A7. Basics of Thermodynamics

1. Interfacial Physics and Thin-Film Processing A7. Basics of Thermodynamics Fall, 2013 Instructor: J.-W. John Cheng Mech. Engr. Dept., Nat’l Chung Cheng Univ.

2. Outline • 1. Thermometer and Zeroth Law • 2. Heat (Enthalpy) and 1st Law • 3. Entropy and 2nd law • 4. Gibbs free energy and equilibrium • 5. Chemical potential • References • [BS99] Bowley, R. and Sanchez, M., Introductory Statistical Mechanics, 2nd ed., 1999, Ch 1 & Ch 2 • [MS95] Moran, M. J. and Shapiro, H. N., Fundamentals of Engineering Thermodynamics, 3rd ed., 1995, Ch 6 & Ch 14

3. 1. Thermometer andZeroth Law

4. Temperature • First, we postulate existence of temperature through experience • In fact, we have T = f(P, V) • “This is a very powerful mathematical statement: it says that the temperature has a unique value for any choice of pressure and volume; it does not depend on any other quantity.” (p. 4, BS99) • Thermal equilibrium • When 2 systems A and B are brought into contact and there is no long a flow of energy between them, • A and B are said to be in thermal equilibrium. • The apparent observation for thermal equilibrium is that A and B have same temperature.

5. Zeroth Law and Thermometer • Zeroth law • If systems A and B are separately in thermodynamic equilibrium with system C, • then systems A and B are in thermodynamic equilibrium with each other. • Thermometer – an implication of Zeroth law • If we want to know if A and B are at the same temperature, we do not need to bring two into contact • It can be answered by observing if they are individually in thermal equilibrium with a third body. • This third body is usually a thermometer (溫度計)

6. 3 Aspects of Thermodynamic Equilibrium • Thermal equilibrium (defined previously) • Mechanical equilibrium • A condition of balance maintained by force balance • Chemical equilibrium • See next page for description • Thermodynamic equilibrium • A system is in thermodynamic equilibrium when thermal, mechanical, and chemical equilibria have been reached. • At thermodynamic equilibrium, the system has well-defined temperature, pressure, and chemical potential.

7. Chemical Equilibrium • Chemical equilibrium concerns systems where the # of particles can change • E.g. a chemical reaction C + D CD • If there are too much C and D, the reaction proceeds to form CD • If there are too much CD, the reaction proceeds to form C and D • In chemical equilibrium, there is a balance between these two rates of reaction, so the numbers of #’s of C, D, and CD remain constant. • E.g. phase changes • Water and ice co-exist at a temperature around 0oC

8. Thermodynamic Coordinates (States) • Thermodynamic Coordinates/States • When in thermodynamic equilibrium, properties of the system only depend on thermodynamic ‘coordinates’, • such as the pressure and volume; • E.g., • consider a pure gas with no chemical reactions between gas particles and having constant number of particles • T = f(P, V) • Thermodynamic coordinates are more commonly referred to as thermodynamic states

9. Functions and Equations of States • Function of States and Equation of States • When a quantity only depends on the present value of thermodynamic coordinates • such as the pressure and volume as that of the temperature T = f(P, V) shown above • we say that the quantity is “a function of states” and • the governing equation “an equation of states.” • Generally, equations of states are very complicated and do not give rise to a simple math formula. • The ideal gas is an exception. PV = nRT  T = (PV)/(nR)

10. Some Definitions • Theory of thermodynamics • Is concerned with systems of a large number of particles which are contained in a vessel of some kind. • Adiabatic wall • An ideal heat-insulating wall • Thermally isolated • Referring to a system is surrounded by adiabatic walls • Diathermal • Referring to a system which allows energy to pass through its walls

11. Some Definitions contd • Isothermal • Any two systems in thermal equilibrium with each other are called isothermal to each other

