1. Chapter 10: Boltzmann Distribution Law (BDL) From Microscopic Energy Levels to Energy Probability Distributions to Macroscopic Properties

2. 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. Recall: requires the probability distribution (i.e. weighting term) for the system.

3. II. BDL Consider a system of N atoms with allowed energy levels E 1, E 2, E 3, …E j … These energies are independent of T. What is the set of equilibrium probabilities for an atom having a particular energy? p 1, p 2, …p j,…, i.e. what is the probability distribution?

4. 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 Σp j = 1  α Σ dp j = 0 is the constraint

5. BDL U = = Σ p j E j  dU = d =Σ(p j dE j + E j dp j ) but dE j = 0 since E j (V, N) and V and N are constant andE j does not depend on T. dU = d = Σ(E j dp j ) S = -k Σp j ℓn p j  dS = -k Σ(1 + ℓn p j )dp j

6. BDL dF = dU – TdS = 0Eqn 10.6 = Σ[E j + kT(1 + ℓn p j * ) + α] dp j * = 0 Solve for p j * = exp(- E j /kT) exp(- α/kT – 1) Σp j * = 1 = exp(- α/kT – 1) Σ exp(- E j /kT) BDL = p j * = [exp(- E j /kT)]/Σ exp(- E j /kT) = [exp(- E j /kT)]/QEqn 10.9(Prob 6) Denominator = Q = partition function Eqn 10.10

7. III. Applications of BDL p(z) = pressure of atm α N(z) α exp (-mgz/kT) p(v x ) = Eqn 10.15 = 1-dimensional velocity distribution = √m/(2πkT) exp (-mv x 2 /2kT) = kT/m for 1-di = kT/2 for 1-di; kT for 2-di; 3kT/2 for 3-di p(v) = [m/(2πkT)] 3/2 exp (-mv 2 /2kT) for 3-di = Eqn 10.17

8. IV. Q = Partition Function Q = Σ exp(- E j /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)

9. 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)

10. Q The ratio E j /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 ℓ th energy level is W(E ℓ )-fold degenerate, then Q = ΣW(E ℓ )exp(- E ℓ /kT) Then p ℓ * = W(E ℓ )exp(- E ℓ /kT)/Q

11. Q A system of two independent and distinguishable particles A and B has Q = q A q B ; in general Q = q N A system of N independent and indistinguishable particles has Q = q N /N! (Prob 5, 8)

12. 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 10.1Prob 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)