Ch 11: The Statistical Mechanics of Simple Gases and Solids

Ch 11: The Statistical Mechanics of Simple Gases and Solids Models for Molecular Motion Correspondence Principle Gases and Solids

I. Classical  Quantum Mechanics • Energy: E = T + V  મ = C d2/dx2 + V(x) • મΨ = EΨ Schroedinger Eqn Eqn 11.4 • મ = Hamiltonian operator = 1-di energy operator; time independent; must be Hermitian; examples of CM functions  QM operators • E = value of energy; eigenvalue; real number; {E} values are the allowed energy values.

CM  QM • Ψ(x) = wave function = math function for system = eigenfunction; time independent • │Ψ2│dx = probability of finding particle in (x, dx+dx) • │Ψ2│ = probability density > 0 • Ψ(x) must be finite, continuous and single-valued • Given system, the goal of QM is to • write મ • apply boundary conditions • find the set of {Ψ} and {E} such that મΨ = EΨ. • 8 steps

II. Quantum Mechanical Models • Particle in the Box = model for translational energy • Simple Harmonic Oscillator (SHO) = model for molecular vibrations • Rigid Rotor (RR) = model for molecular rotations. • Each has simplifying assumptions and yields {Ψ} and {E}.

A. Translational Motion in 1-di • Consider free particle of mass, m, constrained to move in 1-di (x-axis). Let V(x) = 0 from x = -∞ to x = +∞. • મΨ = -(h2/8π2m) Ψ”(x) = εΨ(x) • We need to find a function whose 2nd derivative yields the function back again. • Ψ(x) = sin ax, cos ax, exp(± iax) • ε = a2h2/8π2m; a = real number; ε is continuous.

Translational Motion in 1-di • Now put the particle in a 1-di box where V(x) = 0 between (0, L). But outside the box (x≤0 and x≥L), V(x) = ∞. (2 regions) • Outside box, Ψ(x) = 0 Eqn 11.8 Why? • Boundary conditions: Ψ(0) = 0 = Ψ(L) • મΨ = -(h2/8π2m) Ψ”(x) = εΨ(x) • Solve for eigenfunctions and eigenvalues

Translational Motion in 1-di • Inside box, Ψn(x) = √2/L sin (nπx/L) and energies are εn = n2h2/8mL2 Eqn 11.12 • n = positive integer = quantum number • {εn } are quantized, divergent, nondegenerate. (Fig 11.4, Example 11.1) • Translational temp = Θtrans = h2/(8mL2k) • Θtrans /T << 1  qtr = (2πmkT/h2)½ L • Note as T and L incr, qtr incr. Recall S(T,V)

Correspondence Principle • As T  ∞ and/or n ∞ and/or m ∞, then lim QM = CM • CM is a special case of QM, not an exception or inconsistency. • As n  ∞, │Ψ2│ become more uniform (CM result); see Fig 11.5

Translational Motion in 3-di • Consider a box of length x = a, depth y = b and height z = c with V = 0 inside box and V = ∞ outside box (boundary conditions) • Separate the variables to convert one 3-di problem (Eqn 11.16) into 3 1-di problems. I.e., solve 3 1-di box problems • Write મ(x, y, z) = મx + મy + મz • Write S-Eqn

Translational Motion in 3-di • Write Ψ(x, y, z) = ΨxΨyΨz Each is an independent sin function. • Write ε = εnx + εny + εnz Eqn 11.17 • Quantized energies • Solve for eigenfunctions and eigenvalues. • Write qtr = qxqyqz = (2πmkT/h2)3/2 V • Example 11.2 (qtr = 1030 states/atom) • Prob 11.3, 5, 15

B. Vibrational Motion • Define system, i.e. V(x) • Write મ(x) • Write and solve S-Eqn • Find {Ψn(x)} = NC x HP x AS • Find {εn} • Write qvib and Θvib • Example 11.3 (note error; qvib = 1)

C. Rotational Motion • Define system, i.e. V(θ,φ) • Write મ(θ,φ) • Write and solve S-Eqn • Find {Ψℓ(θ,φ) = Θ(θ) Φ(φ)} = NC x SH • Find {εℓ}; note degeneracy • Write qrot and Θrot (linear and nonlinear) • Example 11.4 (qrot = 72) • Prob 11.10

D. Electronic Energy • This refers to the various allowed electronic states: ground state [lowest energy, 1s2, 2s for Li), first excited state (1s2, 2p for Li*), second excited state…] • qelec = Σ gi exp (-Δεiβ) • Table 11.3 • Finally, q = πqj where j = trans, vib, rot, elec; Table 11.2

III. QM  q  Ideal Gas Properties (micro  macro) • Combine Table 10.1 with partition functions for trans, vib, rot, elect as appropriate: IGL, U, Cv, S (T 11.4), F, μ • Note Q = qN or qN/N! • Distinguish monatomic vs diatomic vs polyatomic gas. • Distinguish linear vs nonlinear molecular gas

Equipartition Theorem • Internal energy is distributed uniformly over each DegF. Full contributions α T.

Problems • 11.1, 11.11, 17

IV. Heat Capacity of Solids • Dulong and Petit: As T  0K, CV  3R • Expt: As T  0K, CV  0 • Einstein: As T  0K, CV  T-2 exp(-hν/β) [accounts for ZPE that can take up energy] • Better Expts: As T  0K, CV  T3 • Debye: As T  0K, CV  T3 [allows coupling between vibrational modes] • Prob 11.9

Ch 11: The Statistical Mechanics of Simple Gases and Solids