1. Címlap Postulatory Thermodynamics Ernő Keszei Loránd Eötvös University Budapest, Hungary http://keszei.chem.elte.hu/

2. Introduction Survey of the laws of classical thermodynamics Postulates of thermodynamics Fundamental equations and equations of state Equilibrium calculations based on postulates Outline

3. Avant propos Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you anymore. Arnold Sommerfeld

4. Problem with teaching thermodynamics Let’s take an example: probability theory Postulates of probability theory: 1. The probability of event A is P(A) > 0 2. If A and B are disjoint events, i. e. A  B = 0, then P(A  B) = P(A) + P(B) 3. For all possible events (the entire sample space S) the equality P(S) = 1 holds Using these postulates, „all possible theorems” can be proved, i. e., all probability theory problems can be solved. Important definition: random experiment and its outcome; a (random) event

5. Fundamentals of (classical) thermodynamics: the laws Two popular textbooks of physical chemistry Atkins P, de Paula J (2009) Physical Chemistry, 9 th edn., Oxford University Press, Oxford Silbey L J, Alberty R A, Moungi G B (2004) Physical Chemistry, 4 th edn., Wiley, New York (Traditional textbook of MIT; typically, a new co-author replaces an old one at each new edition.)

6. Definition of a thermodynamic system Atkins: The system is the part of the world, in which we have special interest. The surroundings are where we make our measurements. Alberty: A thermodynamic system is that part of the physical universe that is under consideration. A system is separated from the rest of the universe by a real or imaginary boundary. The part of the universe outside the boundary is referred to as surroundings. (Introduction: Thermodynamics is concerned with equilibrium states of matter and has nothing to do with time.)

7. The Zeroth Law of thermodynamics Atkins: If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C is also in thermal equilibrium with A. Preceeding statement: Thermal equilibrium is established if no change of state occurs when two objects A and B are in contact through a diathermic boundary. Alberty: It is an experimental fact that if system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other. Preceeding statement: If two closed systems with fixed volume are brought together so that they are in thermal contact, changes may take place in the properties of both. Eventually, a state is reached in which there is no further change, and this is the state of thermal equilibrium.

8. The First Law of thermodynamics Atkins: If we write w for the work done on a system, q for the energy transferred as heat to a system, and ΔU for the resulting change in internal energy, then it follows that ΔU = q + w Alberty: If both heat and work are added to the system, ΔU = q + w For an infinitesimal change in state dU = đq + đw The đ indicates that q and w are not exact differentials.

9. The Second Law of thermodynamics Atkins: No process is possible, in which the sole result is the absorption of heat from a reservoir and its complete conversion into work. (In terms of the entropy:) The entropy of an isolated system increases in the course of spontaneous change: ΔS tot > 0 where S tot is the total entropy of the system and its surroundings. Later (!!): The thermodynamic definition of entropy is based on the expression: Further on: proof of the entropy being a state function, making use of a Carnot cycle.

10. The Second Law of thermodynamics Alberty: The second law in the form we will find most useful: In this form, the second law provides a criterion for a spontaneous process, that is, one that can occur, and can only be reversed by work from outside the system. Previously: (Analyzing three coupled Carnot-cycles, it is stated that…) … there is a state function S defined by

11. The Third Law of thermodynamics Atkins: If the entropy of every element in its most stable state atT = 0 is taken as zero, then every substance has a positive entropy which atT = 0 may become zero, and which does become zero for all perfect crystalline substances, including compounds. Afterwards: It should also be noted that the Third Law does not state that entropies are zero at T = 0: it merely implies that all perfect materials have the same entropy at that temperature. As far as thermodynamics is concerned, choosing this common value as zero is then a matter of convenience. The molecular interpretation of entropy, however, implies that S = 0 at T = 0. … The choice S (0) = 0 for perfect crystals will be made from now on.

12. The Third Law of thermodynamics (Péter Esterházy in a novel on communism) The Kádár-era: filthy land, lying to the marrow, shit as is, in which, aside from this, one could live, aside from the fact that one couldn't put it aside, even though we did put it aside. Atkins: If the entropy of every element in its most stable state atT = 0 is taken as zero, then every substance has a positive entropy which atT = 0 may become zero, and which does become zero for all perfect crystalline substances, including compounds. Afterwards: It should also be noted that the Third Law does not state that entropies are zero at T = 0: it merely implies that all perfect materials have the same entropy at that temperature. As far as thermodynamics is concerned, choosing this common value as zero is then a matter of convenience. The molecular interpretation of entropy, however, implies that S = 0 at T = 0. … The choice S (0) = 0 for perfect crystals will be made from now on.

