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# Chapter 10: Boltzmann Distribution Law (BDL) - PowerPoint PPT Presentation

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)

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### Chapter 10: Boltzmann Distribution Law (BDL)

From Microscopic Energy Levels to Energy Probability Distributions

to Macroscopic Properties

• 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.

• 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?

• 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  αΣdpj = 0 is the constraint

• 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

• 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

• 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

• 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)

• 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)

• 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 ℓthenergy level is W(Eℓ)-fold degenerate, then Q = ΣW(Eℓ)exp(- Eℓ/kT)

• Then pℓ* = W(Eℓ)exp(- Eℓ/kT)/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)

• 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)