CH 240 Chemical Engineering Thermodynamics Spring 2007

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1 CH 240 Chemical Engineering Thermodynamics Spring 2007 Instructor: Nitash P. Balsara, Graduate Assistant: Paul Albertus, Course Description Covers classical thermodynamics and statistical mechanics. Fundamental laws of thermodynamics, thermodynamics potentials and Legendre transforms, equations of state, chemical equilibrium, statistical mechanics, relationship between entropy and probabilities, classical gasses, liquids, and solids, pair correlation functions, Ising model, renormalization group theory, Brownian motion, Langevin equation. Objectives 1. Define the first and second law of thermodynamics. 2. Explain energy minimization and entropy maximization. 3. Differnetiate between extensive and intensive variables. 4. Derive thermodynamic potentials using Legendre transforms. 5. Determine the state of systems by minimizing appropriate thermodynamic potential. 6. Understand the role of equations of state and Maxwell relations 7. Determine the states of systems in chemical equilibrium. 8. Introduce 7 dilemmas that classical mechanics cannot resolve: (I) Typical materials have a large number of independent variables (positions and momenta of atoms or molecules). Why is it possible to propose that the properties of a system can bet expressed in terms of relatively few variables? (II) What is the proof of the validity of the second law? (III) How do you prove the validity of the well-know equation for the Gibbs free energy of ideal mixtures? (IV) How do ideal solids, liquids, and gasses differ? (V) Why is the second virial coefficient related to intermolecular potentials. (VI) Why are the positions of the atoms important for computing the properties of fluids but not their momenta? (VII) Why is universal behavior observed near critical points. Statistical Mechanics 1. Acknowledge that material systems are composed of atoms undergoing highly uncorrelated motion. Define the concept of a microstate and probabilities of achieving specific microstates. 2. Describe Boltzmann's definition of entropy. 3. Derive the equations governing cannonical ensemble. 4. Define other ensembles based on Legendre transforms. 5. Derive the Gibbs entropy formula. 6. Work through application of ensemble equations. Two state model, ideal mixtures, ideal gas (introduce Quantum mechanics), ideal solids.

2 7. Derive equation of state for ideal liquids using spatial correlation functions. 8. Develop the Ising model and conduct Monte Carlo simulations 9. Introduce renormalization group theory. 10. Introduce Brownian motion, temporal correlations, and the Langevin equation. 11. Resolve all of the dilemmas left unanswered by classical thermodynamics. Textbook Introduction to Modern Statistical Mechanics, David Chandler, Oxford University Press, Reference Book Molecular Thermodynamics of Fluid-Phase Equilibria, John Prausnitz, Rudiger Lichenthaler, Edmundo Gomes de Avezedo, Prentice Hall, Grading: Homework: 15 % Mid-term I: 20 % Mid-term II:20% Final: 45 % Course website:

3 1. Classical Thermodynamics (Chandler, Chapters 1 and 2) Topic Content 1. introduction, first law, work, heat, energy, entropy, second law 2. fundamental relationships, S(E,X), E(S,X), variational statements of second law, thermal equilibrium, heat capacity 3. conjugate variables, Legendre transforms, enthalpy, free energies, grand potential, Maxwell relations 4. Gibbs-Duhem equation, rubber band experiment, interfacial energy 5. multiphase equilibria, stability criteria, phase rule 6. chemical, thermal and mechanical equilibrium, ΔS and ΔV at phase transitions, first and second order phase transitions 7. coexisting phases, Maxwell construction, and interfaces 2. Molecular Thermodynamics (Prausnitz, Overview of Chapters 2-11) 8. fugacity and chemical potential, equations of state (EOS), intermolecular potentials (u) 9. relationship between u and EOS, mixture properties, fugacity of pure solids and liquids, fluids mixtures, Raoult's law for grownups, models for activity coefficients, osmotic pressure 10. osmotic pressure as a measure of solute-solute interactions, solubility of solids, intro to statistical mechanics 3. Statistical Mechanics (Chapters 3, 4, 5) 11. defn. of microstates, probability P ν, ensemble averages, time average, microcanonical ensemble, S=klnΩ, 12. the ideal gas, entropy maximization implies finding most probable state 13. finish ideal gas, canonical ensemble, expression for P ν, the partition function Q, <E>, <δe 2 >, 2-state model 14. probabilistic interpretation of heat and work, regular solution theory 15. Flory-Huggins theory, generalized ensembles and partition functions 16. Q for non-interacting systems, ideal gasses from quantum mechanical point of view 17. ideal solids

4 18. classical fluids-example of systems with interactions, expression for Q in terms of r N, p N, splitting of r N and p N terms, Maxwell-Boltzmann distribution, EOS of classical fluid 19. reduced configurational distribution function, g(r), reversible work theorem 20. thermodynamic properties of g(r), second Virial coefficient and intermolecular potential 21. Ising Model (ideal magnets), lattice gas, order parameters, and Monte Carlo simulations 22. transition state sampling and nucleation in Ising magnets 23. random walks and Brownian motion 24. Langevin and Fokker-Plank equations, Brownian motion of a harmonic oscillator, fluctuation dissipation theorem, conclusion 25. renormalization group theory

