MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 1: Assemblies⇒Ensembles, the Ergodic Hypothesis TOPICS COVERED This is a course in building microscopic models for macroscopicphenomena. Most of first half involves idealized systems, where inter-particle interactions can be ignored and where individual particles areadequately described by simple energy level formulas (from QuantumMechanics 5.61 or Classical Mechanics). The second half deals with non-ideality, interacting atoms, as in solids and, in gas phase collisions andchemical reactions. I. Equilibrium Statistical Mechanics (J. W. Gibbs)microscopic basis for macroscopic properties Equilibrium Thermodynamics: 5.60 Quantum mechanics: 5.61 U, H, A, G, S, µ, p, V, T, CV, Cp translation↔particle in a box(nothing microscopic needed)nuclear spinideal gas, ideal solutionrotationphase transitionsvibrationchemical equilibrium electronic electrons, atoms, molecules,Non-equilibriumphotonsChemical kinetics, Arrheniuspermutation symmetryTransport spectroscopyClassical Mechanics: 8.01 Newton’s Laws Kinematics, Phase space Statistical Mechanics: 5.62 (BULK) macro from micro (single molecule properties)idealized micro (no inter-particle interactions)idealized interactions (tricks to build model)models for solids: heat capacity, electrical conductivityKinetic Theory of Gases(Collision Theory) transport (mass, energy, momentum) Transition State Theory5.62 Spring 2008 Lecture 1, Page 2 II. Solid-State Chemistry models for solids prediction of macroscopic properties from microscopic interactions III. Kinetics Models • Kinetic Theory of Gases (Boltzmann) •bulk properties obtained from averages over speeddistributions •less powerful than stat. mech., but simpler to apply in everdaycircumstances •transport properties — relaxation to equilibrium IV. Theories of Reaction Rates bridge between microscopic properties and macroscopic reaction rate:result of many microscopic collisions Collision Theory — based on kinetic theory — fraction of collisionsthat are effective in causing reaction Transition-State Theory — based on stat. mech. probability that aspecial state (transition state) is occupied) reaction dynamics, potential energy surfaces revised 2/6/08 4:18 PM5.62 Spring 2008 Lecture 1, Page 3 Non-Lecture Review of Thermodyamics First Law: dU = d—q + d—w ∫ dU = 0 Find complete set of functions of state and their natural variables: U(S, V, {ni}) dU = TdS – pdV + ∑µidni (Tsurr, pext) i H = U + pVH(S, p, {ni}) dH = TdS + Vdp + ∑µidni i A = U – TS A(T, V, {ni}) dA =–SdT – pdV +∑µidni i G = H – TS = A + pVG(T, p, {ni}) dG =–SdT + Vdp +∑µidni i Many quantities are defined in terms of partial derivatives. ⎛⎜⎝ ∂U ⎛⎜⎝ ⎞⎟⎠V, n{ i ∂H ⎞⎟⎠ T = = ∂S ∂S } p, ni{ } ⎛∂U ⎞ ⎛∂A ⎞ p = −⎜⎝∂V ⎟ S, ni= −⎜∂V ⎟ T, ni⎠{ } ⎝ ⎠{ } revised 2/6/08 4:18 PM5.62 Spring 2008 