13. The Third Law of thermodynamics Alberty: The entropy of each pure element or substance in a perfect crystalline form is zero at absolute zero. Afterwards : We will see later that statistical mechanics gives a reason to pick this value. Atkins: If the entropy of every element in its most stable state atT = 0 is taken as zero, then every substance has a positive entropy which atT = 0 may become zero, and which does become zero for all perfect crystalline substances, including compounds. Afterwards: It should also be noted that the Third Law does not state that entropies are zero at T = 0: it merely implies that all perfect materials have the same entropy at that temperature. As far as thermodynamics is concerned, choosing this common value as zero is then a matter of convenience. The molecular interpretation of entropy, however, implies that S = 0 at T = 0. … The choice S (0) = 0 for perfect crystals will be made from now on.

14. Avant propos After all it seems that Sommerfeld was right… Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you anymore.

15. But thermodynamics is an exact science… Development of axiomatic thermodynamics 1878 Josiah Willard Gibbs Suggestion to axiomatize chemical thermodinamics 1909 Konstantinos Karathéodori (greek matematician) The first system of postulates (axioms) (heat is not a basic quantity) 1966 László Tisza Generalized Thermodynamics, MIT Press (Collected papers, with some added text) 1985 Herbert B. Callen Thermodynamics and an Introduction to Thermostatistics, John Wiley and Sons, New York 1997 Elliott H. Lieb and Jacob Yngvason The Physics and Mathematics of the Second Law of Thermodynamics (15 mathematically sound but simple axioms)

18. Fundamentals of postulatory thermodynamics An important definition: the thermodynamic system The objects described by thermodynamics are called thermodynamic systems. These are not simply “the part of the physical universe that is under consideration ” (or in which we 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 an a posteriori definition; 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.

19. Postulatory thermodynamics A practical simplification: the simple system Simple systems are pieces of matter that are macroscopically 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. (Implicitely assumed in the classical treatment as well.)

20. Postulate 1 of thermodynamics 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 n 1, n 2,…, n K. 1. There exist equilibrium states 2. The equilibrium state is unique 3. The equilibrium state has K + 2 degrees of freedom (in simple systems!) 2. The equilibrium state cannot depend on the ”past history” of the system 3. State variables U, V and n 1, n 2,…, n K determine the state; their functions f(U, V, n 1, n 2,… n K ) are state functions.

21. Postulate 2 of thermodynamics There exists a function (called the entropy and 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. 1. Entropy is defined only for equilibrium states. 2. T he equilibrium state in an isolated composite system will be the one which has the maximum of entropy. Definition: composite system: contains at least two subsystems the two subsystems are sepatated by a wall (constraint)

22. Over what variables is entropy maximal? isolated cylinder fixed, impermeable, thermally insulating piston U α, V α, n α U β, V β, n β In the absence of an internal constraint, a manifold of different systems can be imagined ; all of them could be realized by re-installing the constraint (“virtual states”). Completely releasing the internal constraint(s) results in a well determined state which – over the manifold of virtual states – has the maximum of entropy.

23. Postulate 3 of thermodynamics The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a strictly increasing function of the internal energy. 1. S (U, V, n 1, n 2,… n K ) is an extensive function, i. e., a homogeneous first order function of its extensive variables. 2. There exist the derivatives of the entropy function. 3. The entropy function can be inverted with respect to energy: there exists the function U (S, V, n 1, n 2,… n K ), which can be calculated knowing the entropy function. 4. Knowing the entropy function, any equilibrium state (after any change) can be determined: S = S (U, V, n 1, n 2,… n K ) is a fundamental equation of the system. 5. Consequently, its inverse, U = U (S, V, n 1, n 2,… n K ) contains equivalent information, thus it is also a fundamental equation.

24. Postulate 4 of thermodynamics The entropy of any system is non-negative and vanishes in the state for which the derivative is zero. As, this also means that the entropy is exactly zero at zero temperature. The scale of entropy – contrarily to the energy scale – is well determined. (This makes calculation of chemical equilibrium constants possible.) (“Residual entropy”: no equilibrium !!)

