Classical Statistical Mechanics in the Canonical Ensemble Application

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

Canonical Ensemble in Classical Statistical 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, . . .

The Canonical Distribution in Classical Statistical Mechanics The Partition Function has the form: Z ≡ ∫∫∫d 3 r 1 d 3 r 2…d 3 r. N d 3 p 1 d 3 p 2…d 3 p. N e(-E/k. T) A 6 N Dimensional Integral! • This assumes that we’ve already solved the classical mechanics problem for each particle in the system so that we know the total energy E for the N particles as a function of all positions ri & momenta pi. E E(r 1, r 2, r 3, …r. N, p 1, p 2, p 3, …p. N)

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/(k T)] e /Z B <A>≡ ∫∫∫(A)d 3 r 1 d 3 r 2…d 3 r. N d 3 p 1 d 3 p 2…d 3 p. NP(E) Another 6 N Dimensional Integral!

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 = -∂(ln. Z)/∂β <(ΔE)2> = [∂2(ln. Z)/∂β 2] β = 1/(k. BT), k. B = Boltzmann’s constant. • Entropy: S = k. BβĒ + k. Bln. Z An important, frequently used result!

Summary of the Canonical Ensemble: • Helmholtz Free Energy: F = E – TS = – (k. BT)ln. Z Note that this gives: Z = exp[-F/(k. BT)] d. F = S d. T – Pd. V, so S = – (∂F/∂T)V, P = – (∂F/∂V)T • Gibbs Free Energy: G = F + PV = PV – k. BT ln. Z. • Enthalpy: H = E + PV = PV – ∂(ln. Z)/∂β

Summary of the Canonical Ensemble: • Mean Energy: Ē = – ∂(ln. Z)/∂ = - (1/Z)(∂Z/∂ ) • Mean Squared Energy: 2 2 <E > = ( rpr. Er )/( rpr) = (1/Z)(∂2 Z/∂ 2) • nth Moment: n n n <E > = ( rpr. Er )/( rpr) = (-1) (1/Z)(∂ Z/∂ ) • Mean Square Deviation: <(ΔE)2> = <E 2> - (Ē)2 = ∂2 ln. Z/∂ 2 = -∂Ē/∂. • Constant Volume Heat Capacity CV = (∂Ē/∂T)V = (∂Ē/∂ )(d /d. T) = - k. BT 2∂Ē/ ∂

The Classical Ideal Gas • So, in Classical Statistical Mechanics, the Canonical Probability Distribution is: P(E) = -E/(k. T) [e ]/Z Z ≡ ∫∫∫d 3 r 1 d 3 r 2…d 3 r. N d 3 p 1 d 3 p 2…d 3 p. N e(-E/k. T) • 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….

• 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= n. RT n is the number of moles of gas.

• We’ll do Classical Statistical Mechanics, but very briefly, lets consider the simple Quantum Mechanics of an ideal gas & the take the classical limit. • From the microscopic perspective, an ideal gas is a system of N non interacting Particles of mass m confined in a volume V = abc. (a, b, c are the box’s sides) • Since there is no interaction, each molecule can be considered a “Particle in a Box” as in elementary quantum mechanics.

• 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

• The energy levels for each molecule in the Ideal Gas have the form: (1) l (nx, ny, nx = integers) • 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

• 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:

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

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

• The Entropy can also be obtained: E (5) • As first discussed by Gibbs, 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!