5 Course Summary The course consists of two parts: (1) Classical Thermodynamics, and (2) Statistical Thermodynamics. (1) Classical Thermodynamics The course began by definitions of energy (de), work (dw), and heat (dq). and the first law (de=dq+dw) The work term was generalized to include all kinds of factors including strain in solids, magnetic field effects, and interfacial energy (dw=f i.dx i ). A system is defined by (E,X) where the components of vector X are the appropriate extensive variables that are relevant to the system. We then defined the kinds of problems that can be solved by thermodynamics: Given the initial equilibrium state of the system, how will the system respond if you change one or more constraints? To answer this question we defined a nebulous quantity S(E,X). S is a monotonic function of E and it increases when the system changes under adiabatic conditions. This is the second law. We then defined intensive variables that were partial derivatives of S with respect to E and X i. This led to the fundamental relationship (de=tds+f.dx). If either E(S,X) or S(E,X) is known, in principle, all thermodynamic problems can be solved using the first and second laws. The simplest way to do this is by using the variational statements of the second law. At constant (E,X), S is maximized. At constant (S,X), E is minimized. Temperature is shown to be the quantity that is the same when two systems are in thermal equilibrium. We introduced the notion of fluctuations, i.e. systems in thermal equilibrium exchange energy (δe 1 =-δe 2 0). We defined extensive heat capacities (C) and demonstrated the relationship between C and S. We must acknowledge that typical systems are made up of about molecules with many degrees of freedom. Dilemma #1: Why can systems be defined by a few (e.g. 3) variables (E,X)? What happens to the numerous degrees of freedom that are available to the system? Dilemma #2: Why is S maximized at constant (E,X)? In many cases E and S are not held constant. Instead, intensive variables such as T and p are held constant. Legendre transforms were introduced to solve these

6 kinds of thermodynamic problems. New potentials such as enthalpy (H), Helmholtz energy (A), Gibbs energy (G), and the grand potential (Ω) were developed (e.g. A=E- TS). Maxwell relations relate derivates if extensive and intensive variables [( S/ V) T,n =( p/ T) V,n ]. Thus derivatives that are easy to measure can be related to derivatives that are difficult to measure but important for solving problems. The Gibbs-Duhem relationship enabled deriving an expression for E (E=TS-PV+μ i n i ). Two applications of this development were discussed: (1) systems with interfaces, and (2) stretching of rubber bands. It is important to recognize that classical thermodynamics does not solve problems in isolation. We need input from experiments (e.g. pvt measurements, heat capacity, etc.) and/or statistical models. We then studied the equilibrium of multiphase systems. We defined the nature of thermal (T), chemical (μ), and mechanical (p) equilibrium. Stability criteria (C>0) were developed. The degrees of freedom of multiphase systems are given by Gibbs phase rule. Phase diagrams in various planes (e.g. P-T plane) were discussed. The properties of coexistent phases are related by the Maxwell construction. The properties of interfaces between coexisting phases and the Gibbs adsorption isotherm were developed. We discussed first and second order phase transitions, critical points and universal behavior around the critical point. Dilemma #3: Why is behavior around the critical point universal? Chemical equilibrium in vapor/liquid and solid/liquid coexistence is solved conveniently by introducing fugacity f. f can be gotten from PVT data such as the Virial and van der Waal's equations. Van der Waal developed his equation by assuming that molecules exist and they always attract each other. The attractive intermolecular potential energy Γ is due to induced dipole-induced dipole interactions. Examples of simple expressions Γ are the square-well potential, Leonard-Jones potential, and Stockmayer potential. The second Virial coefficient B is related to Γ through an integral equation. PVT measurements thus provided the first measure of intermolecular interactions. Dilemma #4: Why is B related to Γ? Dilemma #5: Why is it important to study the potential energy between molecules? What about their kinetic energy?

7 Ideal mixtures are defined by ΔG=RT[n 1 lnx 1 +n 2 lnx 2 ]. Dilemma #6: Why is this so? This enables understanding vapor-liquid equilibrium of mixtures (Raoult's law for grown-ups), and osmotic pressure of dilute solutions. Inter-solute interactions can be obtained from osmotic pressure in analogy with the Virial equation. Dilemma #7: How are (a) ideal solids and (b) liquids defined? (c) What is the origin of ideal gas behavior? To answer the dilemmas, and to improve our understanding of matter and its noncontinuum nature, we driven into the field of statistical mechanics. (2) Statistical Mechanics We began with the recognition that defining a classical system requires specifying r N and p N. Molecules are in constant motion and systems are in complex N- dimensional trajectories. Each realization or microstate ν (r N,p N ), a member of an ensemble, has different property G ν. G obs =Sum(P ν G ν ) where P ν is the probability of obtaining state ν. Like the potentials defined above (S, E, H, A, G), the ensembles are defined by certain macroscopic variables that are kept constant. The microcanonical ensemble is obtained at constant N,V,E or (E,X). In this ensemble, all states are assumed to be equally likely (P ν =1/Ω, where Ω is the number of distinguishable microstates at specified N,V,E. Boltzmann declared that S=k B lnω. We showed that S, thus defined, has all of the properties required by the second law. β=1/k B T= lnω/ E. Ω for an ideal gas was derived using a lattice model with assumptions that particles are dilute and non-interacting. We showed that Ω or equivalently S is a very steep function of E (Ω~E 3N/2 ). Thus when E increases slightly Ω increases a lot and when E decreases slightly Ω decreases a lot. Ω of a composite system with the subsystems in thermal equilibrium is given by a sharply peaked function. i.e. Ω or S is maximized! But Ω is the probability of obtaining a given state. At constant (E,X) the most probable state is the maximum entropy state. Dilemmas #1 and #2 are resolved. We also studied fluctuations of E (δe) in the vicinity of equilibrium. In most systems (away from the critical point) δe~n -1/2. Departures from equilibrium approach zero for large N.

