Introduction Postulates of thermodynamics

1. Introduction • Postulates of thermodynamics • Thermodynamic equilibrium in isolatedand isentropic systems • Thermodynamic equilibrium in systemswith other constraints • Thermodynamic processes and engines pp. 1–92 • Thermodynamics of mixtures(multicomponent systems) • Phase equilibria • Equilibria of chemical reactions • Extension of thermodynamics foradditional interactions(non-simple systems) • Elements ofequilibrium statistical thermodynamics • Towards equilibrium– elements of transport phenomena pp. 92–303 • Appendixpp. 305-328

2. Table of contents • Introduction • Postulates of thermodynamics • Thermodynamic equilibrium in isolated and isentropic systems • Thermodynamic equilibrium in systems with other constraints • Thermodynamic processes and engines pp. 1–92 • Thermodynamics of mixtures (multicomponent systems) • Phase equilibria • Equilibria of chemical reactions • Extension of thermodynamics for additional interactions(non-simple systems) • Elements of equilibrium statistical thermodynamics • Towards equilibrium – elements of transport phenomena pp. 92–303 • Appendixpp. 305–328

3. Table of contents Appendix F1. Useful relations of multivariate calculus F2. Changing extensive variables to intensive ones: Legendre transformation F3. Classical thermodynamics: the laws

4. Fundamentals of postulatory thermodynamics An important definition: the thermodynamic system The objects described by thermodynamics are called thermodynamic systems. These are not simply “the partof the physical universe that is under consideration” (or in whichwe have special interest), rather material bodies having a special property; they are in equilibrium. The condition of equilibrium can also be formulated so that thermodynamics is valid for those bodies at rest for which the predictions based on thermodynamic relations coincide with reality (i. e. with experimental results). This is ana posterioridefinition; the validity of thermodynamic description can be verified after its actual application. However, thermodynamics offers a valid description for an astonishingly wide variety of matter and phenomena.

5. Postulatory thermodynamics A practical simplification: the simple system Simple systems are pieces of matter that aremacroscopically homogeneous and isotropic, electrically uncharged, chemically inert, large enough so that surface effects can be neglected, and they are not acted on by electric, magnetic or gravitational fields.Postulates will thus be more compact, and these restrictions largely facilitate thermodynamic description without limitations to apply it later to more complicated systems where these limitations are not obeyed. Postulates will be formulated for physical bodies that are homogeneous and isotropic, and their only possibility to interact with the surroundings is mechanical work exerted by volume change, plus thermal and chemical interactions.

6. Postulates of thermodynamics 1. There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy U, the volume V, and the amounts of the K chemical components n1, n2,…, nK . 2. There exists a function (called the entropy, denoted by S ) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states. 3. The entropy of a composite system is additive over the constituentsubsystems. The entropy is continuous and differentiable and is a strictly increasing function of the internal energy. 4. The entropy of any system is non-negative and vanishes in the state for which the derivative (∂U/∂S)V,n= 0. (I. e., at T = 0.)

7. Summary of the postulates (Simple) thermodynamic systems can be described byK+2 extensive variables.Extensive quantities are their homogeneous linear functions. Derivatives of these functions are homogeneous zero order. Solving thermodynamic problems can be done using differential- and integral calculus of multivariate functions. Equilibrium calculations– knowing the fundamental equations – can be reduced to extremum calculations. Postulates together with fundamental equations can be used directlyto solve any thermodynamical problems.

8. Relations of the functions S and U S(U,V,n1,n2,…nK) is concave, and a strictly monotonousfunction of U In equilibrium,at constant energyU,S is maximal;at constant entropyS,U is minimal.

9. Identifying (first order) derivatives We know:at constantS and n (in closed, adiabatic systems): (This is the volume work.) Similarly:at constantV and n (in closed, rigid wall systems): (This is the absorbed heat.) Properties of the derivative confirm: at constant S andV (in rigid, adiabatic systems): (This is energy change due to material transport) The relevant derivative is called chemical potential:

10. Identifying (first order) derivatives is negative pressure, is temperature, is chemical potential. The total differential can thus be written (in a simpler notation) as:

11. Equilibrium calculations isentropic, rigid, closed system Equilibrium condition:dU= dUα + dU β = 0 S α, V α, n α S β, V β, n β Uα Uβ S α + S β = constant; – dSα = dS β V α + V β = constant; – dV α = dV β impermeable, initially fixed,thermally isolated piston,then freely moving, diathermal Consequences of impermeability (piston): nα = constant; nβ = constant → dn α = 0; dn β = 0 Equilibrium: Tα = Tβ and Pα = Pβ

