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Chapter 2: Extremum Principles Predict Equilibria

Chapter 2 explores the extremum principles governing equilibria in physical systems. Energy tends to minimize while entropy maximizes, leading to various stable, metastable, and unstable states. Mathematical methods identify critical points where forces balance. The first extremum principle states that at equilibrium, net forces sum to zero. The second emphasizes maximizing multiplicity among indistinguishable objects, revealing the most probable states of systems. A lattice model is introduced, outlining assumptions and examples of occupancy arrangements, facilitating a deeper understanding of pressure and chemical potentials.

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Chapter 2: Extremum Principles Predict Equilibria

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  1. Chapter 2: Extremum Principles Predict Equilibria Extremum Principles Equilibrium Model

  2. I. Extremum Principles • Energy (E) tends to reach its minimum. • Entropy (S) tends to reach its maximum. • We can use math to find states that are stable, metastable, unstable, neutral. • When first derivative = 0, there is a min or max. • There is a min when second derivative > 0. • There is a max when the second derivative < 0.

  3. Definitions • Degree of Freedom (variable) vs Constraint (fixed) • Force is a measure of the tendency of a system toward equilibrium; f = -(dV/dx) = 0

  4. II. First Extremum Principle: Equilibrium (Fig 2.5) • At Equilibrium, the net force is zero (f = 0).

  5. III. Second Extremum Principle: Maximize Multiplicity, W • Given N indistinguishable objects each capable of t values (e.g. a coin has t = 2; a die has t = 6), the number of permutations or different sequences is called the multiplicity or W. • W = N!/π ni! Eqn 1.18 • Each sequence is equally possible but as N increases, one particular composition emerges as the most probable. Fig 2.6

  6. Multiplicity • The most probable composition translates into the state most likely to be observed. • This most probable composition is associated with the maximum W value ( or max ln W value). • To find this composition, we must find n* such that d lnW/ dn = 0.

  7. IV. Lattice Model • Assumptions • Atoms and parts of molecules = hard sphere beads. • Space is divided into bead-sized boxes = lattice sites. • The occupancy number of a lattice site is zero or one. • N = # particles • M = # sites and N < M

  8. Examples • # permutations or arrangements of empty and occupied sites = M!/[N! (M-N)!] • Pressure: gas spreads out into largest possible volume • Chemical potential: gases tend to uniform particle distribution or maximum mixing • Elasticity

  9. Stirling’s Approximation • x! ~ (x/e)x p 36 • n! ~ (2πn)1/2 (n/e)n p 56-57 • ℓn n! ~ n ℓn n – n p 56-57 • These expressions become more valid as x or n increases.

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