8 Since we are often interested in keeping intensive variables constant, we need to construct other ensembles. The canonical ensemble is a constant N,V,T ensemble. Here P ν ~exp(-βe ν ), A=-β -1 lnq (governing equation), and P=β -1 ( lnq/ V) (eq of state), where the partition sum Q=Sum[exp(-βE ν )]. The appropriate Legendre transform potential appears naturally in the governing equation. Generalized ensembles were formally developed for situations where other variables are held constant. This led to generalized partition sums Θ and the Gibbs entropy formula. Applications of ensembles considered: (1) Two-state model (2) Ideal solutions, resolved dilemma #6. (3) The ideal gas as particles in a box (introduction to Quantum Mechanics). The particles have many degrees of freedom (dof). Those that are not of interest are separated by Q=Q 1 Q 2 Q 3... due to their independence. (4) Einstein and Debye models for ideal solids resolved dilemma #7(a). Learned about collective modes, independent and fictitious dofs and phonons. In all cases thus far, the dofs of the system were independent. Classical fluids provided our first introduction to systems where the dofs could not be made independent. Here E ν =K(p N )+U(r N ). The distribution of momentum is given by the Maxwell-Boltzmann law (applies to ideal gasses and real liquids). Since the momentum does not depend on volume, P is independent of momentum [P=β -1 ( lnq/ V)]. Equation of state of classical fluids thus depends on U(r N ) only. We introduced reduced configurational distribution functions ρ (2/N) (r 1,r 2 ) and radial distribution functions g(r). The importance of spatial correlations was recognized. The reversible work theorem states that g(r)=exp[-βw(r)], thus relating the structure of fluids and thermodynamics. In the dilute limit βp=ρ+ρ 2 B where B is related to intermolecular potential energy. Dilemma #4 is thus resolved. The Ising model for magnets was another example of a system where dofs are coupled. The energy was related to the order parameter n i of the ith cell and that of its neighbors. Analytical solutions for 1-D and 2-D Ising models have been developed by Ising and Onsager, respectively. The 3-D Ising model can only be solved numerically. The 2D and 3D Ising models provide a quantitative picture of magnetization in the presence and absence of external magnetic fields. This leads to the concept of spontaneous symmetry breaking. By changing the definition of

9 the order parameter, the Ising model can be used to study other systems such as the lattice gas model. The Monte Carlo simulation method was developed to obtain Q for an Ising model numerically. Microscopic reversibility is ensured by the method of Metropolis. Spontaneous fluctuations, symmetry breaking, and coexistence are seen vividly in the simulations. Fluctuations with large length scales are seen near the critical point, demonstrating yet another connection between structure and thermodynamics. The use of Monte Carlo simulations to study first-order phase transitions was described by Albert Pan. Details regarding systems near the critical point were described using Renormalization Group (RG) theory. Here the large length scale of the fluctuations enables the use of block spin transformations to reduce the dof of our system. RG enables calculating energy after successive block spin transformations. If the results of these successive transformations end in a fixed point, one obtains Universal behavior. The reason for universal behavior near critical points is the large length scale of fluctuations. This makes detailed interactions at the molecular level unimportant. These unimportant dofs are integrated out by RG. Dilemma #3 is resolved. We then described Brownian motion, which lies at the heart of fluctuations and phase transitions. We began by a description of Brownian motion as a random walk and showed that the probability P of obtaining a position R at time t was governed by the familiar diffusion equation. The connection between diffusion and underlying random walks was then exposed. Temporal correlations during Brownian motion are captured by the Langevin equation. We concluded our course with a study of Brownian motion in an ideal solid.

Contents. 1 Introduction and guide for this text 1. 2 Equilibrium and entropy 6. 3 Energy and how the microscopic world works 21

Contents. 1 Introduction and guide for this text 1. 2 Equilibrium and entropy 6. 3 Energy and how the microscopic world works 21 Preface Reference tables Table A Counting and combinatorics formulae Table B Useful integrals, expansions, and approximations Table C Extensive thermodynamic potentials Table D Intensive per-particle thermodynamic