Lecture 1, Page 4 ∂H ∂G ⎛ ⎛⎞⎟⎠⎞ V = ⎜⎝ ⎜⎝ ⎟⎠ = ∂p ∂p T, n{ i }S, ni{ } S = −⎜⎛∂A ⎞ ⎛∂G ⎞ ⎝∂T ⎟ V, ni= −⎜∂T ⎠⎟ p, n{ } i ⎠{ } ⎝⎛∂U ⎞ ⎛∂H ⎞ ⎛∂A ⎞ ⎛∂G ⎞ =µi ⎜⎝∂ni ⎟⎠S,V, n{ j =ni }= ⎜⎝∂ni ⎠⎟ S,p, n { j ≠ni }= ⎝⎜∂ni ⎠⎟ T,V, n { j ≠ni }= ⎝⎜∂ni ⎠⎟ T,p, n { j ≠ni } Maxwell relationships (mixed second derivatives), e.g. ⎛ ⎜⎝ ∂2U ∂S∂V ⎛⎞⎟⎠∂2U ∂V∂S ⎞⎟⎠ ∂T ∂V ⎞⎟⎠S, n{ i } ⎛⎜⎝ ⎛⎜⎝ ⎞⎟⎠ ∂p⎜⎝ ⇒ − = = ∂S V, ni{ } { } ni{ } ni⎛∂2G ⎞ ⎛∂2G ⎞ ⎛∂µi ⎞ ⎛∂V ⎞ ⎜ ⎟ = ⎜ ⎟ ⇒⎜ ⎟ = Vi⎝∂p∂ni ⎠T, nj ≠ni ⎝∂ni∂p ⎠T, nj ≠ni ⎝∂p ⎠⎟ T, nj ≠ni = ⎝⎜∂ni ⎠T, n j ≠ni{ } { } { }{ } ⎛∂2G ⎞ ⎛∂2G ⎞ ⎛∂µi ⎞⎟ = −Si. ⎜ ⎠⎝∂T∂ni ⎠⎟ p, n { j ≠ni }= ⎝⎜∂ni∂T⎠⎟ p, n{ j ≠ni } ⇒ ⎝⎜ ∂T p, n { j ≠ni } This allows us to express all Thermodynamic quantities in terms of G and toexpress G in terms of measurable quantities T, V, p, CV, Cp. Suppose we know G(T,p) S = −⎜⎛∂G ⎞ ⎝∂T ⎠⎟ p ∂G ⎛ ⎞ V = ⎜⎝ ⎟⎠ ∂p T H = G − T⎜⎛∂G ⎟⎞ ⎝∂T ⎠p U = G − T⎛⎜∂G ⎞ ⎛∂G ⎞ ⎟⎝∂T ⎟⎠p − p⎜⎝∂p ⎠T A = G − p ⎛⎝∂∂Gp ⎞⎠⎟ T ⎜ revised 2/6/08 4:18 PM5.62 Spring 2008 Lecture 1, Page 5 Cp = −T⎜⎛⎝∂∂2TG2 ⎟⎞⎠p So we can derive all thermodynamic quantities from G(T,p). If we can derive a Statistical Mechanical expression for G(T,p), then we will have allother thermodynamic functions of state. It is also possible to show how all Thermodynamic quantities maybe derivedfrom measurements of p, V, T, Cp, CV. From the natural variables we know the conditions for equilibrium.(Actually this is how we discovered in 5.60 all of the state functions andtheir natural variables.) Quantities HeldConstant Condition for Equilibrium N, p, T G minimized N, V, T A minimized N, p, S H minimized N, V, S U minimized N, U, V closed, isolatedsystem   or N, H,p S maximized (2nd Law) We are going to talk about two kinds of “partition functions” in 5.62. Microcanonical Ω(N, E, V) ⇔ S(N, E, V) Canonical Q(N, T, V) ⇔ A(N, T, V) revised 2/6/08 4:18 PM5.62 Spring 2008 Lecture 1, Page 6 Let’s begin! Goal of Statistical Mechanics: describe macroscopic bulk Thermodynamic properties in terms of microscopic atomic and molecular properties. These microscopic properties are generally measured by spectroscopy. Macroscopic: U, H, A, G, S, µ, p, V, T, CV, Cp complete intensive description of a bulk system: N ≈ 1023 particles e.g. pV = RT only two intensive variables are needed! Gibbs Phase Law F = C – P + 2 # degrees of freedom # components # phases There are only a few things about a bulk system that we can (or need to) measure! Microscopic: N particle monatomic gas If we assume non-interacting particles, we must describe the “state” of each particle in the system. Two ways we might do this. Classical Mechanics: px, py, pz, x, y, z for each particle (p3N ,q3N )  Quantum Mechanics: quantum state (nx, ny, nz) for each particle. Classical Mechanics: N particles, 6N degrees of freedom Quantum Mechanics: N particles, 3N degrees of