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Classical Statistical Mechanics in the Canonical Ensemble: Application to the Classical

Classical Statistical Mechanics in the Canonical Ensemble: Application to the Classical Ideal Gas. Canonical Ensemble in Classical Statistical Mechanics.

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Classical Statistical Mechanics in the Canonical Ensemble: Application to the Classical

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  1. Classical Statistical Mechanicsin the Canonical Ensemble: Application to the Classical Ideal Gas

  2. Canonical Ensemble in ClassicalStatistical Mechanics As we’ve seen, classical phase space for a system with f degrees of freedom is f generalized coordinates & f generalized momenta (qi,pi). The classical mechanics problem is done in the Hamiltonian formulation with a Hamiltonian energy function H(q,p). There may also be a few constants of motion (conserved quantities): energy, particle number, volume, ...

  3. The Canonical Distribution in Classical Statistical Mechanics • The Partition Functionhas the form: • Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT) • A 6N Dimensional Integral! • This assumes that we’ve already solved the classicalmechanics problemfor each particle in the system so that we know the total energy E for the N particles as a function of all positions ri& momentapi. • E  E(r1,r2,r3,…rN,p1,p2,p3,…pN)

  4. CLASSICAL Statistical Mechanics: • Let A ≡any measurable, macroscopic quantity. The thermodynamic average of A ≡<A>. This is what is measured. Use probability theory to calculate <A> : P(E) ≡ e[- E/(kBT)]/Z <A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E) Another 6N Dimensional Integral!

  5. Relationship of Z to Macroscopic Parameters Summary of the Canonical Ensemble (Derivations are in the book! Results are general & apply whether it’s a classical or a quantum system!) Mean Energy: Ē  E = -∂(lnZ)/∂β <(ΔE)2> = [∂2(lnZ)/∂β2] β = 1/(kBT),kB =Boltzmann’s constant. Entropy: S = kBβĒ+ kBlnZ An important, frequently used result!

  6. Summary of the Canonical Ensemble: Helmholtz Free Energy: F = E – TS = – (kBT)lnZ Note that this gives: Z = exp[-F/(kBT)] dF = S dT – PdV, so S = – (∂F/∂T)V, P = – (∂F/∂V)T Gibbs Free Energy: G = F + PV = PV – kBTlnZ. Enthalpy: H = E + PV = PV – ∂(lnZ)/∂β

  7. Summary of the Canonical Ensemble: Mean Energy: Ē = – ∂(lnZ)/∂ = - (1/Z)(∂Z/∂) Mean Squared Energy: <E2> = (rprEr2)/(rpr) = (1/Z)(∂2Z/∂2) nth Moment: <En> = (rprErn)/(rpr) = (-1)n(1/Z)(∂nZ/∂n) Mean Square Deviation: <(ΔE)2> = <E2> - (Ē)2 = ∂2lnZ/∂2 = -∂Ē/∂. Constant Volume Heat Capacity CV = (∂Ē/∂T)V = (∂Ē/∂)(d/dT) = - kBT2(∂Ē/∂)

  8. The Classical Ideal Gas • So, in Classical Statistical Mechanics, the • Canonical Probability Distribution is: • P(E) = [e-E/(kT)]/Z • Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT) • This is the tool we will use in what follows. • As we.ve seen, from the partition function • Z all thermodynamic properties can be • calculated: pressure, energy, entropy….

  9. Consider an Ideal Gas from the point of • view of microscopic physics. It is the • simplest macroscopic system. • Therefore, its useful use it to introduce the • use of the • Canonical Ensemble in Classical • Statistical Mechanics. • The ideal gas Equation of State is • PV= nRT • n is the number of moles of gas.

  10. We’ll do Classical Statistical • Mechanics, but very briefly, lets • consider the simple Quantum • Mechanicsof an ideal gas & the take • the classical limit. • From the microscopic perspective, an • ideal gas is a system of Nnon • Interacting particles of mass m in a • volume V = abc. • (a, b, c are the box’s sides)

  11. Since there is no interaction, each • molecule can be considered a “Particle • in a Box” as in elementary quantum • mechanics. • The energy levels for such a system • have the form: • where nx, ny, nx = integers

  12. The energy levels for each molecule in the • Ideal Gas are: • l • (nx, ny, nx = integers) • (1) • The Ideal Gas molecules are non • interacting, so the gas Partition Function • has the simple form: • Z = (q)N (2) • where q  One Particle Partition Function

  13. Ideal Gas Partition Function: • Z = (q)N (2) • q  One Particle Partition Function • Using (2) in the • Canonical Ensemble • formalism gives the • expressions on the • right for: mean energy • E, equation of state P • & entropy S:

  14. The Partition Function for the 1- • dimensional particle in a “box” • under the assumption that the • energy levels are so closely spaced • that the sum becomes an integral • over phase space can be written: (3)

  15. (3) • For the 3 – dimensional particle in a • “box”, the 3 dimensions are • independent so that the Partition • Function can be written as the product • of 3 terms like equation (3). That is: (4)

  16. Using the Canonical • Ensemble expressions • from before: • The Mean Energy & • the Equation of • State can be obtained • (per mole): • To obtain, for one mole of gas:

  17. The Entropycan also be obtained: E (5) • As first discussed by Gibbs, the Entropy • in Eq. (5) is NOT CORRECT! • Specifically, its dependence on particle • number N is wrong! •  “Gibbs’ Paradox” • in the first part of Ch. 7!

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