25. Summary of the postulates (Simple) thermodynamic systems can be described by K + 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 directly to solve any thermodynamical problems.

26. Relations of the functions S and U S (U, V, n 1, n 2,… n K ) is concave, and a strictly monotonous function of U

27. Fundamental equations in U and S Equilibrium at constant energy (in an isolated system): at the maximum of the function S (U, V, n 1, n 2,… n K ) Equilibrium at constant entropy (in an isentropic system): at the minimum of the function U (S, V, n 1, n 2,… n K ) (In simple systems: isentropic = adiabatic) To find extrema of the relevant functions, we search for the zero values of the first order differentials:

28. 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:

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

30. Fundamental equations and equations of state Equations of state: Energy-based fundamental equation: Entropy-based fundamental equation : U = U(S, V, n)S = S(U, V, n) Its differential form:

31. Some formal relations U = U(S, V, n) is a homogeneous linear function. According to Euler’s theorem: Euler equation Gibbs-Duhem equation We know:

32. Equilibrium calculations isentropic, rigid, closed system impermeable, initially fixed, thermally isolated piston, then freely moving, diathermal S α, V α, n α S β, V β, n β S α + S β = constant; – dS α = dS β Consequences of impermeability (piston): n α = constant; n β = constant → dn α = 0; dn β = 0 V α + V β = constant; – dV α = dV β Equilibrium condition: dU= dU α + dU β = 0 U αU α U βU β Equilibrium: T α = T β and P α = P β

33. Equilibrium calculations isentropic, rigid, closed system S α, V α, n α S β, V β, n β Condition of thermal and mechanical equilibrium in the composit system: U αU α U βU β T α = T β and P α = P β 4 variables S α, V α, S β and V β are to be known at equilibrium. They can be calculated by solving the 4 equations: T α (S α, V α, n α ) = T β (S β, V β, n β ) P α (S α, V α, n α ) = P β (S β, V β, n β ) S α + S β = S ( constant ) V α + V β = V ( constant )

34. Equilibrium at constant temperature and pressure isentropic, rigid, closed system T = T r and P = P r (constants) S r, V r, n r T r, P r S, V, n T, P equilibrium condition: the „internal system” is closed n r = constant and n = constant d (U+U r ) = d U + T r dS r – P r dV r = 0 S r + S = constant; – dS r = dS V r + V = constant; – dV r = dV d (U+U r ) = d U + T r dS r – P r dV r = d U + T r dS – P r dV = 0 T = T r and P = P r d (U+U r ) = d U – TdS + PdV = d (U – TS + PV ) = 0 minimizing U + U r is equivalent to minimizing U – TS + PV Equilibrium condition at constant temperature and pressure: minimum of the Gibbs potential G = U – TS + PV

35. Summary of equilibrium conditions Via intensive variables: identity of these variables in all phases φ Thermal equilibrium: T φ  T,  Mechanical equilibrium: P φ  P,  Chemical equilibrium : μ φ  μ i,  i Extension is simple for variables characterizing other interactions: E. g. electrostatic equilibrium: Ψ φ  Ψ,  (Ψ φ : electric potential of phase φ) For chemical equilibrium, there is a condition for individual components; for all components that can freely move between the subsystems (phases) of a composite system.

36. Summary of equilibrium conditions Other (entropy-like) potential functions can also be applied if needed. Constraints Condition of equilibrium Mathematical condition Condition of stability U and V constant maximum of S (U, V, n) S and V constant maximum of U (S, V, n) S and P constant maximum of H (S, P, n) T and V constant maximum of F (T, V, n) T and P constant maximum of G (T, P, n) Via extensive variables: extrema of these variables in the system

37. Postulatory thermodynamics is easy to understand Postulates are based on quantities characteristic of the system only Relevant quantities (as internal energy and entropy) are defined in the postulates Postulates are ready to use in equilibrium calculations Derivation of auxiliary thermodynamic functions (as free energy and Gibbs potential) is straightforward Exact mathematical treatment of equilibria is easy Conclusions

38. Thus, it is worth both teaching teaching and learning postulatory thermodynamics thermodynamics !