12. Equilibrium calculations isentropic, rigid, closed system Condition of thermal andmechanical equilibriumin the composit system: S α, V α, n α S β, V β, n β Uα Uβ Tα = Tβ and Pα = Pβ 4 variables Sα , Vα , S β and V β are to be known at equilibrium. They can be calculated by solving the 4 equations: T α (Sα, V α, nα) = T β (Sβ, V β, nβ) P α (Sα, V α, nα) = P β (Sβ, V β, nβ) Sα + Sβ = S(constant) Vα + Vβ = V(constant)

13. Equilibrium at constant temperature and pressure isentropic, rigid, closed systemT= T randP=Pr(constants) equilibrium condition: the „internal system” is closedn r = constant andn= constant d(U+Ur) = dU+TrdSr – PrdVr = 0 S r, V r, n rT r, Pr S, V, n T, P Sr + S= constant; – dSr = dS V r + V= constant; – dV r = dV d(U+Ur) = dU+TrdSr – PrdVr = dU+TrdS– PrdV= 0 T=Trand P = Pr d(U+Ur) = dU–TdS+ PdV= d(U–TS+ PV) = 0 minimizingU+Uris equivalent to minimizing U–TS+ PV Equilibrium condition at constant temperature and pressure: minimum of the Gibbs potential G = U – TS+ PV

14. Rankine vapor cycle and engines heat engine refrigerator

15. Fugacty and interrelation of activities Illustration of thethermodynamic definitionof fugacity

16. Fugacity and interrelation of activities Relation of the activities fi(referenced to infinite dilution) and γi(referenced to pure substancefor the same system

17. Overview of different activities

18. Phase diagram of a van der Waals fluid Equilibrium condition: and

19. P(V,T ) phase diagram of a pure substance contractingwhen freezing

20. P(V,T ) phase diagram of a pure substance expanding when freezing

21. Thermodynamics of phase separation 2 components,liquid-liquid molar Gibbs potential (g) of (heterogeneous)mechanical dispersion and (homogeneous)mixture Common tangents

22. Thermodynamics of phase separation 2 components,solid-liquid

23. Thermodynamics of phase separation 2 components,solid-liquid

24. Other binary solid-liquid phase diagrams compound formation peritectic reaction syntectic reaction monotectic reaction

25. Three-componentphase diagrams 3D diagram 2D projection 3D diagram 2D projection

26. Factors influencing chemical equilibria Example: 1 ½ H2 + ½ N2 NH3 reaction mixing

27. Extension for additional interactions surface effects(elements of surface chemistry) electrically charged phases(elements of electrochemistry)

28. Energydistributionincanonicalensembles density function ofmultiparticleenergy distribution density function ofsingle particleenergy distribution

29. General interpretation of entropy Misunderstandings due to the interpretation as “order–disorder” ordered disordered greater entropy smaller entropy

30. Viscuous flow as momentum transfer

31. Lagrange-transformation (Appendix) the envelope of the tangent linesdetermines the curve

32. Special terms and notation explained The words diabatic, adiabatic and diathermalhave Greek origin. The Greek noun διαßασις [diabasis] designates a pass through,e. g., a river, and its derivative διαßατικος [diabatikos] means the possibility that something can be passed through. Adding the prefix α- expressing negation, we get the adjective αδιαßατικος [adiabatikos] meaning non-passability. In thermodynamic context, diabatic means the possibility for heat to cross the wall of the container, while adiabatic has the opposite meaning, i. e.the impossibility for heat to cross. …. The name comes from the German freie Energie(free energy). It also has another name, Helmholtz potential, to honor Hermann Ludwig Ferdinand von Helmholtz (1821-1894) German physician and physicist. Apart from F, it is denoted sometimes by A, the first letter of the German word Arbeit = work, referring to the available useful work of a system.

33. Summing up Összefoglalás • easy-to-follow basis of thermodynamics • postulates ready-to-use in equilibrium calculations • detailed discussion of multicomponent systems • sound thermodynamic foundations ofphase transitions & related equilibria chemical reactions (homogeneous & heterogeneous) surface chemistry electrochemistry • exact explanation of statistical thermodynamics • elements of nonequilibrium thermodynamics (transport) • Appendix: calculus + laws of classical thermodynamics

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