12. Exact Differential (A Math Tool) • Consider a function of states, G = g(x, y) • The total derivative is defined dG as • Expressing dG = A(x,y)dx + B(x,y)dy, we have • Conversely, a change dG =A(x,y)dx + B(x,y)dy is called an exact differential if we have

13. Inexact Differential • Inexact Differential • When a change dG =A(x,y)dx + B(x,y)dy with • The change is called an inexact differential • To differentiate from exact differential, we will put a bar on top of it

14. Application of Exact Differential • Consider a 2-D force field • Q. When does there exist a potential function u(x,y) s.t. Hint:

15. 2. Heat (Enthalpy) and 1st Law

16. First Law • Internal energy U is a function of state. • It includes potential energy, kinetic energy, and others • First law • Energy is conserved if heat Q is taken into account. • U = W + Q • U: internal energy, W: external work, Q: external heat

17. Enthalpy (焓) • Consider a system • Under a pressure P, a change dV implies that external world does work to the system by -PdV • Resulting in an increase in internal energy of the system. • This change in internal energy is described as follows Note the bar on top of Q reminds us that dQ is not an exact differential • Enthalpy (under constant pressure) • Above deduction from 1st law of thermodynamics implies a new useful variable, the enthalpy H = U + PV (for isobaric process)

18. Heat Capacities • General concept of heat capacity • The amount of heat absorption dQ required for dT increase in temperature of the system • Note that dQ is condition dependent; different condition gives rise to different value of C • Cv: heat capacity at constant volume • CP: heat capacity at constant pressure • Note CP CV • The difference is small for liquid and solid in comparison to that for gas because gas expands significantly with temperature increase

19. 3. Entropy and 2nd Law

20. Motivation for Entropy • 1st law states • Note that external heat change is an inexact differential • It is interesting to know that • by multiplying an integrating factor an inexact differential can sometimes become exact • E.g. * * Performing a path integration of eyzdf along a particularly chosen path

21. Motivation for Entropy contd • A quest • “Can we find an integrating factor which multiplies dQ and produces an exact differential? • If we can do this then we can construct a new function of states and call it the entropy.” (p.25)

22. Entropy of Ideal Gas • Ideal gas is • A collection of n moles of gas molecules whose internal energy is the total kinetic energy of the gas • and satisfy the following equation of states • 1st law says

23. Entropy of Ideal Gas contd • Consider an integrating factor, 1/T Independent of process path!

24. Entropy of Ideal Gas contd • Above integration result implies existence of new function of states, S • Entropy S of ideal gas at temperature T & volume V w.r.t. a reference entropy S0

25. Formal Definition of Entropy • Definition of Entropy (unit: J/K) • For a reversible process, the entropy S of the system is defined to be • Thus, the 1st law for reversible processes can be expressed as Note this equation is only valid for reversible processes * “A reversible process is defined as one which may be exactly reversed to bring the system back to its initial state with no other change in the surroundings.” (p. 16) * The subscript rev in dQrev is to remind that the underlying process is a reversible one.

26. 2nd Law of Thermodynamics • Clausius inequality as 2nd law of thermodynamics* • For a general process, reversible or irreversible, we have • And cycle = 0, when process is reversible; • cycle > 0, when process is irreversible • cycle“is a measure of the effect of the irreversibilities present within the system executing the cycle.” • or as “the entropy introduced by internal irreversibilities during the cycle.” (p. 203, [MS95]) * In most textbook, the 2nd law refers to the principle of entropy increase and derive Clausius inequality as a corollary. But in some books, the 2nd law starts with Clausius inequality and derive principles of entropy increase as a corollary.

27. Entropy Balance Equation • Entropy balance eq. • The first implication of the Clausius ineq. • Consider a scenario of a cycle consisting of • a forward path, either irreversible or reversible, from state 1  state 2 and • a reversible return path from state 2  state 1 • From Clausius inequality, we have entropy balance eq.

