probability Distribution
• Microstate: a specific configuration of
atoms (Fig 10.1 has 5 microstates)
• Macrostate: a collection of microstates at
the same energy (Fig 10.1 has 2
macrostates); they differ in a property
determined by finding <Property>.
• Recall: <Prop> requires the probability
distribution (i.e. weighting term) for the
system.
II. BDL
• Consider a system of N atoms with
allowed energy levels E1, E2, E3, …Ej…
• These energies are independent of T.
• What is the set of equilibrium probabilities
for an atom having a particular energy? p1,
p2, …pj,…, i.e. what is the probability
distribution?
BDL
• Let T, V, N be constant  dF = 0 is the
condition for equilibrium.
• dF = dU –TdS – SdT = dU – TdS = 0
• Find dU and dS; plug into dF and minimize
• Σpj = 1  α Σ dpj = 0 is the constraint
BDL
• U = <E> = Σ pjEj 
• dU = d<E> =Σ(pjdEj + Ejdpj) but dEj = 0
since Ej(V, N) and V and N are constant
andEj does not depend on T.
• dU = d<E> = Σ(Ejdpj)
• S = -k Σpjℓn pj 
• dS = -k Σ(1 + ℓn pj)dpj
BDL
• dF = dU – TdS = 0
Eqn 10.6
= Σ[Ej + kT(1 + ℓn pj*) + α] dpj* = 0
• Solve for pj* = exp(- Ej /kT) exp(- α/kT – 1)
• Σpj* = 1 = exp(- α/kT – 1) Σ exp(- Ej /kT)
• BDL = pj* = [exp(- Ej /kT)]/Σ exp(- Ej /kT) =
[exp(- Ej /kT)]/Q Eqn 10.9
(Prob 6)
• Denominator = Q = partition function
Eqn 10.10
III. Applications of BDL
• p(z) = pressure of atm α N(z)
α exp (-mgz/kT)
• p(vx) = Eqn 10.15 = 1-dimensional velocity
distribution = √m/(2πkT) exp (-mvx2/2kT)
<vx2> = kT/m for 1-di
<Ek> = kT/2 for 1-di; kT for 2-di; 3kT/2 for 3-di
• p(v) = [m/(2πkT)]3/2 exp (-mv2/2kT) for 3-di
= Eqn 10.17
IV. Q = Partition Function
• Q = Σ exp(- Ej /kT)
• Tells us how particles are distributed or
partitioned into the accessible states.
(Prob 3)
• As T increases, higher energy states are
populated and Q  number of accessible
states. This is also true for energy levels
that are very close together and easily
populated. (Fig 10.c)
Q
• An inverse statement can be made: As T
decreases, higher energy states are
depopulated and with the lowest state
being the only one occupied. In this case,
Q  1. This is also true for energy levels
that are very far together and only the j = 1
level is populated. (Fig 10.5a)
Q
• The ratio Ej/kT determines if we are in the
high T (ratio is low) range or low T (ratio is
high) range. (Fig 10.6)
• If the ℓth energy level is W(Eℓ)-fold
degenerate, then Q = ΣW(Eℓ)exp(- Eℓ/kT)
• Then pℓ* = W(Eℓ)exp(- Eℓ/kT)/Q
Q
• A system of two independent and
distinguishable particles A and B has Q =
qA qB; in general Q = qN
• A system of N independent and
indistinguishable particles has Q = qN/N!
• (Prob 5, 8)
V. From Partition Function to
Thermodynamic Properties
• Recall the role of Ψ ( energy, angular
momentum, position, … i.e. properties) in
QM. To some extent, the role of Q ( U,
S, G, H… i.e. thermo. prop.s) is similar.
• Table 10.1
Prob 11
• Ensemble: collection of all possible
microstates. Canonical (constant T, V,N),
Isobaric-isothermal (T,p,N), Grand
canonical (T,V,μ), microcanonical (U,V<N)