28. Entropy Balance Equation contd • Entropy balance in differential form • 1st term of RHS • entropy transfer accompanying heat transfer; • positive value means transferring into system • 2nd term of RHS, cycle  0 always • entropy produced within the system by the action of irreversibilities • Interpretation of entropy balance eq • (An entropy change) = (entropy transfer due to heat transfer) + (entropy induced by action of irreversibilities)

29. Principle of Increase of Entropy • Principle of Entropy Increase • Another implication of the Clausius inequality • System under consideration • Consider an enlarged system comprising a system of interest and that portion of the surroundings affected by the system as it undergoes a process. • Since all energy and mass transfers taking place are included within the enlarged system, • The enlarged system is considered thermally isolated, i.e., dQ = 0

30. Principle of Increase of Entropy contd • Entropy balance of enlarged system gives • which implies • Since   0 in all actual processes, the only processes that can occur in nature are those with entropy increase of the isolated system • The above is the so-called the principle of increase of entropyfor thermally isolated system

31. Themodynamic Equilibrium of Isolated Systems • Implied by the principle of entropy increase, • the entropy of an isolated system increases as it approaches the state of equilibrium, and • the equilibrium state is attained when the entropy reaches a maximum

32. Microscopic Definition of Entropy • Above discussion is the phenomenological definition of entropy, the classical way • Modern statistical thermodynamics gives the following microscopic definition • Let  denote the total number of possible microscopic states available to a system • the entropy of the system is defined as • Principle of entropy increase implies equilibrium is characterized with max disorder, i.e., largest 

33. 4. Gibbs Free Energy and Equilibrium

34. Why Is Gibbs Free Energy Necessary? • Principle of entropy increase tells that • An isolated system reaches its thermodynamic equilibrium when its entropy is maximum. • Limitation of principle of entropy increase is that it is applicable only to isolated systems • How to describe thermodynamic behavior of a more general system?

35. 2 Balance Eq’s of Thermodynamic Sys • (1) Energy balance equation (1st law) dU = dW + dQ • U: internal energy, W: external work, Q: external heat • (2) Entropy balance equation (2nd law) • Gibbs free energy is a clever application of these two balance equations

36. Predicting Process Direction • 1st law for isobaric (恆壓) process (1) • Entropy balance in differential form (2) • (1), (2)  the only process allowed must satisfy (3) • (3) can be used to study direction of process change • i.e., the system will change with the direction which would result in negative value of LHS of (3)

37. Gibbs Free Energy • Above inequality suggests a new function of states, the Gibbs free energy • Definition of Gibbs Free Energy, G

38. Gibb’s Criterion for Equilibrium • Inequality of slide 35 in terms of Gibbs free energy (4) • For isothermal and isobaric processes, we have • Thus, the process proceeds to state with lower G, and • the equilibrium state occurs at min Gibbs free energy, i.e., when

39. 5. Chemical Potential

40. Dependence on Size of the System • Intuitively, when the number of moles of the particle increases, G also increases. • Specifically, we have (without proof) and • Any function of states is linearly proportional to the size of the system, like G, is called an extensive property • Extending to multi-component system, we have and

41. Chemical Potential, i From last slide, • Taking partial derivation w.r.t.  gives • Define chemical potential as

42. iBeing Independent of Size • Note that our assumption of extensiveness on G will give for single component system • iis independent of the size of the system • i.e., chemical potential is a so-called intensive property of the system

43. Chemical Potential of Ideal Gas Mixture • Consider a binary mixture as an illustration • Let n1 and n2 be the numbers of moles of gases 1 and 2, respectively • From previous discussion related to the ideal gas, we know • Gibbs free energy of binary mixture of ideal gases is • Note • By comparison, we will obtain i

44. Equilibrium Criterion in Terms of Chemical Potential • As noted before, equilibrium criterion of an isothermal and isobaric process is • Thus, this equilibrium criterion of an isothermal and isobaric process can be reformulated as